Georgios Kydonakis

Djibril Gueye

Yohann Bouilly

Pierre-Alexandre Arlove

Martin Mion-Mouton

Valdo

Julie Giorgio

Alex Despo

Arturo Bere

Finitely generated normal subgroups of Kähler groups Je tiens tout d'abord à remercier Thomas Delzant et Pierre Py, car sans leur soutien, leur encadrement et leur disponibilité, ce travail n'aurait pas été possible. Je les remercie tous deux de m'avoir proposé ce projet, pour l'enthousiasme avec lequel ils ont dirigé mes recherches et pour toutes les rencontres et discussions enrichissantes au sein et en dehors de l'IRMA qui ont abouti à ce travail. Je tiens également à les remercier de tout le soutien qu'ils m'ont apporté avant de commencer ce projet, de m'avoir aider à venir en France pour e ectuer le M2 puis la thèse, Pierre pour toutes les discussions intéressantes que nous avons eues au Mexique et Thomas pour avoir dirigé mon mémoire de M2. Enfin, je tiens à les remercier pour l'aide qu'ils m'ont apportée à mon arrivée à Strasbourg et pour l'accompagnement dont ils ont fait preuve pendant la période di cile du confinement. Je tiens également à remercier Benoît Claudon et Claudio Llosa Isenrich d'avoir accepté d'être les rapporteurs de ma thèse, pour leur lecture détaillée et attentive de ce travail et pour tous leurs précieux commentaires. Je remercie également Julie Déserti et Vincent Koziarz d'avoir accepté d'être membres du jury.

J'aimerais remercier tous les chercheurs et tout le personnel de l'IRMA d'avoir rendu mon séjour à l'IRMA très agréable. Je remercie tous les membres de l'équipe de géométrie pour leur enthousiasme dans les groupes de travail et les séminaires. Je remercie également Tatiana Beliaeva, Alexander Thomas et Marianne Rydzek pour l'organisation du cercle mathématique. Un grand merci à tous les doctorants de l'IRMA que j'ai recontré au cours de ma thèse, pour les discussions et les moments passés dans et hors du laboratoire. Et

Introduction en français

Une variété kählérienne est une variété complexe munie d'une métrique hermitienne dont la partie imaginaire est une 2-forme fermée. Un groupe est dit groupe kählérien s'il peut être réalisé comme le groupe fondamental d'une variété kählérienne compacte. Une référence classique sur ce sujet est [START_REF] Amorós | Fundamental groups of compact Kähler manifolds[END_REF] et une référence plus récente est [START_REF] Burger | Fundamental groups of Kähler manifolds and geometric group theory[END_REF].

Il existe de nombreuses restrictions topologiques sur les variétés kählériennes compactes. Par exemple, la théorie de Hodge permet de munir le premier groupe de cohomologie à coe cients dans R d'une variété kählérienne compacte d'une structure complexe (voir chapitre 6 de [START_REF] Voisin | Théorie de Hodge et géométrie algébrique complexe[END_REF]). Ceci implique que le rang de l'abélianisation d'un groupe kählérien doit être pair. Puisque tout sous-groupe d'indice fini d'un groupe kählérien est à nouveau un groupe kählérien, cette dernière a rmation doit également être valable pour tout sousgroupe d'indice fini d'un groupe kählérien. De cette manière, on peut voir qu'un groupe libre n'est jamais kählérien car pour un tel groupe, on peut toujours trouver un sousgroupe d'indice fini dont l'abélianisation a un rang impair. Voici quelques exemples de groupes kählériens.

• Le groupe fondamental d'une surface de Riemann fermée est un groupe kählérien car toute 2-forme sur un tel espace est fermée.

• Pour tout entier positif n, le groupe Z 2n est kählérien car il peut-être réalisé comme le groupe fondamental du tore complexe C n /Z 2n . Dans ce cas, toute forme hermitienne définie positive sur C n induit une métrique kählérienne sur C n /Z 2n . • L'espace projectif complexe CP n muni de la métrique de Fubini-Study est un variété kählérienne. Ainsi, toute sous-variété complexe de CP n est munie d'une structure kählérienne. Les groupes fondamentaux de telles variétés sont appelés groupes projectifs et en particulier ils sont des groupes kählériens1 .

• Dans [START_REF] Serre | Sur la topologie des variétés algébriques en caractéristique p[END_REF], Serre a prouvé que tous les groupes finis sont des groupes projectifs et donc des groupes kählériens.

• Le produit direct de groupes kählériens et les sous-groupes d'indice fini de groupes kählériens, sont à nouveau des groupes kählériens. Ceci découle du fait que la métrique donnée par le produit des métriques kählériennes et la métrique du tiré en arrière sous un difféomorphisme local holomorphe d'une métrique kählérienne, sont à nouveau des métriques kählériennes.

• Les espaces symétriques Hermitiens et leurs quotients par des sous-groupes discrets sans torsion de leur groupe d'isométries holomorphes, sont des variétés kählériennes. Alors, les réseaux uniformes sans torsion du groupe d'isométries holomorphes d'espaces symétriques Hermitiens sont des groupes kählériens. Une astuce de Kollár, permet d'étendre ce résultat aux réseaux uniformes avec de la torsion (voir [START_REF] Arapura | Fundamental groups of smooth projective varieties, Current topics in complex algebraic geometry[END_REF] pour une preuve de cette dernière a rmation). Un travail de Toledo [START_REF] Toledo | Examples of fundamental groups of compact Kähler manifolds[END_REF] montre que de nombreux réseaux non-uniformes d'isométries holomorphes d'espaces symétriques Hermitiens sont également des groupes kählériens. Dans l'étude des groupes kählériens il y a deux types de résultats : il y a des résultats négatifs qui disent que certaines familles de groupes ne contiennent pas de groupes kählériens. Mentionnons (parmi les nombreux résultats de ce type) les suivants : Carlson et Toledo ont prouvé dans [START_REF] Carlson | Harmonic mappings of Kähler manifolds to locally symmetric spaces[END_REF] qu'un réseau dans le groupe d'isométries de l'espace hyperbolique réel de dimension au moins 3, n'est jamais kählérien. Delzant (en suivant certains résultats d'Arapura-Nori [START_REF] Arapura | Solvable fundamental groups of algebraic varieties and Kähler manifolds[END_REF], de Brudnyi [START_REF] Brudnyi | Solvable matrix representations of Kähler groups[END_REF] et de Campana [START_REF] Campana | Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler[END_REF]) a prouvé dans [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] qu'un groupe kählérien résoluble doit être virtuellement nilpotent. En étudiant la cohomologie L 2 des variétés et des groupes kählériens, Gromov a prouvé dans [START_REF] Gromov | Sur le groupe fondamental d'une variété kählérienne[END_REF] qu'un groupe kählérien infini est un groupe à un bout. Une preuve plus détaillée de ce dernier résultat peut être trouvée dans le travail de Arapura, Bressler et Ramachandran [START_REF] Arapura | On the fundamental group of a compact Kähler manifold[END_REF]. D'autre part, il existe des résultats positifs, c'est-à-dire des constructions de groupes kählériens avec des propriétés intéressantes. Mentionnons l'existence de groupes kählériens non résiduellement finis [START_REF] Toledo | Projective varieties with non-residually finite fundamental group[END_REF] ou de groupes kählériens nilpotents qui ne sont pas virtuellement Abéliens [START_REF] Campana | Remarques sur les groupes de Kähler nilpotents[END_REF]. Dans cette thèse, nous contribuerons à ces deux lignes de recherche.

Contexte général de ce travail

Supposons que nous avons une suite exacte courte

1 / / G / / / / Q / / 1, (1)
où est un groupe kählérien et G est un groupe de type fini. Étant donné une telle suite exacte courte, nous pouvons demander :

Question 1. Que peut-on dire de l'action par conjugaison de sur G ?

Nous allons restreindre notre étude aux suites exactes courtes comme [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF], où G et Q sont des groupes infinis. L'action par conjugaison de sur G induit un homomorphisme de groupes de dans le groupe d'automorphismes de G.

≠ae Aut(G) " ' ae (x ' ae "x" ≠1 ).

Rappelons que le groupe d'automorphismes intérieurs de G, noté par Inn(G), est composé des automorphismes de G induits par la conjugaison d'un élément de G, c'est-àdire d'automorphismes de la forme x ' ae gxg ≠1 pour un certain g dans G. Le groupe d'automorphismes intérieurs est normal dans Aut(G) et le groupe quotient Aut(G)/Inn(G), noté Out(G), est appelé le groupe d'automorphismes extérieurs de G. Une suite exacte courte comme [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF], induit un homomorphisme de groupes ae Out(G) qui se factorise par Q, c'est-à-dire, le diagramme suivant

/ / Out(G) Q ; ;
commute. L'homomorphisme Q ae Out(G) obtenu de cette manière est appelé la monodromie de la suite exacte courte [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF].

La géométrie complexe fournit des exemples de suites exactes courtes comme [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF]. Par exemple, une fonction holomorphe surjective à fibres connexes f : X ae S entre un variété 2. Restrictions sur la monodromie 3 complexe compacte X et une surface de Riemann fermée S de genre positif, induit une suite exacte courte

1 / / G / / fi 1 (X) fú / / fi orb 1 (S) / / 1,
où G est l'image du groupe fondamental de la fibre générique dans fi 1 (X) et S est munie d'une structure d'orbifold qui prend en compte les multiplicités des fibres singulières (voir Section 1.4.1). Dans [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF], Dimca, Papadima et Suciu ont étudié certaines propriétés de finitude du groupe fondamental d'une fibre générique d'une fonction holomorphe comme avant, dans le cas particulier où X est un produit direct de surfaces de Riemann de genre supérieur ou égal à deux, et S est de genre un. Cette étude sur les propriétés de finitude des groupes fondamentaux des fibres génériques, a été poursuivie par Llosa Isenrich [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] et par Bridson et Llosa Isenrich [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF]. Ils construisent de nouveaux groupes kählériens avec des propriétés de finitude intéressantes. On se pose alors la question suivante.

Question 2. Pour une fonction holomorphe surjective à fibres connexes f : X ae S entre une variété complexe compacte X et une surface de Riemann fermée S de genre positif, que peut-on dire des propriétés de finitude du groupe fondamental de sa fibre générique ?

Nous nous intéresserons au cas où X est une variété kählérienne compacte. Ce travail donnera des réponses partielles aux Questions 1 et 2.

Restrictions sur la monodromie

Un problème classique est l'étude de la monodromie : étant donné une submersion holomorphe propre f : X ae Y (qui n'est pas un fibré holomorphe localement trivial) à fibres de dimension 1, on peut étudier la monodromie fi 1 (Y ) ae MCG(S), où S est la fibre générique de f qui est une surface topologique et MCG(S) est le groupe modulaire de S. Dans [START_REF] Shiga | On monodromies of holomorphic families of Riemann surfaces and modular transformations[END_REF], Shiga a prouvé le résultat suivant.

Théorème (Shiga). Soit f : X ae B une submersion holomorphe qui n'est pas un fibré holomorphe localement trivial d'une surface complexe compacte sur une surface hyperbolique avec fibre générique une surface hyperbolique fermée S. Alors, l'image de la monodromie fi 1 (B) ae MCG(S) est infinie et elle ne peut préserver aucune classe d'isotopie d'une courbe fermée simple dans S.

L'un des résultats qui a motivé le travail présenté dans le Chapitre 1, est le résultat suivant de Bregman et Zhang (voir [START_REF] Bregman | On Kähler extensions of abelian groups[END_REF]).

Théorème (Bregman-Zhang). Soit S une surface fermée de genre g Ø 2,

un groupe kählérien et k un entier positif tel qu'il existe une suite exacte courte

1 / / fi 1 (S) / / P / / Z k / / 1.
Alors, il existe un sous-groupe d'indice fini 1 de contenant fi 1 (S), tel que la suite exacte courte restreinte

1 / / fi 1 (S)
/ / 1 P / / P ( 1 ) ƒ Z k / / 1 est un produit direct.

Introduction en français

Nous étudierons ici les suites exactes courtes comme [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF], où G est le groupe fondamental d'une surface fermée orientée S de genre g Ø 2, sans supposer que S est la fibre d'une submersion holomorphe. Une des clés de cette étude est le fait que tout élément de Out(fi 1 (S)) peut être représenté par un di éomorphisme de la surface S (voir Dehn [START_REF] Dehn | Papers on group theory and topology, Translated from the German and with introductions and an appendix by John Stillwell, With an appendix[END_REF] pour la preuve originale et Nielsen [START_REF] Nielsen | Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen[END_REF] pour la première preuve publiée). Nous donnerons également quelques restrictions sur les suites exactes courtes comme [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF], où G se décompose comme produit amalgamé ou extension HNN.

Présentation des résultats sur les groupes kählériens et les groupes de type fini agissant sur des arbres

Nous présentons ici les résultats du préprint [START_REF] Nicolás | On finitely generated normal subgroups of Kähler groups[END_REF]. Notre hypothèse principale est qu'un groupe kählérien , admet comme sous-groupe normal un groupe de type fini G qui agit sur un arbre T . Dans de nombreux cas, nous verrons que l'action de G sur T s'étend à , ce qui nous permettra d'appliquer le résultat suivant de Gromov et Schoen (voir [START_REF] Gromov | Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one[END_REF]).

Théorème (Gromov-Schoen). Soit X une variété kählérienne compacte dont le groupe fondamental agit sur un arbre qui n'est pas isomorphe à une droite ni à un point. Supposons que l'action est minimale et sans point fixe au bord. Alors, il existe une fonction surjective holomorphe à fibres connexes de X sur un orbifold hyperbolique fermé , induisant la suite exacte courte

1 / / N / / / / fi orb 1 ( ) / / 1,
de sorte que la restriction de l'action à N est triviale.

Voir la Section 1.4.1 pour la définition du groupe fondamental orbifold fi orb 1 ( ). Le problème d'étendre l'action de G sur un arbre au groupe Aut(G) a été étudié dans la littérature. Dans plusieurs cas, on peut prouver des résultats d'extension. Ceci est le cas si le groupe G est l'un des exemples suivant :

1. Un produit libre AúB avec A et B des groupes indécomposables, non cycliques infinis, agissant sur son arbre de Bass-Serre.

2. Le groupe de Baumslag-Solitar Èx, t|tx p t ≠1 = x q Í, agissant sur son arbre de Bass-Serre, où p, q sont des entiers avec p, q > 1, et tels qu'aucun n'est un multiple de l'autre (voir Gilbert, Howie, Metaftsis et Raptis [START_REF] Gilbert | Tree actions of automorphism groups[END_REF] et Pettet [START_REF] Pettet | The automorphism group of a graph product of groups[END_REF]).

Un groupe hyperbolique à un bout sans torsion, avec un groupe d'automorphismes

extérieurs infini et qui n'est pas virtuellement un groupe de surface, agissant sur son arbre JSJ (Sela [START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF], voir aussi Bowditch [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF]).

Afin d'appliquer ces résultats à l'étude des groupes kählériens, supposons que le groupe G n'a pas de centre, que l'arbre T n'est pas une droite ni un point et que l'action de G sur T est minimale, fidèle et sans points fixes au bord de T . Soit l'image du morphisme ae Aut(G) induit par l'action par conjugaison de sur G. Sous ces conditions nous prouvons le résultat suivant dans le Chapitre 1.

Théorème A. Supposons qu'il existe un sous-groupe d'indice fini 0 de contenant G ƒ Inn(G), de sorte que l'action de G sur T puisse être étendue en une action G-compatible de sur T . Alors G est virtuellement un groupe de surface. De plus, il existe un sous-groupe d'indice fini 1 de contenant G tel que la suite exacte courte restreinte Ce résultat peut être vu comme une version topologique du résultat de Shiga. Nous verrons également que nous pouvons retrouver le résultat de Bregman et Zhang, en utilisant des résultats classiques sur le groupe de monodromie d'une surface hyperbolique et l'hyperbolisation des variétés de dimension 3. D'autres applications sont obtenues.

1 / / G /

Théorème C. Soit G un groupe qui se décompose comme un produit libre non trivial A ú B avec A et B indécomposables et non cycliques infinis. Alors G ne se plonge pas comme sous-groupe normal dans un groupe kählérien.

Remarque. Pour l'arbre de Bass-Serre d'un produit libre G = A ú B ú C, il existe des automorphismes de G qui ne peuvent pas être étendus. Par exemple siest un élément de B, l'automorphisme défini par a ' ae a, b ' ae b, and c ' ae -c-≠1 pour tout a oe A, b oe B et c oe C (appelé automorphisme de Fouxe-Rabinovitch), ne s'étend pas à l'arbre de Bass-Serre dont la graphe de groupes correspondant est :

A B C
Cela implique que notre preuve du Théorème C ne s'étend pas à un produit libre d'au moins trois facteurs. Théorème D. Soit G le groupe de Baumslag-Solitar Èx, t|tx p t ≠1 = x q Í, où p, q sont des entiers avec p, q > 1, et tels qu'aucun n'est un multiple de l'autre. Alors G ne se plonge pas comme sous-groupe normal dans un groupe kählérien.

Théorème E. Supposons que G est un groupe hyperbolique sans torsion à un bout, qui se plonge comme sous-groupe normal dans un groupe kählérien . Si Out(G) est infini, alors

G est virtuellement un groupe de surface.

Nous observons que dans ce dernier théorème il n'y a pas d'hypothèse sur la fonction de monodromie. Nous avons seulement besoin de savoir que Out(G) est infini.

Groupes kählériens et propriétés de finitude

Nous présentons ici quelques résultats sur les propriétés de finitude des groupes kählériens. Une introduction détaillée aux propriétés de finitude sera donnée dans la Section 2.2. En particulier, les propriétés de finitude FP n et F n y seront définies.

Soit X un variété complexe compacte connexe de dimension complexe n Ø 2 et soit S une surface de Riemann fermée de genre positif. Une fonction holomorphe surjective à fibres connexes f : X ae S est appelée un pinceau irrationnel. Dans [START_REF] Kapovich | On normal subgroups in the fundamental groups of complex surfaces[END_REF], Kapovich a prouvé le résultat suivant Théorème (Kapovich). Soient X une surface kählérienne compacte asphérique, S une surface de Riemann fermée de genre positif et f : X ae S un pinceau irrationnel dont les fibres singulières sont de multiplicité un. Alors f est une submersion ou le noyau de la fonction induite au niveau des groupes fondamentaux f ú : fi 1 (X) ae fi 1 (S), n'est pas finiment présenté.

Ceci était le premier résultat qui faisait un lien entre l'existence de points critiques d'un pinceau irrationnel et les propriétés de finitude du noyau de la fonction induite au niveau des groupes fondamentaux. Comme conséquence de ce résultat, Kapovich a prouvé l'existence de surfaces hyperboliques complexes compactes dont les groupes fondamentaux admettent un sous-groupe normal de type fini qui n'est pas finiment présentable. Autrement dit, Kapovich a donné les premiers exemples de réseaux uniformes non cohérents dans PU(2, 1). Rappelons qu'un groupe est dit cohérent si chaque sous-groupe de type fini est aussi finiment présentable. D'autres exemples de groupes kählériens non cohérents ont été donnés par Kapovich [START_REF] Kapovich | Noncoherence of arithmetic hyperbolic lattices[END_REF] et Py [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF]. L'un des principaux ingrédients de ces exemples est le résultat suivant, essentiellement dû à Kapovich (voir [START_REF] Kapovich | Noncoherence of arithmetic hyperbolic lattices[END_REF] et [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF]), qui découle de la combinaison du Théorème 4 dans [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF] (dû à Kapovich) avec le résultat de Bregman et Zhang cité précédemment.

Théorème. Soit un groupe kählérien qui peut être réalisé comme le groupe fondamental d'une surface kählérienne compacte asphérique X avec premier nombre de Betti positif. Supposons que ne possède aucun sous-groupe abélien de type fini dont le normalisateur a un indice fini dans . Alors, au moins un des cas suivants se produit :

1. est non cohérent.

Il existe un revêtement fini de X, qui admet une submersion holomorphe sur une

surface hyperbolique compacte à fibres hyperboliques connexes.

Friedl et Vidussi donnent un ra nement de ces résultats sur la cohérence des groupes kählériens dans [START_REF] Friedl | Vidussi Virtual algebraic fibrations of Kähler groups[END_REF]. Outre l'étude de la cohérence de certains groupes, on peut également étudier l'existence de sous-groupes normaux qui ne sont pas de type fini. Par exemple, rappelons qu'un sous-groupe normal d'un groupe libre non abélien ou d'un groupe de surface, est finiment engendré si et seulement s'il est d'indice fini (voir [START_REF] Schreier | Die Untergruppen der freien Gruppen[END_REF] pour la preuve originale dans le cas des groupes libres et les lemmes 3.3 et 3.4 dans [START_REF] Catanese | Fibred Kähler and quasi-projective groups, Special issue dedicated to Adriano Barlotti[END_REF]). En général, les sous-groupes coabéliens normaux des groupes kählériens qui ne sont pas de type fini, sont liés aux fibrations sur des surfaces de Riemann. En étudiant l'invariant de Bieri-Neumann-Strebel des groupes kählériens, Delzant a prouvé dans [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] le résultat suivant.

Groupes kählériens et propriétés de finitude

Théorème (Delzant). Considérons la suite exacte courte

1 / / G / / / / Q / / 1,
où est un groupe kählérien, Q est un groupe abélien et G n'est pas de type fini. Alors, si X est une variété kählérienne compacte qui réalise comme son groupe fondamental, il existe une fonction holomorphe surjective à fibres connexes de X sur un orbifold hyperbolique de dimension 2.

Nous renvoyons le lecteur à [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] pour les définitions d'un orbifold hyperbolique de dimension 2 et d'une fonction holomorphe d'une variété complexe compacte sur un tel espace. Lorsque Q = Z, le théorème ci-dessus a été établi précédemment par Napier et Ramachandran dans [START_REF] Napier | Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces[END_REF] en utilisant des techniques di érentes.

Nous allons introduire quelques résultats récents sur la construction de groupes kählériens ayant des propriétés de finitude exotiques et le contexte qui a motivé ces constructions. Dans [45, §0.3.1], Kollár a posé la question de savoir si un groupe projectif est toujours commensurable (à noyaux finis près) à un groupe admettant un espace classifiant qui soit une variété quasi-projective. Puisque toute variété quasi-projective a le type d'homotopie d'un complexe fini [30, p. 27], une réponse positive à cette question impliquerait que tout groupe projectif est commensurable à un groupe ayant un espace classifiant fini (voir Section 2.2.1 pour la définition de la commensurabilité). Cependant, une réponse négative à la question de Kollár a été donnée par Dimca, Papadima et Suciu dans [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]. Dans [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF], les auteurs ont prouvé les deux résultats suivants (voir Théorème C et §2 dans [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]).

Théorème (Dimca, Papadima and Suciu). Si n Ø 3 et si f : X ae S est un pinceau irrationnel avec des points critiques isolés, alors le groupe fondamental d'une fibre lisse de f se plonge dans celui de X et coïncide avec le noyau de l'homomorphisme induit au niveau des groupes fondamentaux f ú : fi 1 (X) ae fi 1 (S).

Remarquez que sous l'hypothèse de ce dernier résultat, si on suppose que X est une variété kählérienne, on obtient un sous-groupe normal de type fini du groupe fondamental de X, qui est aussi un groupe kählérien.

Théorème (Dimca, Papadima and Suciu).

Soit X = 1 ◊ • • • ◊ n un produit direct de
n surfaces de Riemann de genre supérieur à 1 et soit S de genre 1. Si f : X ae S est un pinceau irrationnel avec des points critiques isolés, alors H n (ker(f ú ), Q) est de dimension infinie.

En combinant ces deux derniers résultats, Dimca, Papadima et Suciu ont donnée une réponse négative à la question de Kollár. En e et, dans la situation de ce dernier résultat, le noyau de l'homomorphisme f ú : fi 1 (X) ae fi 1 (S), qui est le groupe fondamental d'une fibre lisse de f si n Ø 3, ne peut pas être de type FP n car le groupe H n (ker(f ú ), Z) n'est pas de type fini (voir Proposition 87). La propriété d'être de type FP n est invariante par la relation de commensurabilité [START_REF] Bieri | Homological dimension of discrete groups[END_REF][START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]. Par conséquent, aucun groupe commensurable au noyau de f ú ne peut avoir un espace classifiant fini.

Suite à l'article [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF], d'autres exemples de groupes projectifs ayant des propriétés de finitude exotiques ont été construits et étudiés par Llosa Isenrichh [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] et par Bridson et Llosa Isenrich [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF]. Tous les exemples étudiés dans [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF][START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] sont soit des sous-groupes de produits directs de groupes de surface ou des extensions de tels sous-groupes, comme dans [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF].

Introduction en français

Présentation de résultats sur les groupes kählériens ayant des propriétés de finitude exotiques

Nous présentons ici nos résultats sur la construction de nouveaux groupes kählériens ayant des propriétés de finitude exotiques. Voir [START_REF] Nicolás | Py Irrational pencils and Betti numbers[END_REF] pour une prépublication issue de ce travail, à paraître dans les Annales de la Faculté des Sciences de Toulouse. Il s'agit d'un travail en collaboration avec Pierre Py.

Soit f : X ae S un pinceau irrationnel avec dim C X = n Ø 2. Nous supposons que les points critiques de f sont isolés et que f n'est pas une submersion; son ensemble de points critiques est alors non vide. Soit ' X ae X le revêtement de X tel que fi 1 ( ' X) ƒ ker(f ú ). Les résultats principaux sont les suivants :

Théorème F. Le groupe d'homologie H n ( ' X, Q) est de dimension infinie.
Théorème G. Si X est asphérique, le groupe H n (ker(f ú ), Q) est de dimension infinie. En particulier ker(f ú ) n'est pas de type FP n .

Dans le cas particulier où X est un produit direct de surfaces de Riemann et que S est de genre 1, nous retrouvons à partir du Théorème G le second résultat de Dimca, Papadima et Suciu cité précédemment. Il est intéressant de chercher d'autres exemples de variétés projectives (ou kählériennes fermées) munies d'une fonction holomorphe à laquelle on peut appliquer les Théorèmes F et G. Une façon de construire de nouveaux exemples de groupes kählériens ayant des propriétés de finitude exotiques est d'utiliser la surface [START_REF] Cartwright | Enumeration of the 50 fake projective planes[END_REF]) pour une définition et des propriétés de cette surface kählérienne compacte) que nous désignerons par Y . La fonction d'Albanese de cette surface complexe h : Y ae E est une fonction holomorphe dont le but est une courbe elliptique. Il a été prouvé que ses singularités sont isolées (voir [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF]) et non dégénérées (voir [START_REF] Koziarz | Stability of the Albanese fibration on the Cartwright-Steger surface[END_REF] et [START_REF] Rito | Surfaces with canonical map of maximum degree[END_REF]). On peut donc considérer le produit Y b de Y avec lui-même b fois et la fonction

h + • • • + h : Y b ae E. ( 2
)
Cela fournit des exemples naturels auxquels on peut appliquer les Théorèmes F et G. En désignant par < PU(2, 1) le groupe fondamental de la surface de Cartwright-Steger, la dernière construction avec le Théorème G implique immédiatement :

Théorème H. Le produit direct de b copies de contient un sous-groupe normal coabélien

N qui est de type FP 2b≠1 mais qui satisfait que H 2b (N, Q) est de dimension infinie.

Introduction

A Kähler manifold is a complex manifold endowed with a Hermitian metric whose imaginary part is a closed 2-form. A group is called a Kähler group if it can be realized as the fundamental group of a compact Kähler manifold. We refer the reader to [START_REF] Amorós | Fundamental groups of compact Kähler manifolds[END_REF] for a classical reference on this subject and to [START_REF] Burger | Fundamental groups of Kähler manifolds and geometric group theory[END_REF] for a more recent survey.

There are many topological restrictions on compact Kähler manifolds. For instance, Hodge Theory enables to endow the first cohomology group of a compact Kähler manifold with coe cients in R with a complex structure (see Chapter 6 of [START_REF] Voisin | Théorie de Hodge et géométrie algébrique complexe[END_REF]). This implies that the rank of the Abelianization of a Kähler group must be even. Since any finite index subgroup of a Kähler group is again Kähler, the latter assertion must hold as well for any finite index subgroup of a Kähler group. One can see in this way that a free group is never Kähler since for such a group one can always find a finite index subgroup whose Abelianization has odd rank. Let us see some examples of Kähler groups.

• The fundamental group of a closed Riemann surface is a Kähler group since any 2-form on such a space is closed.

• For any positive integer n, the group Z 2n is Kähler since it is the fundamental group of the complex torus C n /Z 2n . In this case, any positive-definite Hermitian form on C n induces a Kähler metric on C n /Z 2n . • The complex projective space CP n endowed with the Fubini-Study metric is a Kähler manifold. Therefore, any complex submanifold of CP n is endowed with a Kähler structure. The fundamental groups of such manifolds are called projective groups; they are in particular Kähler groups1 .

• In [START_REF] Serre | Sur la topologie des variétés algébriques en caractéristique p[END_REF], Serre proved that all finite groups are projective groups and therefore Kähler groups.

• The direct product of Kähler groups and finite index subgroups of Kähler groups are again Kähler groups. This follows from the fact that the product metric of Kähler metrics and the pullback metric under a holomorphic local di eomorphism of a Kähler metric are again Kähler metrics.

• Hermitian symmetric spaces and their quotients by torsion-free discrete subgroups of their group of holomorphic isometries, are Kähler manifolds. Then, torsion-free uniform lattices of holomorphic isometries of Hermitian symmetric spaces are Kähler groups. A trick due to Kollár allows to extend this result to uniform lattices with non-trivial torsion (see [START_REF] Arapura | Fundamental groups of smooth projective varieties, Current topics in complex algebraic geometry[END_REF] for a proof of the latter assertion). A work of Toledo [START_REF] Toledo | Examples of fundamental groups of compact Kähler manifolds[END_REF] shows that many non-uniform lattices of holomorphic isometries of Hermitian symmetric spaces are also Kähler groups.

There are two types of results in the study of Kähler groups: there are some negative results which say that certain families of groups do not contain Kähler groups. Let us mention (among many other such results) the following: Carlson and Toledo proved in [START_REF] Carlson | Harmonic mappings of Kähler manifolds to locally symmetric spaces[END_REF] that a lattice in the isometry group of the real hyperbolic space of dimension at least 3 Introduction is never Kähler. Delzant (following certain results of Arapura-Nori [START_REF] Arapura | Solvable fundamental groups of algebraic varieties and Kähler manifolds[END_REF], Brudnyi [START_REF] Brudnyi | Solvable matrix representations of Kähler groups[END_REF] and Campana [START_REF] Campana | Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler[END_REF]) proved in [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] that a solvable Kähler group must be virtually nilpotent. By studying the L 2 cohomology of Kähler manifolds and Kähler groups, Gromov proved in [START_REF] Gromov | Sur le groupe fondamental d'une variété kählérienne[END_REF] that an infinite Kähler group has one end. A more detailed proof of the latter result can be found in the work of Arapura, Bressler and Ramachandran [START_REF] Arapura | On the fundamental group of a compact Kähler manifold[END_REF]. On the other hand there are positive results, i.e. some constructions of Kähler groups with interesting properties. Let us mention the existence of non-residually finite Kähler groups [START_REF] Toledo | Projective varieties with non-residually finite fundamental group[END_REF] or of non-virtually Abelian nilpotent Kähler groups [START_REF] Campana | Remarques sur les groupes de Kähler nilpotents[END_REF]. In this thesis we will contribute to these two lines of research.

General context of this work

We assume that we have a short exact sequence

1 / / G / / / / Q / / 1, (1)
where is a Kähler group and G is a finitely generated group. Given such a short exact sequence we can ask:

Question 1. What can be said about the conjugation action of on G ?

We will restrict our study to short exact sequences as in [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF] where G and Q are infinite groups. The conjugation action of on G induces a group homomorphism from to the group of automorphisms of G ≠ae Aut(G) " ' ae (x ' ae "x" ≠1 ).

Recall that the group of inner automorphisms of G, denoted by Inn(G), is composed of the automorphisms of G induced by the conjugation by an element of G, i.e. automorphisms of the form x ' ae gxg ≠1 for some g in G. The group of inner automorphisms is normal in Aut(G) and the quotient group Aut(G)/Inn(G) denoted by Out(G) is called the outer automorphism group of G. A short exact sequence as in (1) induces a group homomorphism ae Out(G) that factors through Q, i.e. the following diagram

/ / Out(G) Q ; ;
commutes. The homomorphism Q ae Out(G) obtained in this way is called the monodromy of the short exact sequence [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF].

The main examples of short exact sequences as in (1) come from complex geometry. For instance, a surjective holomorphic map with connected fibers f : X ae S between a compact complex manifold X and a closed Riemann surface S of positive genus, induces a short exact sequence

1 / / G / / fi 1 (X) fú / / fi orb 1 (S) / / 1,
where G is the image of the fundamental group of the generic fiber in fi 1 (X) and S is endowed with an orbifold structure that considers the multiplicities of singular fibers (see Section 1.4.1). In [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF], Dimca, Papadima and Suciu studied some finiteness properties of the fundamental group of a generic fiber of a holomorphic map as before, in the particular case when X is a direct product of Riemann surfaces of genus greater or equal than two and S is of genus 1. This study on finiteness properties of fundamental groups of generic fibers was continued by Llosa Isenrich [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] and by Bridson and Llosa Isenrich [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF].

They construct new Kähler groups with interesting finiteness properties. Then one has the following question.

Question 2. For a surjective holomorphic map with connected fibers f : X ae S between a compact complex manifold X and a closed Riemann surface S of positive genus, what can be said of the finiteness properties of the fundamental group of its generic fiber ?

We will be interested in the case when X is a compact Kähler manifold. This work will give partial answers to Questions 1 and 2.

Restrictions on monodromy

A classical problem is the study of the monodromy: given a proper holomorphic submersion f : X ae Y (which is not a locally trivial holomorphic fiber bundle) with fibers of dimension 1, one can study the monodromy fi 1 (Y ) ae MCG(S), where S is the generic fiber of f , which is a topological surface and MCG(S) is the mapping class group of S. In [START_REF] Shiga | On monodromies of holomorphic families of Riemann surfaces and modular transformations[END_REF], Shiga proved the following result.

Theorem (Shiga). Let f : X ae B be a holomorphic submersion which is not a locally trivial holomorphic fiber bundle from a compact complex surface onto a hyperbolic surface with generic fiber a closed hyperbolic surface S. Then the monodromy fi 1 (B) ae MCG(S) has infinite image and it cannot preserve any isotopy class of a simple closed curve in S.

One of the results that motivated the work presented in Chapter 1, is the following result of Bregman and Zhang (see [START_REF] Bregman | On Kähler extensions of abelian groups[END_REF]).

Theorem (Bregman-Zhang). Let S be a closed surface of genus g Ø 2, be a Kähler group and k be a positive integer such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Z k / / 1.
Then there is a finite index subgroup 1 of containing fi 1 (S) such that the restricted short exact sequence

1 / / fi 1 (S) / / 1 P / / P ( 1 ) ƒ Z k / / 1
splits as a direct product.

Here we will study short exact sequences as in [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF], where G is the fundamental group of a closed oriented surface S of genus g Ø 2 and without assuming that S is the fiber of a holomorphic submersion. One of the keys of this study is the fact any element of Out(fi 1 (S)) can be represented by a di eomorphism of the surface S (see Dehn [START_REF] Dehn | Papers on group theory and topology, Translated from the German and with introductions and an appendix by John Stillwell, With an appendix[END_REF] for the original proof and Nielsen [START_REF] Nielsen | Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen[END_REF] for the first published proof). We will also give some restrictions on short exact sequences as in [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF] where G splits as an amalgamated product or an HNN extension.

Introduction

Presentation of results on Kähler groups and finitely generated groups acting on trees

Let us present the results of the preprint [START_REF] Nicolás | On finitely generated normal subgroups of Kähler groups[END_REF]. Our main assumption is that a Kähler group admits as a normal subgroup a finitely generated group G acting on a tree T . In many cases we will see that the action of G on T extends to , which will allow us to apply the following result of Gromov and Schoen (see [START_REF] Gromov | Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one[END_REF]).

Theorem (Gromov-Schoen). Let X be a compact Kähler manifold whose fundamental group acts on a tree which is not isomorphic to a line nor a point. Suppose that the action is minimal with no fixed points on the boundary. Then there is a surjective holomorphic map with connected fibers from X to a closed hyperbolic orbifold inducing the short exact sequence

1 / / N / / / / fi orb 1 ( ) / / 1,
such that the restriction of the action to N is trivial.

See Section 1.4.1 for the definition of the orbifold fundamental group fi orb 1 ( ). The problem of extending the action of G on a tree to the group Aut(G) has been studied in the literature. In several cases, one can prove extension results. This is the case if the group G is of one of the following forms:

1. A free products A ú B with A and B indecomposable groups, not infinite cyclic, acting on their Bass-Serre tree.

2. The Baumslag-Solitar group Èx, t|tx p t ≠1 = x q Í, acting on its Bass-Serre tree, where p, q are integers with p, q > 1 and such that neither is a multiple of the other (see Gilbert, Howie, Metaftsis and Raptis [START_REF] Gilbert | Tree actions of automorphism groups[END_REF] and Pettet [START_REF] Pettet | The automorphism group of a graph product of groups[END_REF]).

3. A one-ended hyperbolic groups without torsion, with infinite outer automorphism group and which is not virtually a surface group, acting on their JSJ tree. (Sela [START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF], see also Bowditch [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF]).

In order to apply these results to the study of Kähler groups, let us assume that the group G is centerless, the tree T is not a line nor a point and that the action of G on T is minimal, faithful and without fixed points on the boundary of T . Let be the image of the morphism ae Aut(G) induced by the conjugation action of on G. Under these conditions we prove the following result in Chapter 1.

Theorem A. Suppose that there is a finite index subgroup 0 of containing G ƒ Inn(G) such that the action of G on T can be extended to a G-compatible action of 0 on T . Then G is virtually a surface group. Moreover, there is a finite index subgroup 1 of containing G such that the restricted short exact sequence

1 / / G / / 1 P / / P ( 1 ) / / 1
splits as a direct product.

See Section 1.3.1.b for the definition of G-compatible action. The first application of this result is the following.

Theorem B.

Let S be a closed surface of genus g Ø 2 and be a Kähler group such that there is a short exact sequence

1 / / fi 1 (S) / / / / Q / / 1.
Suppose that the conjugation action of on fi 1 (S) preserves the conjugacy class of a simple closed curve in S. Then there is a finite index subgroup 1 of containing fi 1 (S) such that the restricted short exact sequence

1 / / fi 1 (S) / / 1 / / Q 1 / / 1
splits as a direct product (where Q 1 is the image of 1 in Q). In particular the monodromy

Q ae Out(fi 1 (S)) is finite.
This result can be seen as a topological version of Shiga's result. We will also see that we can recover the result by Bregman and Zhang by using classical results on the mapping class group and the hyperbolization of three manifolds. More applications are obtained.

Theorem C. Let G be a group that splits as a non-trivial free product A ú B with A and B indecomposable, not infinite cyclic. Then G does not embed as a normal subgroup in a Kähler group.

Remark. For the Bass-Serre tree of a free product G = A ú B ú C, there are automorphisms of G which cannot be extended. For instance ifis a fixed element of B, the automorphism such that a ' ae a, b ' ae b, and c ' ae -c-≠1 for all a oe A, b oe B and c oe C (called automorphism of Fouxe-Rabinovitch), does not extend to the Bass-Serre tree whose corresponding graph of groups is:

A B C
This implies that our proof of Theorem C does not extend to a free product of at least three factors.

Theorem D. Let G be the Baumslag-Solitar group

Èx, t|tx p t ≠1 = x q Í, where p, q are integers with p, q > 1 and such that neither one is a multiple of the other. Then G does not embed as a normal subgroup in a Kähler group.

Theorem E. Suppose that G is a one-ended torsion-free hyperbolic group that embeds as a normal subgroup in a Kähler group . If Out(G) is infinite, then G is virtually a surface group.

We observe that in this last theorem there is no hypothesis on the monodromy map. We only need to know that Out(G) is infinite.

Kähler groups and finiteness properties

Let us now present some results on finiteness properties of Kähler groups. A detailed introduction to finiteness properties will be given in Section 2.2. In particular, the finiteness properties FP n and F n will be defined there.

Let X be a connected compact complex manifold of complex dimension n Ø 2 and let S be a closed Riemann surface of positive genus. A surjective holomorphic map with connected fibers f : X ae S is called an irrational pencil. In [START_REF] Kapovich | On normal subgroups in the fundamental groups of complex surfaces[END_REF], Kapovich proved the following result.

Theorem (Kapovich). Let X be an aspherical compact Kähler surface, S be a closed Riemann surface of positive genus and f : X ae S an irrational pencil whose singular fibers have multiplicity one. Then f is a submersion or the kernel of the induced map on fundamental groups f ú : fi 1 (X) ae fi 1 (S), is not finitely presented.

This was the first result relating the existence of critical points of an irrational pencil to the finiteness properties of the kernel of the induced map on fundamental groups. As a consequence of this result, Kapovich proved the existence of compact complex hyperbolic surfaces whose fundamental groups admit a finitely generated normal subgroup which is not finitely presentable. In other words, Kapovich gave the first examples of noncoherent uniform lattices in PU [START_REF] Amorós | Fundamental groups of compact Kähler manifolds[END_REF][START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF]. Recall that a group is called coherent if every finitely generated subgroup is also finitely presentable. Further examples of noncoherent Kähler groups were given by Kapovich [START_REF] Kapovich | Noncoherence of arithmetic hyperbolic lattices[END_REF] and Py [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF]. One of the main ingredients for these examples is the following result essentially due to Kapovich (see [START_REF] Kapovich | Noncoherence of arithmetic hyperbolic lattices[END_REF] and [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF]), which follows by combining Theorem 4 in [START_REF] Py | Some noncoherent, nonpositively curved Kähler groups[END_REF] (due to Kapovich) with the result of Bregman and Zhang quoted earlier.

Theorem. Let be a Kähler group that can be realized as the fundamental group of an aspherical compact Kähler surface X with positive first Betti number. Assume that has no finitely generated Abelian subgroup whose normalizer has finite index in . Then, at least one of the following cases occurs:

1. is noncoherent.

2. There is a finite cover of X that admits a holomorphic submersion onto a compact hyperbolic surface with connected hyperbolic fibers.

Friedl and Vidussi give a refinement of these results about coherence of Kähler groups in [START_REF] Friedl | Vidussi Virtual algebraic fibrations of Kähler groups[END_REF]. Besides studying the coherence of certain groups, one can also study the existence of non-finitely generated normal subgroups. For instance, recall that a normal subgroup of a non-Abelian free group or a surface group is finitely generated only if it has finite index (see [START_REF] Schreier | Die Untergruppen der freien Gruppen[END_REF] for the original proof in the case of free groups and Lemmas 3.3 and 3.4 in [START_REF] Catanese | Fibred Kähler and quasi-projective groups, Special issue dedicated to Adriano Barlotti[END_REF]). In general, non-finitely generated normal coabelian subgroups of Kähler groups are related to fibrations on Riemann surfaces. By studying the Bieri-Neumann-Strebel invariant of Kähler groups, Delzant proved in [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] the following result.

Theorem (Delzant). Let us consider the short exact sequence

1 / / G / / / / Q / / 1,
where is a Kähler group, Q an Abelian group and G is not finitely generated. Then, if X is a compact Kähler manifold that realizes as its fundamental group, there exists a surjective holomorphic map with connected fibers from X to a 2-dimensional hyperbolic orbifold.

Kähler groups and finiteness properties

We refer the reader to [START_REF] Delzant | L'invariant de Bieri-Neumann-Strebel des groupes fondamentaux des variétés kählériennes[END_REF] for the definitions of a 2-dimensional hyperbolic orbifold and a holomorphic map from a compact complex manifold to such a space. When Q = Z, the above theorem was previously established by Napier and Ramachandran in [START_REF] Napier | Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces[END_REF] using di erent techniques.

Let us introduce some recent results on the construction of Kähler groups with exotic finiteness properties and the context that motivated these constructions. In [45, §0.3.1], Kollár asked whether a projective group is always commensurable (up to finite kernels) to a group admitting a classifying space which is a quasi-projective variety. Since any quasiprojective variety has the homotopy type of a finite complex [30, p. 27], a positive answer to this question would imply that any projective group is commensurable to a group having a finite classifying space (see Section 2.2.1 for the definition of commensurability). However, a negative answer to Kollár's question was given by Dimca, Papadima and Suciu in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]. In [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] the authors proved the following two results (see Theorem C and §2 in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]).

Theorem (Dimca, Papadima and Suciu). If n Ø 3 and if f : X ae S is an irrational pencil with isolated critical points, then the fundamental group of a smooth fiber of f embeds into that of X and coincides with the kernel of the induced homomorphism f ú : fi 1 (X) ae fi 1 (S).

Notice that under the hypothesis of the latter result, if we assume that X is Kähler, one obtains a finitely generated normal subgroup of the fundamental group of X, which is again a Kähler group.

Theorem (Dimca, Papadima and Suciu).

Let X = 1 ◊ • • • ◊ n be a direct product of
n Riemann surfaces of genus greater than 1 and let S have genus 1. If f : X ae S is an irrational pencil with isolated critical points then H n (ker(f ú ), Q) has infinite dimension.

Combining the latter two results, Dimca, Papadima and Suciu answered negatively Kollár's question. Indeed in the situation of the latter result, the kernel of the induced homomorphism f ú : fi 1 (X) ae fi 1 (S), which is the fundamental group of a smooth fiber of f if n Ø 3, cannot be of type FP n as the group H n (ker(f ú ), Z) is not finitely generated (see Proposition 87). The property of being of type FP n is invariant by the commensurability relation [START_REF] Bieri | Homological dimension of discrete groups[END_REF][START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]. Therefore, no group commensurable to the kernel of f ú can have a finite classifying space.

Building on the work [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF], further examples of projective groups with exotic finiteness properties were constructed and studied by Llosa Isenrich [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] and by Bridson and Llosa Isenrich [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF]. All the examples studied in [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF][START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] are either subgroups of direct products of surface groups or extensions of such subgroups as in [START_REF] Bridson | Kodaira fibrations, Kähler groups, and finiteness properties[END_REF].

Presentation of results on Kähler groups with exotic finiteness properties

Here we present our results on the construction of new Kähler groups with exotic finiteness properties. See [START_REF] Nicolás | Py Irrational pencils and Betti numbers[END_REF] for a preprint of this work which will appear in the Annales de la Faculté des Sciences de Toulouse. This is a joint work with Pierre Py.

Let f : X ae S be an irrational pencil with dim C X = n Ø 2. We assume that the critical points of f are isolated and that f is not a submersion; its critical set is then a nonempty finite set. Let ' X ae X be the covering space of X such that fi 1 ( ' X) ƒ ker(f ú ). The main results are the following:

Introduction

In the special case when X is a product of Riemann surfaces and S has genus 1, we recover from Theorem G the second result of Dimca, Papadima and Suciu quoted before. It is of course interesting to look for more examples of projective (or closed Kähler) manifolds endowed with a holomorphic map to which one can apply Theorems F and G. One way to build new examples of Kähler groups satisfying an exotic finiteness property is to use the Cartwright-Steger surface (we refer to [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF][START_REF] Cartwright | Enumeration of the 50 fake projective planes[END_REF] for a definition and properties of this compact Kähler surface) that we will denote by Y . The Albanese map of this complex surface h : Y ae E is a holomorphic map whose target is an elliptic curve and it has been proven that its singularities are isolated (see [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF]) and nondegenerate (see [START_REF] Koziarz | Stability of the Albanese fibration on the Cartwright-Steger surface[END_REF] and [START_REF] Rito | Surfaces with canonical map of maximum degree[END_REF]). We can thus consider the product Y b of Y with itself b times and the map

h + • • • + h : Y b ae E.
(

) 2
This provides natural examples to which one can apply Theorems F and G. Denoting by

< PU(2, 1)
the fundamental group of the Cartwright-Steger surface, the latter construction together with Theorem G immediately implies:

Theorem H. The direct product of b copies of contains a coabelian normal subgroup N which is of type FP 2b≠1 but satisfies that H 2b (N, Q) has infinite dimension.
The group N appearing above, is the kernel of the morphism on fundamental groups induced by the map (2). The fact that N is of type FP 2b≠1 follows from the results in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF]; we will explain it again in this text. In particular, Theorem H implies that N is not of type FP 2b . In Chapter 2, we prove that no finite index subgroup of N , embeds in a direct product of surface groups. This implies that the groups constructed in this way are of a di erent nature compared to the examples from [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF][START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF].

Introduction

Let G be a finitely generated group. Bass-Serre Theory (see [START_REF] Scott | Topological methods in group theory, Homological group theory[END_REF] and [START_REF] Serre | Trees, Springer Monographs in Mathematics[END_REF]) establishes a dictionary between decompositions of G as an amalgamated product or an HNN extension and actions of G on a simplicial tree without inversions which are transitive on the set of edges. Of course, the theory also deals with more complicated graphs of groups but we will mainly deal with amalgamated products and HNN extensions.

Question. Given a short exact sequence of finitely generated groups

1 / / G / / / / Q / / 1, (1.1.1)
where G acts on a tree T , under which conditions can we (virtually) extend this action to ?

If the center of G acts trivially on T , we get an induced action of the group of inner automorphisms of G on T , and then this question can be approached by looking at the inclusion Inn(G) Òae Aut(G) and by trying to extend the action on T to a larger subgroup of Aut(G). Several authors have studied the latter question. Karrass, Pietrowski and Solitar [START_REF] Karrass | Automorphisms of a free product with an amalgamated subgroup, Contributions to group theory[END_REF] studied the case of an amalgamated product A ú C B, where C is maximal among all its conjugates in A and B. Pettet [START_REF] Pettet | The automorphism group of a graph product of groups[END_REF] studied the general situation of a graph of groups with a more restrictive condition on the edge stabilizers of its Bass-Serre tree ("edge group incomparability hypothesis") which is equivalent to the conjugate maximal condition of Karrass, Pietrowski and Solitar when there is one orbit of edges. A particular case of this situation was studied by Gilbert, Howie, Metaftsis and Raptis [START_REF] Gilbert | Tree actions of automorphism groups[END_REF] where they proved that the action of the Baumslag-Solitar group on its Bass-Serre tree can be extended to the whole group of automorphisms. In the context of a one-ended torsion-free hyperbolic group Chapter 1. Kähler groups acting on trees G, Sela [START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF] proved the existence of a "canonical" tree T on which G acts named the JSJ tree. This means that Aut(G) contains a subgroup of finite index such that the action of G on T extends to such subgroup. By studying the action of Aut(G) on ˆG, Bowditch [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF] constructed a finer tree T B such that the action of G on T B can be extended to Aut(G).

We will apply these results to study finitely generated normal subgroups of Kähler groups. The main ingredient to apply these results about actions on trees that extend to a group of automorphisms is a classical result of Gromov and Schoen [START_REF] Gromov | Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one[END_REF] about Kähler groups acting on trees (see Theorem 47).

For the reader's convenience, we will state again four theorems from the introduction that will be proved in this chapter. We will study short exact sequences as in (1.1.1), where is a Kähler group and G is a surface group, i.e., when G can be realized as the fundamental group of a closed surface S of genus g Ø 2.

Theorem B. Suppose that the conjugation action of on fi 1 (S) preserves the conjugacy class of a simple closed curve in S. Then there is a finite index subgroup 1 of containing fi 1 (S) such that the restricted short exact sequence

1 / / fi 1 (S) / / 1 / / Q 1 / / 1 splits as a direct product (where Q 1 is the image of 1 in Q).
In particular the monodromy

Q ae Out(fi 1 (S)) is finite.
In Section 1.5 we will see that using Theorem B one can recover Bregman and Zhang's result on Kähler extensions of Abelian groups by surface groups and that this result can be extended to Kähler extensions by surface groups whose monodromy maps have Abelian image (see Corollaries 54 and 59).

Let us go back to the study of extensions as in (1.1.1) where G need not be a surface group. The folllowing three results are an application of Gromov and Schoen's theorem and the works [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF][START_REF] Gilbert | Tree actions of automorphism groups[END_REF][START_REF] Karrass | Automorphisms of a free product with an amalgamated subgroup, Contributions to group theory[END_REF][START_REF] Pettet | The automorphism group of a graph product of groups[END_REF][START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF]. Recall that a group H is indecomposable if for any decomposition H = H 1 ú H 2 as a free product, H 1 or H 2 is trivial.

Theorem C. Let G be a group that splits as a non-trivial free product A ú B with A and B indecomposable, not infinite cyclic. Then G does not embed as a normal subgroup in a Kähler group.

Theorem D. Let G be the Baumslag-Solitar group

Èx, t|tx p t ≠1 = x q Í,
where p, q are integers with p, q > 1 and such that neither one is a multiple of the other. Then G does not embed as a normal subgroup in a Kähler group.

Finally, our last result of this chapter deals with the case where the normal subgroup

G is a one-ended hyperbolic group.

Theorem E. Suppose that G is a one-ended torsion-free hyperbolic group that embeds as a normal subgroup in a Kähler group . If Out(G) is infinite, then G is virtually a surface group.

We observe that in this last theorem there is no hypothesis on the monodromy map ae Out(G). We only need to know that Out(G) is infinite.

The structure of this chapter is the following. In Section 1.2, we give the definitions of an amalgamated product and HNN extensions and we recall some basic aspects of Bass-Serre Theory. We study as well the particular case of decompositions of surface groups over maximal cyclic subgroups and the Bass-Serre trees associated to these decompositions. In Section 1.3, we explain how to extend the action of a group on a tree to a subgroup of its group of automorphisms. This section is mainly expository. It gives a more geometric approach to the results of Karras, Pietrowski, Solitar and Pettet. It is only Section 1.3.4 that deals with surface groups, that we will use to prove Theorem B (in Section 1.4). Hence the reader only interested in the proof of Theorems A B, C, D and E can skip Sections 1.3.1, 1.3.2 and 1.3.3. In Section 1.4, we apply these results to the study of Kähler groups admitting as a normal subgroup a group acting on a tree. Theorem A is proved in Section 1.4.

2).

• To prove Theorem D we apply Gilbert, Howie, Metaftsis and Raptis's work (Theorem 43).

• Finally, to prove Theorem E we apply directly the work of Bowditch and Sela (Theorem 45). In Section 1.5 we study the monodromy map of a Kähler extension by a surface group and establish variations on Theorem B.

Groups acting on trees

In this section we introduce the definitions of an amalgamated product and an HNN extension and we state the principal results of Bass-Serre Theory for such groups. We refer the reader to [START_REF] Scott | Topological methods in group theory, Homological group theory[END_REF] and [START_REF] Serre | Trees, Springer Monographs in Mathematics[END_REF] for an introduction to this subject. We will also describe the decompositions of a surface group as an amalgamated product or as an HNN extension over a cyclic group (corresponding to a simple closed curve) and we will give a geometric construction of the Bass-Serre tree for such groups as a dual tree embedded in the hyperbolic plane.

Amalgamated products and HNN extensions

1.2.1.a Definitions

The free product of two groups A and B, denoted by A ú B can be described as the set of words

g 1 g 2 • • • g n ,
where g i oe A \ {1} or g i oe B \ {1} for all i = 1, . . . , n and such that if g i oe A, then g i+1 oe B and if g i oe B, then g i+1 oe A. The operation of this group is given by the juxtaposition of words (up to simplification) and the empty word represents the identity element. Given presentations of A and

B A = ÈX | RÍ and B = ÈY | SÍ, (1.2.1)
we obtain that the free product A ú B is the group given by the presentation

ÈX Û Y | R Û SÍ.
ÈX Û Y | R Û S, i A (c)i ≠1 B (c) 'c oe CÍ.
Notice that this construction is interesting only if C is properly contained in A and B, otherwise the amalgamated product is isomorphic to A or B.

Using the general construction of amalgams given by Serre in [START_REF] Serre | Trees, Springer Monographs in Mathematics[END_REF], one can prove the non-triviality of groups defined by certain presentations. An instance of this is the following result of Higman, B.H. Neumann and H. Neumann (see [66, p. 8]). Proposition 3. Let C be a subgroup of a group A and let ◊ : C ae A be an injective homomorphism. Then, there exists a group G containing A and an element t of G \ A such that tct ≠1 = ◊(c) for all c in C. Furthermore, if A is countable (or finitely generated, or torsion-free) one can choose G to be a group with the same property.

In the latter proposition, the subgroup generated by A and t is called HNN extension of A relative to ◊ and it is denoted by Aú C,◊ . Given a presentation of A as in (1.2.1) and an injective homomorphism ◊ : C ae A as in Proposition 3, we get that the HNN extension

Aú C,◊ is given by ÈX, t | R, tct ≠1 ◊(c ≠1 ) 'c oe CÍ.
A priori, a group given by the latter presentation could be trivial, but Proposition 3 guarantees the non-triviality of such a group. Definition 4. We say that a group G splits over a group C if it decomposes as an amalgamated product A ú C B or as an HNN extension Aú C,◊ .

1.2.1.b Examples

• If A and B are the free groups of rank n and m respectively, then the free product A ú B is the free group of rank n + m.

• More interesting examples coming from topology are given by Van Kampen's Theorem. If X is the union of two open sets U and V whose intersection is path connected, then the fundamental group of X is given by the amalgamated product of the fundamental groups of U and V along the fundamental group of U fl V (where all the fundamental groups are based at a point in U fl V ), if the inclusions U fl V Òae U and U fl V Òae V induce injections on fundamental group.

• Given a group A and an automorphism ◊ : A ae A, the HNN extension Aú A,◊ coincides with the semidirect product A o Z: the morphism Aú A,◊ ae Z that sends t to the generator of Z and all the elements of A to zero, has the group A as its kernel. Therefore we obtain a short exact sequence 1 ae A ae Aú A,◊ ae Z ae 1.

Since the map Z ae Aú A,◊ that sends the generator of Z to t is a section of the latter short exact sequence, we obtain that Aú A,◊ splits as the semidirect product A o Z.

• Let p, q be two non-zero integers. The Baumslag-Solitar group G(p, q) is defined as the HNN extension given by the presentation

Èx, t | tx p t ≠1 = x q Í.
Notice that if p = 1 = q we obtain the fundamental group of a torus and if p = 1 = ≠q, we obtain the fundamental group of the Klein bottle. Proposition 3 guarantees the non-triviality of such groups: let A = Z and C = pZ. If we denote by x the generator of Z, then in multiplicative notation, x p is the generator of pZ. Now, let ◊ : pZ ae Z be the injective homomorphism that sends x p to x q (in multiplicative notation). Hence, the Baumslag-Solitar group is precisely the HNN extension Zú pZ,◊ , which by Proposition 3 is non-trivial.

• In Section 1.2.3, we will see that any simple closed curve in a closed surface of genus g Ø 2, induces a decomposition of the fundamental group of such a surface as an amalgamated product or as an HNN extension over a cyclic group.

Bass-Serre dictionary

Recall that a (simplicial) tree is a connected graph (V, E), where V is the set of vertices and E is the set of edges with the additional property that for any edge e in E, the graph (V, E \ {e}) obtained by removing the edge e from the graph (V, E), is not connected.

An isomorphism between two trees T = (V, E) and

T Õ = (V Õ , E Õ ) is a pair of bijections Ï V : V ae V Õ and Ï E : E ae E Õ such that if e = {v, w} is an edge of T , then {Ï V (v), Ï V (w)} = Ï E ({v, w}).
The latter condition means that the bijection between the sets of vertices of the trees T and T Õ respect the adjacency relation.

An automorphism of a tree T is an isomorphism between T and itself. Given a tree T and a group G, we abbreviate the expression G acts on T by automorphisms by G acts on T .

Notations. Suppose that G acts on a tree T and let e and v be an edge and a vertex of T respectively. We will denote by G e and G v the subgroups of G given by the respective stabilizers of e and v. The dictionary established by Bass-Serre theory is given by the following results (see [START_REF] Scott | Topological methods in group theory, Homological group theory[END_REF][START_REF] Serre | Trees, Springer Monographs in Mathematics[END_REF]).

Theorem 5 (Serre). Let G be a group acting on a tree T without inversions in such a way that the action is transitive on the set of edges. Let e = (v 1 , v 2 ) be an edge of T .

If v 1 and v 2 are in di erent orbits, then G splits as the amalgamated product

G v 1 ú Ge G v 2 .
2. If v 1 and v 2 are in the same orbit, then G splits as the HNN extension G v 1 ú Ge,◊ where ◊ : G e ae G v 1 is the monomorphism given by the conjugation by an element t in

G \ G v 1 that sends v 1 to v 2 .
Theorem 6 (Serre). Let G be a group that splits as an amalgamated product or as an HNN extension. Then, G acts on a tree T without inversions and this action is transitive on the set of edges. Recall that the boundary of a tree is given by the set of infinite paths without backtracking starting at a fixed point of the tree (this is one definition among many others).

• If G splits as an amalgamated product A ú C B and C is properly contained in A and B, then the action of G on the boundary of T has no fixed points. All the infinite paths without backtracking starting at A, have as their first edge an element aC in the set of left cosets A/C. Since the action of A on A/C is transitive and C is properly contained in A, none of these paths is fixed under the action of A and thus neither under the action of G.

• If G splits as an HNN extension Aú C,◊ , and C and ◊(C) are properly contained in A, then the action of G on the boundary of T has no fixed points. The argument is very similar to the case of an amalgamated product. In this case the infinite paths starting at A have as their first edge an element of the form aC or atC for some a in A, and the action of A is transitive on each of the following subsets of edges {aC | a oe A} and {atC | a oe A}.

See Proposition 4.13 of [START_REF] Minasyan | Acylindrical hyperbolicity of groups acting on trees[END_REF] for a characterization of a group that splits as the fundamental group of a graph of groups whose action on the corresponding Bass-Serre tree fixes a point on the boundary

Decomposition of a surface group

Definition 7. A group G is called a surface group if it can be realized as the fundamental group of a closed oriented surface of genus g Ø 2.

Let " be a simple closed curve in a closed oriented surface S of genus g Ø 2. The curve is called separating, if we obtain 2 connected components when we cut S along ", and nonseparating if we obtain 1 connected component. Van Kampen's Theorem allows us to express the fundamental group of S in terms of the fundamental groups of the surfaces obtained after cutting S along ".

1.2.3.a Cutting along a separating curve

Let " be a separating curve in a closed surface S of genus g Ø 2 and let S 1 and S 2 be small neighborhoods of the surfaces obtained after cutting S along ", such that S 1 fl S 2 is a small neighborhood of " that deformation retracts onto ". We fix a base point x 0 on ". Then, if we denote by A, B and C the fundamental groups based at x 0 of S 1 , S 2 and S 1 fl S 2 respectively, by Van Kampen's Theorem we obtain that

fi 1 (S, x 0 ) ƒ A ú C B.
Notice that A and B are free groups and since S 1 fl S 2 deformation retracts onto ", we get that C is a cyclic group generated by the homotopy class of ". Hence, for each nonnullhomotopic separating curve of S we obtain a decomposition of the fundamental group of S as an amalgamated product of two free groups along a cyclic group.

1.2.3.b Cutting along a nonseparating curve

Let " be a nonseparating curve in a closed surface S of genus g Ø 2. As before, we fix a base point x 0 on " and let • be a loop as in Figure 1.2.5. Let S 1 be the surface obtained after cutting S along " and let S 2 be a small neighborhood of " and • . Now, one can apply Van Kampen's Theorem to S 1 and S 2 . For this, one must choose a "copy" of " (containing a "copy" of x 0 ) in the surface S 1 . Let A, B and C be the fundamental groups based at x 0 of S 1 , S 2 and S 1 fl S 2 respectively. Then, we obtain that

fi 1 (S, x 0 ) ƒ A ú C B.
We will simplify the latter expression by understanding the maps on fundamental groups fi 1 (S 1 fl S 2 ) ae fi 1 (S 1 ) and fi 1 (S

Èa 1 , b 1 , . . . , a g≠1 b g≠1 , c 1 , c 2 | [a 1 , b 1 ] • • • [a g≠1 , b g≠1 ]c 1 c 2 Í. x 0 a 1 a 2 b 1 a 1 1 b 1 1 b 2 a 1 2 b 1 2 ⌧ 1 Figure 1.2.6: 4g ≠ 1-polygon without two disks
If we denote by c and t the homotopy classes of " and • in S 2 , we get that fi 1 (S 2 , x 0 ) is a free group of rank 2 generated by c and t, since S 2 deformation retracts onto the wedge-sum of the loops defined by " and • .

Finally, notice that the intersection between S 1 and S 2 is given by the union of a neighborhood of the boundary components of S 1 with a neighborhood of the path • 1 (see Figure 1.2.7). Hence fi 1 (S 1 fl S 2 , x 0 ) is a free group of rank 2, since S 1 fl S 2 deformation retracts onto two loops joined by a path, where one of these loops is precisely ".

From this, we deduce that the inclusions S 1 fl S 2 Òae S 1 and S 1 fl S 2 Òae S 2 identify c with c 1 and tc 1 t ≠1 with c ≠1 2 . Thus, we obtain the following presentation of fi 1 (S, x 0 )

S 1 \ S 2
Èa 1 , b 1 , . . . , a g≠1 b g≠1 , c 1 , c 2 , t | [a 1 , b 1 ] • • • [a g≠1 , b g≠1 ]c 1 c 2 , tc 1 t ≠1 c 2 Í.
If we denote by C the cyclic subgroup of fi 1 (S 1 , x 0 ) generated by c 1 , we obtain that fi 1 (S, x 0 ) is given by the HNN extension

fi 1 (S 1 , x 0 )ú C,◊
, where

◊ : C ae fi 1 (S 1 , x 0 ) c 1 ' ae tc 1 t ≠1 .

Dual tree of a simple closed curve in a surface group

As we saw in Section 1.2.3, the fundamental group of a closed surface S of genus g Ø 2 splits over the cyclic subgroup generated by a simple closed curve. By Bass-Serre Theory, associated to such a splitting there exists a tree endowed with an action of the fundamental group of such a surface. Here we will give the construction of a dual tree associated to a simple closed curve in a closed surface of genus g Ø 2 and we will see that this tree coincides with the Bass-Serre tree associated to the decompositions studied in Section 1.2.3. This allows us to have a better understanding of the Bass-Serre tree for such decompositions of a surface group. This material is very classical, here we simply give a self-contained exposition of it.

Let S be a Riemann surface of genus greater than one. By the Uniformization Theorem we have that S is biholomorphic to H 2 / , where is a discrete subgroup of PSL(2, R). The application H 2 ae H 2 / is its universal covering space and it induces a hyperbolic structure on S. Let us recall the classical identification of the fundamental group of S with the group

< PSL(2, R) = Isom + (H 2 ). If we denote by p : (H 2
, z 0 ) ae (S, x 0 ) the universal covering based map of S, then we have a map

fi 1 (S, x 0 ) ae Aut(H 2 ) (1.2.2) [ ‡] ' ae Ï [ ‡] ,
where

Ï [ ‡] is the unique automorphism of H 2 such that Ï [ ‡] (z 0 ) = ' ‡(1)
, with ' ‡ a lift of ‡ starting at the point z 0 . By the uniqueness of path liftings we have that for loops ‡, • :

[0, 1] ae S " ‡• (1) = Ï [ ‡] (' • (1)) = Ï [ ‡] ¶ Ï [• ] (z 0 ), which implies that (1.2.
2) is a group homomorphism, its image is the discrete group and it is an isomorphism from fi 1 (S, x 0 ) onto . From now on we identify these two groups.

As in Section 1.2.3, let " be a simple closed curve in S based at x 0 . Recall that every non-nullhomotopic simple closed curve in S is homotopic to a unique closed geodesic (see 1.2. Groups acting on trees 27 [START_REF] Farb | A primer on mapping class groups[END_REF] p. 24). Hence, we may assume that " is a closed geodesic for the hyperbolic structure on S.

Let E be the set of bi-infinite geodesics in H 2 , whose images under p are equal to " and let V be the set of connected components of H 2 \ fiE. We say that two elements of V are related if they are separated by an element of E. In other words, V, W in V are related if there exists l in E such that V fi l fi W is connected. This defines a symmetric binary relation on V. Hence, V endowed with this relation defines a graph, where the set of edges is precisely given by E (see Figure 1.2.8). For all x in fi 1 (S, x 0 ), let l x be the element of E passing through Ï x (z 0 ). Then we get that l x = Ï x (l 1 ). This follows from the fact that p ¶ Ï x and thereby p ¶ Ï x (l 1 ) = p(l 1 ) = ", i.e. Ï x (l 1 ) is a bi-infinite geodesic in H 2 passing through Ï x (z 0 ), whose image under p is equal to ". Hence, the action of fi 1 (S, x 0 ) preserves the set E, and by continuity it preserves the set V with its symmetric binary relation.

A combinatorial path in the graph (V, E) between two vertices V, W of (V, E) is given by finitely many connected components

V 1 , . . . , V k+1 of H 2 \ fiE, such that • V 1 = V , V k+1 = W . • For all i = 1, . . . , k, V i " = V i+1 . • For all i = 1, . . . , k, there is a geodesic l i such that V i fi l i fi V i+1 is connected. As a consequence, we get that V 1 fi l 1 fi V 2 fi • • • fi V k fi l k fi V k+1 is connected.
This observation allows us to prove the following result Proposition 8. The graph (V, E) is a tree.

Proof. Let l be an element of E defining an edge between the connected components V and W of H 2 \ fiE. Then V and W are in di erent connected components of H 2 \ l and by the latter observation there is no combinatorial path from V to W in the graph (V, E \ {l}), which implies the result.

2 The tree (V, E) is called the dual tree of S associated to ". Now, we will see that this tree coincides with the Bass-Serre tree associated to the splitting of fi 1 (S, x 0 ) over the cyclic Chapter 1. Kähler groups acting on trees subgroup generated by the homotopy class of ", in such a way that the action of fi 1 (S, x 0 ) on the dual tree is consistent with the action of fi 1 (S, x 0 ) on the Bass-Serre tree. Lemma 9. The stabilizer of the bi-infinite geodesic l 1 under the action of fi 1 (S, x 0 ) on H 2 is equal to the cyclic group C generated by the homotopy class of ".

Proof. The cyclic group C stabilizes l 1 since the concatenation of " (or ") with itself n times, lifts to a geodesic arc contained in l 1 for all positive integer n. Now, let x be an element of fi 1 (S) such that

l x = Ï x (l 1 ) = l 1 .
Recall that l x is the bi-infinite geodesic in H 2 passing through Ï x (z 0 ) whose image under p is equal to ". If ‡ is a closed curve in S based at x 0 whose homotopy class is equal to x, we get that

Ï x (z 0 ) = Ï [ ‡] (z 0 ) = ' ‡(1) oe l 1 .
Hence, the geodesic arc contained in l 1 joining z 0 and Ï [ ‡] (z 0 ) is homotopic to ' ‡ relative to {0, 1}, which implies that [ ‡] = x is contained in C, and the result follows.

2 As a consequence of the latter result, we obtain a bijection between the set of edges of the dual tree (V, E) and the set of edges of the Bass-Serre tree T :

Corollary 10. The map E ae fi 1 (S, x 0 )/C Ï x (l 1 ) = l x ' ae xC
is well-defined and it is a bijection. Proof. By Lemma 9, we get that the map E ae fi 1 (S, x 0 )/C is well-defined and it is injective. The surjectivity of such a map follows directly by definition.

2 We recall a topological result that will be useful to define the bijection between the set of vertices of the dual tree (V, E) and the set of vertices of the Bass-Serre tree T .

Proposition 11. Let Y be a connected smooth manifold and X a connected smooth submanifold of Y . Let us fix a base point x 0 in X. If we denote by i : (X, x 0 ) ae (Y, x 0 ) the inclusion map and by p : ( ' Y , x0 ) ae (Y, x 0 ) the universal covering space of Y , then i ú : fi 1 (X, x 0 ) ae fi 1 (Y, x 0 ) is injective if and only if, the connected component of p ≠1 (X) containing x0 is simply connected.

Remark 12. Under the hypothesis of the latter result, if we denote by '

X the connected component of p ≠1 (X) containing x0 , we get that the lifts based at

x0 of i ú (fi 1 (X, x 0 )) are contained in ' X, i.e. i ú (fi 1 (X, x 0 )) stabilizes ' X under the action of fi 1 (Y, x 0 ) on ' Y . Hence, if '
X is simply connected, Proposition 11 allows us to consider fi 1 (X, x 0 ) as a subgroup of fi 1 (Y, x 0 ) that stabilizes ' X.

Corollary 13. If X is a connected component of S \ ", then fi 1 (X, x 0 ) ae fi 1 (S, x 0 ) is injective.

Proof. Every connected component of H 2 \ fiE is convex (for the hyperbolic metric) since it is the intersection of a countable set of half-planes which are convex. Therefore, every connected component of H 2 \fiE is simply connected and the result follows from Proposition 11.

Let us fix X a connected component of S \". By Corollary 13, we can consider fi 1 (X, x 0 ) as a subgroup of fi 1 (S, x 0 ). Let V 1 be the connected component of H 2 \ fiE containing all the lifts based at z 0 of the elements of fi 1 (X, x 0 ). We denote by V X the set of connected components of H 2 \ fiE whose image under p is equal to X. Observe that the action of fi 1 (S, x 0 ) on H 2 preserves V X . For all x in fi 1 (S, x 0 ), we write

V x = Ï x (V 1 ).
Lemma 14. The stabilizer of V 1 under the action of fi 1 (S, x 0 ) on H 2 is equal to fi 1 (X, x 0 ).

Proof. By the proof of Corollary 13, we get that p V 1 : V 1 ae X is the universal covering space of X and by Remark 12 we get that fi 1 (X, x 0 ) stabilizes V 1 . Now, let x be an element of fi 1 (S, x 0 ) such that

V x = Ï x (V 1 ) = V 1 . Then, we have that l x = Ï x (l 1 ) is a boundary component of V 1 containing Ï x (z 0 ). If we denote by ' ‡ the geodesic arc in H 2 joining z 0 with Ï x (z 0 ), we get that ' ‡ is contained in V 1 (by the convexity of V 1 .) Hence ‡ = p ¶ ' ‡ defines a closed curve in X such that Ï x (z 0 ) = Ï [ ‡] (z 0 )
. Therefore x is an element of fi 1 (X, x 0 ) and the result follows.

2 As a consequence of this result we obtain the following corollary that will allow us to give the bijection between the vertices of the dual tree (V, E) and the Bass-Serre tree T . We omit the proof since it is the same as the proof of Corollary 10.

Corollary 15. The map

V X ae fi 1 (S, x 0 )/fi 1 (X, x 0 ) V x ' ae xfi 1 (X, x 0 )
is well-defined and it is a bijection.

We conclude this section by studying separately the cases when " is a separating curve and when it is a nonseparating curve.

1.2.4.a Case of a separating curve

As before, we denote by S 1 and S 2 the surfaces obtained after cutting S along " and by A and B their respective fundamental groups based at x 0 . Let V 1 and W 1 be the connected components of H 2 \ fiE that contain the lifts based at z 0 of A and B respectively. Then V 1 and W 1 are separated by l 1 . We write

V x = Ï x (V 1 ) and W x = Ï x (W 1 ).
Then we have that

V = {V x | x oe fi 1 (S, x 0 )} Û {W x | x oe fi 1 (S, x 0 )},
and by Corollary 15, we have a bijection

V ae fi 1 (S, x 0 )/A Û fi 1 (S, x 0 )/B (1.2.3) V x ' ae xA W x ' ae xB.
Chapter 1. Kähler groups acting on trees

Notice that since l 1 separates V 1 and W 1 , then by continuity, l x separates V x and W x for all x in fi 1 (S, x 0 ). Hence V x and W x are vertices in the dual tree (V, E) joined by the edge l x . Finally, by Corollary 10, the map

E ae fi 1 (S)/C (1.2.

1.2.4.b Case of a nonseparating curve

As before, we denote by S 1 the surface obtained after cutting S along " and by A the fundamental group of S 1 based at x 0 . Now, let V 1 be the connected component of H 2 \ fiE that contains the lifts based at z 0 of A. We write

V x = Ï x (V 1 ).
Then we have that

V = {V x | x oe fi 1 (S, x 0 )},
and by Corollary 15, we have a bijection

V ae fi 1 (S, x 0 )/A (1.2.5)

V x ' ae xA Recall that p V 1 : V 1 ae S 1
is the universal covering space of S 1 . Since S 1 has two boundary components, we get that V 1 has two types of boundary components, corresponding to the preimages of the boundary components of S 1 . Recall that t is given by the homotopy class of a simple closed curve • in S that becomes a path in S 1 joining its two boundary components. Notice that the lift of this path based at z 0 is a curve contained in V 1 joining z 0 with Ï t (z 0 ). Hence l 1 (which contains z 0 ) and l t (which contains Ï t (z 0 )) are boundary components of V 1 of di erent type. Furthermore, the two types of boundary components of V 1 are given by {l a | a oe A} and {l at | a oe A}.

Notice that l t separates the connected components V 1 and V t and l 1 separates de connected components V t ≠1 and V 1 . More generally, by continuity, we get that l x separates the connected components V xt ≠1 and V x , i.e. V xt ≠1 and V x are vertices in the dual tree (V, E) joined by the edge l x . Finally, by Corollary 10, the map

E ae fi 1 (S, x 0 )/C (1.

Extending actions on the Bass-Serre tree

As explained at the beginning of this chapter, this section is mainly expository and serves as a (geometric) introduction to [START_REF] Gilbert | Tree actions of automorphism groups[END_REF][START_REF] Karrass | Automorphisms of a free product with an amalgamated subgroup, Contributions to group theory[END_REF][START_REF] Pettet | The automorphism group of a graph product of groups[END_REF]].

Extending an action to a group of automorphisms

Let G be a finitely generated group that splits as an amalgamated product or an HNN extension and let T be its Bass-Serre tree. If the center Z(G) of G acts trivially on T , the action of G on T factors through the quotient G/Z(G). Since this quotient is isomorphic to the group Inn(G) of inner automorphisms of G, we obtain an induced action of Inn(G) on T . The aim of this section is to extend (under suitable hypothesis) this induced action of Inn(G) on T to a larger group of automorphisms of G.

1.3.1.a Conditions to obtain a trivial action of the center of the group

The following two results exhibit su cient conditions to guarantee that the center of G acts trivially on T .

Lemma 16. If there is a vertex v of T whose stabilizer is its own normalizer in G, then Z(G) acts trivially on T .

Proof. Since for a subgroup of G the property of being its own normalizer is preserved by conjugation we get that the stabilizer of any vertex in the orbit of v is its own normalizer. By definition Z(G) commutes with all the elements of G. Hence, it normalizes the stabilizer of any vertex of T . In particular Z(G) normalizes the stabilizer of any vertex in the orbit of v. Therefore, using the hypothesis of the Lemma, we get that Z(G) is contained in the stabilizer of any vertex in the orbit of v, i.e. the action of G on T restricted to Z(G) fixes each vertex in the orbit of v. Hence Z(G) fixes every path between any two vertices in the orbit of v and as a consequence, it fixes the whole tree T . 2

Remark 17. In the latter proof we used the fact that if a group G acts on a tree T , and there is an element x of G that fixes two vertices v and w of T , then x fixes the path between Chapter 1. Kähler groups acting on trees v and w. This follows from the fact that the action of G on T is by automorphisms. Then, the action of G on T sends paths to paths, and the assertion follows from observing that the geodesic paths in a tree are unique.

For any two vertices v and v Õ of T we will denote by [v, v Õ ] the combinatorial geodesic joining v with v Õ . Lemma 18. Suppose that there is an edge of T whose stabilizer is properly contained in the stabilizers of its vertices. Then the stabilizer of every vertex of T is its own normalizer.

Proof. Since the action of G on T is transitive on the set of edges, the stabilizer of any edge is properly contained in the stabilizers of its vertices. Given a vertex v of T and an element x of G, we know that the stabilizer of x • v is xG v x ≠1 . Now, let v be a vertex of T and x an element of G that normalizes G v . We need to prove that x • v = v. Let us assume by contradiction that x • v " = v and let e be the first edge of the path

[v, x • v]. Then, the stabilizer G [v,x•v] of the path [v, x • v] is contained in G e . Since x normalizes G v ,
G v µ G [v,x•v] µ G e ( G v ,
which is a contradiction.

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = xG v 2 x ≠1
, and Ï(G e ) = xG e x ≠1 .

Lemma 20. Suppose that the stabilizer of any vertex of T is its own normalizer and let

Ï : G ae G be an element of Aut T (G). Then, Ï induces an isometry Ï : T ae T such that for all g in G

Ï(g • ) = Ï(g) • Ï( ). (1.3.1)
Moreover, if Ï is the inner automorphism given by the conjugation by x, then Ï is given by the action of x on T .

Remark 21. By Lemma 16, we get that the hypothesis of Lemma 20 on the vertex stabilizers implies that the center of G acts trivially on T .

Proof of Lemma 20. Let us fix an edge e = (v 1 , v 2 ). We warn the reader that the following construction of Ï depends on the choice of e. We write

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = xG v 2 x ≠1
, and Ï(G e ) = xG e x ≠1 .

We define an isometry Ï :

T ae T as follows. If v is a vertex of T we write v = g • v i for i = 1, 2 and define Ï(v) = Ï(g)x • v i . If f is an edge of T we write f = g • e and define Ï(f ) = Ï(g)x • e.
1.3. Extending actions on the Bass-Serre tree These definitions are independent of the choice of the element g. In the case of an HNN extension, we must check that the action on a vertex v is independent of whether we represent v as the image of v 1 or v 2 (as v 1 and v 2 are in the same orbit). But if v 2 = t • v 1 one checks that t ≠1

x ≠1 Ï(t)x normalizes G v 1 , and thus by the hypothesis of the lemma we get that t ≠1

x ≠1 Ï(t)x lies in G v 1 . Using this observation one proves that Ï(v) is well defined. In this way we obtain two bijections, one of the set of vertices of T and one of the set of edges of T , which define an isometry of T . If there is another element y in G such that

Ï(G v 1 ) = yG v 1 y ≠1 , Ï(G v 2 ) = yG v 2 y ≠1
, and Ï(G e ) = yG e y ≠1 then x ≠1 y is in the normalizer of G v 1 and G v 2 . Since G v 1 and G v 2 are their own normalizers we get that x ≠1 y is in the intersection of G v 1 and G v 2 , which is G e . This implies that Ï is well defined and it has the desired property

Ï(g • ) = Ï(g) • Ï( ).
Finally if Ï is the inner automorphism given by the conjugation by x we get that

Ï(g • e) = Ï(g)x • e = (xgx ≠1 )x • e = x • (g • e).
Similarly, for i = 1, 2 we obtain that Ï(g Proof. Let us fix an edge e = (v 1 , v 2 ). By Lemma 20, it su ces to show that for all Ï,  in Aut T (G) the isometries Ï and  defined as before satisfy:

• v i ) = x • (g • v i ),
Ï ≠1 = Ï ≠1 and Ï ¶ Â = Ï ¶ Â. Let Ï be an element in Aut T (G) such that Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = xG v 2 x ≠1 , and Ï(G e ) = xG e x ≠1 .
Then, if y = Ï ≠1 (x ≠1 ) we get that

Ï ≠1 (G v 1 ) = yG v 1 y ≠1 , Ï ≠1 (G v 2 ) = yG v 2 y ≠1
, and Ï ≠1 (G e ) = yG e y ≠1 .

Therefore the isometry Ï ≠1 is given by

Ï ≠1 (g • v i ) = Ï ≠1 (g)y • v i for i = 1, 2; Ï ≠1 (g • e) = Ï ≠1 (g)y • e.
A direct computation shows that Ï ≠1 = Ï ≠1 . Now, let Ï and  be two elements of Aut(G) such that

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = xG v 2 x ≠1
, and Ï(G e ) = xG e x ≠1 , and

Â(G v 1 ) = yG v 1 y ≠1 , Â(G v 2 ) = yG v 2 y ≠1
, and Â(G e ) = yG e y ≠1 .

Then, if z = Â(x)y we get that

 ¶ Ï(G v 1 ) = zG v 1 z ≠1 ,  ¶ Ï(G v 2 ) = zG v 2 z ≠1
, and  ¶ Ï(G e ) = zG e z ≠1 .

Hence the isometry  ¶ Ï is given by

 ¶ Ï(g • v i ) =  ¶ Ï(g)z • v i for i = 1, 2;  ¶ Ï(g • e) =  ¶ Ï(g)z • e.
Once again, a direct computation shows that  ¶ Ï =  ¶ Ï. 2 Note that the isometry Ï : T ae T defined in Lemma 20 depends on the selected edge e = (v 1 , v 2 ) when Ï is not an inner automorphism of G. Therefore, the action of Aut T (G) on T also depends on this edge.

Definition 24. We denote by T the subgroup of Aut(G) that preserves the conjugacy class of each vertex stabilizer and each edge stabilizer of the Bass-Serre tree T of G, i.e. an automorphism Ï of G is an element of T if for any edge e = (v 1 , v 2 ) there are elements

x, y, z in G such that Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = yG v 2 y ≠1
, and Ï(G e ) = zG e z ≠1 .

It is clear that Aut T (G) is always contained in T . In the following, we will see some situations where both groups almost coincide.

First examples of extended actions

1.3.2.a Malnormal condition

A subgroup H of a group G is called malnormal if xHx ≠1 fl H is trivial for all x in G \ H.
Theorem 25. Let G be a group that splits as an amalgamated product A ú C B and let T be its Bass-Serre tree. If A and B are malnormal subgroups of G and C is a non-trivial subgroup properly contained in A and in B, then the induced action of Inn(G) extends to a G-compatible action of T on T .

Proof. By Lemma 20 and Lemma 23 it su ces to show that T is contained in Aut T (G):

Let Ï be an automorphism of G contained in T and let x, y, z be elements in G such that

Ï(A) = xAx ≠1
, Ï(B) = yBy ≠1 , and Ï(C) = zCz ≠1 .

Then C is contained in z ≠1 xAx ≠1 z and z ≠1 yBy ≠1 z. By the malnormality of A and B in G, we get that z ≠1

x and z ≠1 y are elements of A and B respectively. Hence

Ï(A) = zAz ≠1 , Ï(B) = zBz ≠1
, and Ï(C) = zCz ≠1 .

Therefore T is contained in Aut T (G).

1.3.2.b Free product of two groups

Let G be a group that splits as a free product A ú B and let T be its Bass-Serre tree. In this case the group T is the subgroup of Aut(G) such that for all Ï in T there are elements

x, y in G such that Ï(A) = xAx ≠1 and Ï(B) = yBy ≠1 .

Lemma 26. Let x be an element of G such that A and xBx ≠1 generate G. Then there is an element a of A such that xBx ≠1 = aBa ≠1 .

Proof. The result is clear if x is an element of A or B. We claim that x cannot be of the form

x = b 1 a 1 b 2 • • • b n a n ,
with a i and b i non-trivial elements of A and B respectively for all i. Otherwise, since A and xBx ≠1 generate G, there would be elements -1 , . . . , -k+1 of A and elements -1 , . . . ,k of B such that

b 1 = -1 (x-1 x ≠1 )-2 • • • -k (x-k x ≠1 )-k+1 ,
withi " = 1 for i = 2, . . . , k andi " = 1 for i = 1, . . . , k, which is impossible since the element on the right side of the latter equation cannot be reduced to a word of length one.

Finally, if x = -y-for someoe A, -oe B and y oe G, we obtain that A and yBy ≠1 generate G. We deduce from this and the previous argument that x must be of the form If we denote by  the inner automorphism of G given by the conjugation by x ≠1 we get that

 ¶ Ï(A) = A and  ¶ Ï(B) = x ≠1 yBy ≠1
x. Since  ¶ Ï is an automorphism of G we have that A and x ≠1 yBy ≠1

x generate G. It follows from Lemma 26 that there is an element a in A such that x ≠1 yBy ≠1 x = aBa ≠1 . Hence,

Ï(A) = (xa)A(xa) ≠1 and Ï(B) = (xa)B(xa) ≠1 ,
and therefore T is contained in Aut T (G). 2 Notice that if A and B are indecomposable, not infinite cyclic and non isomorphic to each other, then T coincides with Aut(G) (see [65, p. 152]). As a consequence of Proposition 27, we obtain the following result: Theorem 28. Let G be a group that splits as a free product A ú B with A and B indecomposable, not infinite cyclic and non isomorphic to each other. Then, the induced action of Inn(G) on T extends to a G-compatible action of Aut(G) on T .

Remark 29. If G = A ú A, the same argument proves that Aut(G) contains a subgroup of index 2 which extends the action of G on its Bass-Serre tree. The group Aut(G) itself acts on the first barycentric division of the Bass-Serre tree of G.

Edge stabilizer conditions

In order to study a more general situation, we will suppose that G splits over a subgroup C and that the action of G on its Bass-Serre tree T satisfies the following conditions: B1 The stabilizer of an edge is non-trivial and properly contained in the stabilizers of its vertices. Moreover, the normalizer of an edge stabilizer is contained in the stabilizer of one of its vertices.

B2

The stabilizers of two edges e and e Õ with a common vertex v are either identical or their intersection is trivial. If they are identical there is an element g in the stabilizer of v such that e Õ = g • e. In particular g is in the normalizer of the stabilizer of e.

Let us fix an edge e = (v 1 , v 2 ) of T . By condition B1 we may assume that the normalizer of G e is contained in G v 1 . For any two vertices v and v Õ of T we will denote by d(v, v Õ ) the number of edges in the path [v, v Õ ]. Let G 0 be the kernel of the map

G ≠ae Z/2Z g ' ae Y _ ] _ [ 0 if d(v 1 , g • v 1 ) is even; 1 if d(v 1 , g • v 1 ) is odd.
Note that for every vertex v of T and for all g in G 0 , d(v, g • v) is even. If G splits as an amalgamated product then G = G 0 . If G splits as an HNN extension then G 0 is a subgroup of G of index 2 which defines a cohomology class in H 1 (G, Z/2Z). More precisely, if G splits as the HNN extension Aú C,◊ , we have that G 0 is the kernel of the map ÷ : G ae Z/2Z such that ÷(t) = 1 and ÷(a) = 0 for all a in A.

Lemma 30.

Let Ï : G ae G be an automorphism of G. Suppose that for i = 1, 2 there is an element

g i in G 0 such that Ï(G v i ) = g i G v i g ≠1 i . 1. If G e is contained in Ï(G v 1 ), then g 1 is an element of G v 1 . 2. If G e is contained in Ï(G v 2 ) ,then there is an element h in the normalizer of G e such that hg 2 is an element of G v 2 .
Proof.

1. If, to obtain a contradiction, we suppose that g 1 is not an element of

G v 1 , then the path [v 1 , g 1 • v 1 ] is non-trivial. Note that the stabilizer of g 1 • v 1 is precisely Ï(G v 1 ) = g 1 G v 1 g ≠1 1 . Therefore, since G e is contained in both G v 1 and Ï(G v 1 ), we get that G e stabilizes the path [v 1 , g 1 • v 1 ].
In particular, it stabilizes the first two edges of this path: e Õ = (v 1 , v Õ ) and e ÕÕ = (v Õ , v ÕÕ ). By condition B2, the stabilizers of e Õ and of e ÕÕ are equal to G e and there is a non-trivial element g in the stabilizer of v Õ such that g • e Õ = e ÕÕ . Using the fact that the stabilizers of e, e Õ and e ÕÕ are equal, we conclude that g is in the normalizer of G e and therefore in G v 1 . Hence g stabilizes e Õ which contradicts the fact that the path [v 1 , g 1 • v 1 ] is without backtracking.

2. Suppose that g 2 is not an element of G v 2 . Let e Õ and e ÕÕ be the first two edges of the path

[v 2 , g 2 • v 2 ].
As before, the stabilizers of e, e Õ and e ÕÕ are equal. The proof of the first part of this lemma implies that e = e Õ and that the path [v 2 , g 2 • v 2 ] can not contain more than two edges. Hence this path has exactly two edges and the vertex between v 2 and g 2 • v 2 is v 1 . Finally, condition B2 implies that there is an element h in the stabilizer of

v 1 such that h • (g 2 • v 2 ) = v 2 .

Corollary 31. The action of G on T is faithful and the center Z(G) of G is trivial.

Proof. Let x be an element of G that fixes every vertex of T and let g be an element of

G v 2 \ G e .
If we denote by Ï the inner automorphism of G given by the conjugation by g then we have that

Ï(G v 1 ) = gG v 1 g ≠1 , Ï(G v 2 ) = G v 2 ,
and Ï(G e ) = gG e g ≠1 .

Since x fixes every vertex of T , it fixes in particular the path [v 1 , g • v 1 ] which is the path whose edges are e and g • e. If we suppose that x is not the identity we get that G e fl G g•e is non-trivial and by condition B2 we get that G e = G g•e = gG e g ≠1 . Therefore G e is contained in Ï(G v 1 ) and by Lemma 30 g is an element of G v 1 . This would imply g is contained in G v 1 fl G v 2 = G e which contradicts the choice of g. Hence, we conclude that x is the identity. This implies that Z(G) is trivial since Lemma 16, Lemma 18 and condition B1 imply that Z(G) acts trivially on T . 2 In the following, we will denote by 0 T the subgroup of Aut(G) given by the following property: an automorphism Ï : G ae G is an element of 0 T if for any edge e = (v 1 , v 2 ) of T , there are elements x, y, z in the same left coset of G 0 in G such that

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = yG v 2 y ≠1
, and Ï(G e ) = zG e z ≠1 .

Note that if xG v 1 x ≠1 = aG v 1 a ≠1 for some a in G, by condition B1 and Lemma 18 we have that a ≠1

x is in G v 1 . Therefore, since G v 1 is contained in G 0 we get that xG 0 = aG 0 and thereby the left coset xG 0 is well defined. Similarly, we get that the left cosets yG 0 and zG 0 are well defined.

The main result of this section is the following:

Theorem 32. Let G be a group that splits over a subgroup C and let T be its Bass-Serre tree. If the action of G on T satisfies conditions B1 and B2, then the induced action of Inn(G) on T extends to a G-compatible action of 0 T on T . Proof. By Lemma 20 and Lemma 23 it su ces to show that 0 T is contained in Aut T (G): Let Ï be an automorphism of G contained in 0 T and let x, y, z be elements in the same left coset of G 0 in G such that

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = yG v 2 y ≠1
, and Ï(G e ) = zG e z ≠1 .

Then G e is contained in

z ≠1 xG v 1 x ≠1 z and z ≠1 yG v 2 y ≠1 z. Note that z ≠1
x and z ≠1 y are elements in G 0 . By the first part of Lemma 30 z ≠1

x is an element of G v 1 . By the second part of Lemma 30, there is an element h in the normalizer of G e (thereby in G v 1 ) such that

hz ≠1 y is in G v 2 .
Hence we get that

Ï(G v 1 ) = zh ≠1 G v 1 hz ≠1 , Ï(G v 2 ) = zh ≠1 G v 2 hz ≠1 , and Ï(G e ) = zh ≠1 G e hz ≠1 .
Therefore 0 T is contained in Aut T (G). 2
Lemma 33. The group 0 T is a subgroup of T of index at most 2. Since T = 0 T when G splits as an amalgamated product, it su ces to show that this is true when G splits as an HNN extension. With this aim we will suppose for the rest of this subsection that G splits as the amalgamated product G v 1 ú Ge,◊ where ◊ : G e ae G v 1 is the monomorphism given by the conjugation of an element t in G \ G v 1 . To prove Lemma 33 we will need the following result:

Lemma 34. If Ï is an automorphism of G contained in T then Ï(G 0 ) = G 0 .
Proof. Note that G 0 can be obtained as the kernel of the map ÷ : G ae Z/2Z such that ÷(t) = 1 and ÷(g) = 0 for all g in G v 1 . This follows from the fact that for all x in G the set of vertices of

T joined to x • v 1 is {xgt ' |g oe G v 1 and ' = ±1}.
Every element in G can be written as

g 1 t n 1 g 2 t n 2 • • • g k t n k g k+1 ,
where g i is in G v 1 for all i = 1, . . . , n + 1 and n i is a non-zero integer for all i = 1, . . . , n.

Hence, if Ï is an automorphism of G contained in T and x is an element of G such that Ï(G v 1 ) = xG v 1 x ≠1 , we get that for all i = 1, . . . , n + 1 there is an element

g Õ i in G v 1 such that Ï(g i ) = xg Õ i x ≠1 . Therefore, Ï(g 1 t n 1 g 2 t n 2 • • • g k t n k g k+1 ) = (xg Õ 1 x ≠1 Ï(t) n 1 • • • Ï(t) n k xg Õ k+1 x ≠1 ).
Hence we get that ÷ ¶ Ï(g

1 t n 1 g 2 t n 2 • • • g k t n k g k+1 ) = ÷(Ï(t) n 1 +•••+n k ). This implies that ÷(Ï(t)) = 1, otherwise Ï(G) would be contained in G 0 which is not possible since Ï is an automorphism of G and G 0 is a subgroup of index 2. We conclude that ÷ = ÷ ¶ Ï and thereby Ï(G 0 ) is contained in G 0 . The same argument shows that Ï ≠1 (G 0 ) is contained in G 0 which implies the result. 2
Proof of Lemma 33. Let Ï be an automorphism of G contained in T and let x, y, z be elements in G such that

Ï(G v 1 ) = xG v 1 x ≠1 , Ï(G v 2 ) = yG v 2 y ≠1
, and Ï(G e ) = zG e z ≠1 .

(1.

3.2)

First of all notice that the left cosets xG 0 and yG 0 are equal: since

G v 2 = tG v 1 t ≠1 then Ï(G v 2 ) = Ï(t)Ï(G v 1 )Ï(t ≠1 ) = Ï(t)xt ≠1 G v 2 tx ≠1 Ï(t ≠1 ).
By Lemma 34 and its proof, t and Ï(t) are not contained in G 0 . This implies that x is in

G 0 if and only if Ï(t)xt ≠1 is in G 0 , if
T ≠ae Z/2Z Ï ' ae Y _ ] _ [ 0 if z ≠1 x oe G 0 ; 1 if z ≠1 x / oe G 0 .
Let us see thatis a homomorphism:

• It is well defined: let Ï be a automorphism of G as in (1.3.2). If there are elements a, c in G such that

Ï(G v 1 ) = aG v 1 a ≠1 and Ï(G e ) = cG e c ≠1 ,
then by Lemma 18, a ≠1

x is in G v 1 µ G 0 and by condition B1 c ≠1 z is in the normalizer of G e which is contained in G v 1 µ G 0 . Hence xG 0 = aG 0 and zG 0 = cG 0 which implies that z ≠1 xG 0 = c ≠1 aG 0 . Thereforeis well defined.

• It is an homomorphism: suppose that Ï and  are automorphisms of G contained in T and let x, z, a, c be elements in G such that

Ï(G v 1 ) = xG v 1 x ≠1 and Ï(G e ) = zG e z ≠1 , Â(G v 1 ) = aG v 1 a ≠1 and Â(G e ) = cG e c ≠1 . Then we get that  ¶ Ï(G v 1 ) = Â(x)aG v 1 a ≠1 Â(x ≠1 ) and  ¶ Ï(G e ) = Â(z)cG e c ≠1 Â(z ≠1 ). Note that c ≠1 Â(z ≠1 x)a = c ≠1 (Â(z ≠1 x))c(c ≠1 a) and therefore by Lemma 34, c ≠1 Â(z ≠1 x)a is in G 0 if and only if (z ≠1 x)(c ≠1 a) is in G 0 which implies that -( ¶ Ï) = -(Â)-(Ï).
Finally we conclude that the kernel ofis a subgroup of T of index at most 2 which is contained in 0 T . 2

Surface groups

Let S be a closed surface of genus g Ø 2 and let " be a simple closed curve in S. We know from Section 1.2.3 that the fundamental group of S splits over the cyclic subgroup C generated by the homotopy class of ". In the following, we will omit the base point of the fundamental group of S to simplify notation.

Remark 35. The group of automorphisms of fi 1 (S) that preserves the conjugacy class of

" is a subgroup of index at most 2 of the subgroup of fi 1 (S) made of the automorphisms

Ï : fi 1 (S) ae fi 1 (S) such that Ï(C) = xCx ≠1 (1.3.3)
for some x in fi 1 (S). This follows from the fact that if an automorphism Ï satisfies (1.

3.3) then Ï(["]) must be equal to either x["]x ≠1 or to x["] ≠1

x ≠1 .

Lemma 36.

Let " be a simple closed curve in S and let T be the Bass-Serre tree of fi 1 (S) associated to ". The group of automorphisms of fi 1 (S) that preserves the conjugacy class of C contains Aut T (fi 1 (S)) as a subgroup of index at most 2. Hence T contains Aut T (fi 1 (S)) as a subgroup of index at most 2.

Proof. Let be the subgroup of Aut(fi 1 (S)) that preserves the conjugacy class of C. We have the inclusions:

Aut T (fi 1 (S)) µ T µ .
Hence the second point of the lemma follows from the first one. By the Dehn-Nielsen-Baer Theorem (see [START_REF] Farb | A primer on mapping class groups[END_REF]) we have an isomorphism between the (extended) mapping class group of S and the outer automorphism group of fi 1 (S) MCG ± (S) = Di ± (S)/Di 0 (S) ae Out(fi 1 (S)) = Aut(fi 1 (S))/Inn(fi 1 (S)), which takes the class of a di eomorphism f : S ae S to the automorphism induced by f on fi 1 (S) (well-defined up to conjugacy). Let us fix an automorphism Ï of fi 1 (S) contained in and let f : S ae S be a di eomorphism inducing Ï.

We fix a base point x 0 on " and a collar neighborhood U " of ". Since the curve f ¶ " is isotopic to either " or ", one can replace f by a di eomorphism f 1 isotopic to f such that f 1 preserves " globally and fixes x 0 . Let Di (S, ") be the group of di eomorphisms of S preserving " globally, preserving each connected component of U " \ " and fixing x 0 . Then, Chapter 1. Kähler groups acting on trees a di eomorphism contained in Di (S, ") induces an automorphism of fi 1 (S) that preserves the subgroup C and the fundamental groups of the connected components of S \ ". In particular, this induced automorphism lies in Aut T (fi 1 (S)).

The elements of which are induced by di eomorphisms of Di (S, ") form a subgroup 0 of index at most 2. Hence if Ï oe 0 we obtain that Ï = f ú ¶ I for some I oe Inn(fi 1 (S)) and for some f oe Di (S, "). Therefore Ï oe Aut T (fi 1 (S)).

2 Since S is a closed surface of genus g Ø 2, given a non-nullhomotopic simple closed curve " in S, we get that the connected components obtained after cutting S along " are surfaces of positive genus with boundary components. Therefore, the fundamental group of such surfaces are free groups of rank greater or equal than 2, containing the cyclic subgroup generated by the homotopy class of ". Hence, by Lemmas 16 and 18 we get that the stabilizer of any vertex of T is its own normalizer. Hence, we obtain the following result as a consequence of Lemmas 23 and 36.

Proposition 37. The action of fi 1 (S) on T extends to a subgroup of index at most 2 of the group of automorphisms of fi 1 (S) that preserves the conjugacy class of ". Moreover, this extension is fi 1 (S)-compatible.

Maximal families of edge stabilizers

Definition 38. Let F be a family of subgroups of G. We say that F is maximal if for any

2 subgroups H, K in F , K < H implies K = H.
Let G be a group acting on a tree T and let v be a vertex of T . We say that an edge of T is v-incident if one of its vertices is v. We will be interested in the following conditions: C1 The stabilizer of an edge of T is properly contained in the stabilizers of its vertices.

C2

For any vertex v in T the family of stabilizers of v-incident edges is maximal.

Remark 39. Recall that condition C1 implies that the stabilizer of any vertex is its own normalizer and therefore the center of G acts trivially on T (see Remark 21).

Here we will see that if a group G splits as an amalgamated product or an HNN extension and the action of G on its Bass-Serre tree satisfies conditions C1 and C2, then the induced action of Inn(G) on T extends to a larger group of automorphisms of G. The case of an amalgamated product is due to Karras, Pietrowski and Solitar (see [START_REF] Karrass | Automorphisms of a free product with an amalgamated subgroup, Contributions to group theory[END_REF]) and the case of an HNN extension is due to Pettet (see [START_REF] Pettet | The automorphism group of a graph product of groups[END_REF]). These results generalize the results of Sections 1.3.2 and 1.3.3.

Theorem 40 (Karrass,Pietrowski,Solitar,[START_REF] Karrass | Automorphisms of a free product with an amalgamated subgroup, Contributions to group theory[END_REF]). Let G be a group that splits as an amalgamated product A ú C B and let T be its Bass-Serre tree. If the action of G on T satisfies conditions C1 and C2, then the induced action of Inn(G) on T extends to a G-compatible action of T on T .

Theorem 41 (Pettet,[START_REF] Pettet | The automorphism group of a graph product of groups[END_REF]). Let G be a group that splits as an HNN extension Aú C,◊ and let T be its Bass-Serre tree. If the action of G on T satisfies conditions C1 and C2, then the induced action of Inn(G) on T extends to a subgroup of T of index at most 2. Moreover, this extension is G-compatible.

Extending actions on the Bass-Serre tree

Remark 42. The original statements of Theorems 40 and 41 only give information about the decomposition of T as an amalgamated product in the case of Theorem 40, and as an HNN extension in the case of Theorem 41, but we can deduce from the proofs of these results that such decompositions are induced by an action of T on T that extends the one of G on T .

In [START_REF] Pettet | The automorphism group of a graph product of groups[END_REF], Pettet gives a more general result than the one stated above. Pettet studied groups that split as the fundamental group of a graph of groups with a more restrictive condition on the edge stabilizers of its Bass-Serre tree (edge group incomparability hypothesis), which coincides with the maximality of the families of v-incident edge stabilizers when G splits as an HNN extension. In [START_REF] Gilbert | Tree actions of automorphism groups[END_REF], Gilbert, Howie, Metaftsis and Raptis studied the particular situation when the edge stabilizers are cyclic. In particular they proved the following result on Baumslag-Solitar groups:

Theorem 43 (Gilbert, Howie, Metaftsis, Raptis). Let G be the Baumslag-Solitar group

Èx, t|tx p t ≠1 = x q Í,
where p, q are integers with p, q > 1 and such that neither one is a multiple of the other.

Then the induced action of Inn(G) on T extends to a G-compatible action of Aut(G) on T .

Remark 44.

If G splits as a free product A ú B, then for any vertex v in its Bass-Serre tree T , the family of stabilizers of v-incident edges is maximal. Hence, one can recover Theorem 28 as a consequence of Theorem 40.

One-ended hyperbolic groups

Interesting examples of groups with a non-trivial splitting are given by one-ended hyperbolic groups whose group of outer automorphisms is infinite. In Section 1.3.4 we saw that a surface group splits over a cyclic subgroup. The last assertion holds as well for a group which is virtually a surface group. When the group is a one-ended hyperbolic group which is not virtually a surface group, the theory of JSJ decompositions guarantees the existence of a non-trivial splitting over a virtually cyclic subgroup under the assumption that its group of outer automorphisms is infinite (see for instance Theorem 1.4 in [START_REF] Levitt | Automorphisms of hyperbolic groups and graphs of groups[END_REF]). The notion of JSJ decomposition was introduced in [START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF] by Sela for one-ended hyperbolic groups without torsion. In [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF], Bowditch constructed a slightly di erent JSJ decomposition for one-ended hyperbolic groups (not necessarily torsion-free) by studying cut points on the boundary of the group. Both decompositions are "canonical" and they coincide when the group is torsion-free. In the rest of this section, G will denote a one-ended torsion-free hyperbolic group.

Theorem 45 (Sela, Bowditch). Suppose that Out(G) is an infinite group and that G is not virtually a surface group. Then there is a non-trivial splitting of G, given by a minimal action of G on a tree T without inversions and finite quotient T /G. Moreover, the edge stabilizers are all cyclic subgroups and the set of vertex stabilizers is Aut(G)-invariant. In particular, there is a finite index subgroup of Aut(G) that preserves this splitting i.e. that preserves the conjugacy class of each vertex stabilizer and each edge stabilizer.

This splitting of G is called the JSJ splitting. The adjacency relation on the set of vertices induces a well-defined binary relation on the set of vertex stabilizers. The JSJ splitting is called canonical since Aut(G) preserves the set of vertex stabilizers with the Chapter 1. Kähler groups acting on trees induced binary relation. In fact, by considering the action of G on ˆG, Bowditch proved in [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF] that the extended action of Aut(G) on T is G-compatible. We refer to [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF] and [START_REF] Levitt | Automorphisms of hyperbolic groups and graphs of groups[END_REF] for a detailed description of this splitting. As in the case of a group that splits over a subgroup C, we obtain that the action of G on T is minimal and without fixed points on the boundary.

Kähler extensions and actions on trees

In this section we will prove our main results.

Applying Gromov and Schoen's Theorem

Let be a Kähler group that fits in a short exact sequence

1 / / G / / P / / Q / / 1, (1.4.1)
where G is a finitely generated group with trivial center acting on a tree T . Suppose that T is not isomorphic to a line nor a point and that the action of G on T is minimal, faithful and without fixed points on the boundary. Note that since the center of G is trivial, the action of G on T coincides with the induced action of Inn(G) on T . Let be the image of the morphism ae Aut(G) induced by the conjugation action of on G. The main result of this section is the following: Theorem A. Suppose that there is a finite index subgroup 0 of containing Inn(G) such that the action of G on T can be extended to a G-compatible action of 0 on T . Then G is virtually a surface group. Moreover, there is a finite index subgroup 1 of containing G such that the restricted short exact sequence

1 / / G / / 1 P / / P ( 1 ) / / 1
splits as a direct product.

To prove Theorem A we will need a result of Gromov and Schoen [START_REF] Gromov | Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one[END_REF] on Kähler groups acting on trees (see also Sun [71]). For the reader's convenience we recall some properties of holomorphic maps between compact complex manifolds and closed Riemann surfaces. A surjective holomorphic map with connected fibers f : X ae S from a compact complex manifold X to a closed Riemann surface S induces an orbifold structure as follows. For every point p in S, let m(p) be the greatest common divisor of the multiplicities of the irreducible components of the fiber f ≠1 (p) and let be the set of points in S such that m(p) Ø 2. Note that is finite since it is contained in the set of critical values of f . Hence, if = {p 1 , . . . , p k }, by assigning the multiplicity m i = m(p i ) to each p i , the orbifold induced by f is given by S endowed with the set of marked points {(p i , m i ) | i = 1, . . . , k}, which is denoted by

= {S; (p 1 , m 1 ), . . . , (p k , m k )}.
For all i = 1, . . . , k let " i be a loop around p i given by the boundary of a small enough disk such that p i is the only singular value contained in the disk. If we denote by c i the homotopy class of " i in fi 1 (S \ ), the fundamental group of the orbifold is given by

fi orb 1 ( ) = fi 1 (S \ )/ π c m 1 1 , . . . , c m k k ∫,
is bijective. The injectivity follows from the fact that the restriction of ⁄ ≠1 ¶ to G is the identity. For the surjectivity, if (x 0 , q 0 ) is an element of G ◊ P ( 1 ), by taking y in 1 such that P (y) = q 0 and x in G such that ⁄(x) = (y), we get that ÷(x 0 x ≠1 y) = (x 0 , q 0 ). 2

Remark 48. In the latter proof, we used the fact that if the action of G on T is minimal without fixed points on the boundary, then any extension of this action has these properties as well.

Applications

Here we will apply Theorem A and the results of Section 1.3 to prove Theorems B, C, D and E. For this, we need to check that the action with which we start with is minimal, faithful and without fixed points on the boundary. One can verify from the discussion in Section 1.2.2, that for a surface group (see also Section 1.2.3), a free product of two groups and a Baumslag-Solitar group as in Theorem D, the actions on their respective Bass-Serre tree are minimal and without fixed points on the boundary of the tree. For one-ended hyperbolic groups which are not virtually a surface group, this is a consequence of the theory of JSJ decompositions (see [START_REF] Bowditch | Cut points and canonical splittings of hyperbolic groups[END_REF][START_REF] Sela | Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II[END_REF]). Hence, we only need to show faithfulness in all cases.

Proposition 49. Let S be a closed surface of genus g Ø 2 and let C be the cyclic subgroup of fi 1 (S) generated by the homotopy class of a simple closed curve ". Then the action of fi 1 (S) on the Bass-Serre tree associated to the splitting of fi 1 (S) over C is faithful.

Proof. Let T be the Bass-Serre tree associated to the splitting of fi 1 (S) over C. Recall that T coincides with the dual tree associated to " (see Section 1.2.4). This identification is given by a bijection between the edges of T and a set of disjoint bi-infinite geodesics in

H 2 .
Hence, an edge stabilizer of T coincides with the stabilizer of a bi-infinite geodesic in H 2 under the action of fi 1 (S) on H 2 . From this, we conclude that the intersection of the stabilizers of two di erent edges of T is trivial, which implies the faithfulness of the action of G on T . 2

Proof of Theorem B. Let " be the simple closed curve in S, whose conjugacy class is preserved by the conjugation action of on fi 1 (S). Let C be the cyclic subgroup of fi 1 (S) generated by the homotopy class of " and let T be the Bass-Serre tree associated to the splitting of fi 1 (S) over C. Therefore, the image of ae Aut(fi 1 (S)) denoted by is contained in the subgroup of automorphisms of fi 1 (S) that preserves the conjugacy class of ". By Proposition 37, the action of fi 1 (S) on T can be extended to a fi 1 (S)-compatible action of a subgroup of of index at most 2 on T . By Proposition 49 we have that the action of fi 1 (S) on T is faithful and the result follows from Theorem A. 2

Remark 50. Let G be a group that splits over a subgroup C and let T be its Bass-Serre tree. Suppose that for an edge of T , its stabilizer is properly contained in its vertex stabilizers.

Suppose as well that for any vertex v of T , the family of stabilizers of v-incident edges is maximal. Let be a Kähler group containing G as a normal subgroup such that the conjugation action of on G preserves the conjugacy classes of edge stabilizers and vertex stabilizers of T . If the action of G on its Bass-Serre tree is faithful, then as a consequence of Theorems 40, 41 and A we have that G must be virtually a surface group.

Theorem C follows from Theorems 28 and A, and the following proposition.

Proposition 51. If a surface group is embedded in a free product AúB, then it is embedded in A or it is embedded in B (up to conjugacy). In particular, a non-trivial free product of two groups is not virtually a surface group.

Proof. First, we will prove that a non-trivial free product is not isomorphic to a surface group. Let us assume by contradiction that the free product C úD is isomorphic to a surface group. Recall that an infinite index subgroup of a surface group is a free group. Hence, since C and D are both infinite index subgroups of C ú D, we obtain that they are both free groups. Thus, the free product C ú D is also a free group, which is a contradiction. Note that the same argument shows that a surface group is not isomorphic to a free product with more than two free factors. The general case follows from the Kurosh subgroup Theorem (see [65, p. 151]), which states that a subgroup of the free product A ú B is given by the free product of a free group with subgroups of conjugates of A and B. Hence, if a surface group embeds in A ú B, by the Kurosh subgroup Theorem and the previous argument it embeds as a subgroup of a conjugate of A or as a subgroup of a conjugate of B.

2 Theorem D follows from Theorem 43 and A, and the following two propositions:

Proposition 52. The Baumslag-Solitar group G(p, q) = Èx, t|tx p t ≠1 = x q Í
is not virtually a surface group.

Proof. If |p| " = |q| and if we suppose that G(p, q) is a surface group, we would have that G(p, q) acts on H 2 . Hence, if we denote by ¸the translation length of x in H 2 , as a consequence of the relation tx p t ≠1 = x q , we would have that |p|¸= |q|¸which is impossible. If H is a finite index subgroup of G(p, q), there exists an integer k such that x k and t k are contained in H. From the unique relation of the Baumslag-Solitar group, we deduce that tx np t ≠1 = x nq for every integer n. Then, an argument by induction shows that

t k x k•p k t ≠k = x k•q k .
Hence, if we suppose that H is a surface group and we denote by ¸the translation length of x k in H 2 , we would have that |p k |¸= |q k |¸, which is impossible.

If |p| = |q|, the subgroup generated by x p is normal in G(p, q). Hence its intersection with any finite index subgroup H of G(p, q) is normal in H. This cannot happen if H is a surface group since such a group does not admit a non-trivial Abelian normal subgroup. 2 Proposition 53. The Baumslag-Solitar group

G(p, q) = Èx, t|tx p t ≠1 = x q Í,
where p, q are di erent integers with p, q > 1, acts faithfully on its Bass-Serre tree.

Proof. The kernel N of the action of G(p, q) on its Bass-Serre tree is contained in every edge stabilizer. In particular, it is contained in Èx p Í. Assume by contradiction that N is non-trivial. Then, N is generated by x kp for some nonzero integer k. Since N is a normal subgroup of G(p, q), we get that tx kp t ≠1 = x kq is an element of Èx kp Í. Therefore kq = mkp and p divides q. By symmetry, we obtain that q divides p and thereby p = q, which is a contradiction.

2 Finally, Theorem E follows from Theorems 45 and A, and the fact that the action of a one-ended hyperbolic group with infinite group of outer automorphisms acts faithfully on its JSJ tree. The latter assertion follows from a similar argument to the one of Proposition 49 (in this case the intersection of the stabilizers of two di erent edges without common vertices is trivial).

Lemma 56 (Birman-Lubotzky-McCarthy). Let A be an Abelian subgroup of the mapping class group of a closed oriented surface of genus g Ø 2 generated by reducible elements

{• 1 , . . . , • k }. Then, there is a non-empty canonical admissible set A such that • i (A ) = A for all i = 1, . . . , k.
The second lemma is the following. It will be a consequence of Thurston's hyperbolization Theorem (see [START_REF] Otal | Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3[END_REF]), together with a classical result of Carlson and Toledo [START_REF] Carlson | Harmonic mappings of Kähler manifolds to locally symmetric spaces[END_REF]. Lemma 57. Let S be a closed surface of genus g Ø 2, be a Kähler group and k be a positive integer such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Z k / / 1.
Then the image of the monodromy map  : Z k ae MCG(S) does not contain pseudo-Anosov elements.

Proof. Let us assume by contradiction that the image of  contains a pseudo-Anosov element • . It is well known that the cyclic subgroup generated by a pseudo-Anosov element is a finite index subgroup of its centralizer (see [START_REF] Mccarthy | Normalizers and centralizers of pseudo-Anosov mapping classes[END_REF]). Hence, by passing to a finite index subgroup of if necessary (which is also a Kähler group), we may assume that the image of  is the cyclic subgroup generated by • . Let {e 1 , . . . , e k } be a basis of Z k such that {e 1 , . . . , e k≠1 } generates the kernel of  and Â(e k ) = • . We denote by 0 the subgroup of generated by fi 1 (S) and P ≠1 (e k ).

Claim 58. is isomorphic to 0 ◊ Z k≠1 .
Claim 58 implies the result since by Thurston's hyperbolization Theorem (see [START_REF] Otal | Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3[END_REF]), 0 is a cocompact lattice in the group of orientation preserving isometries of the hyperbolic 3-space and by Carlson and Toledo's Theorem (see [START_REF] Carlson | Harmonic mappings of Kähler manifolds to locally symmetric spaces[END_REF]), the projection of onto the cocompact lattice 0 factors through a surface group . Hence, there is a commutative diagram

ƒ 0 ◊ Z k P 1 / / ◊ % % 0 ? ?
where P 1 is the projection onto 0 . This leads to the desired contradiction, since ◊( 0 ) is a subgroup of isomorphic to 0 (as a consequence of the fact that P 1 0 is the identity map), and every subgroup of is the fundamental group of a (closed or open) surface, but 0 is neither free nor isomorphic to the fundamental group of a closed oriented surface since

H 3 ( 0 , Z) is non-trivial. 2
Proof of Claim 58. Observe that for each e i in the kernel of Â, there is a unique element g i in such that g i centralizes fi 1 (S) and P (g i ) = e i . Since the commutator [g i , g j ] is an element of fi 1 (S) which centralizes fi 1 (S) we get that the group K generated by {g 1 , . . . , g k≠1 } is isomorphic to Z k≠1 and by construction [K, fi 1 (S)] is trivial. Using the fact that fi 1 (S) is a normal subgroup of and that K centralizes fi 1 (S) we obtain that [K, ] centralizes fi 1 (S). Finally, since [K, ] is in the kernel of P (which is fi 1 (S)) we conclude that [K, ] is in the center of fi 1 (S) which is trivial. Therefore the subgroups K and 0 commute and the result follows from observing that = K 0 and K fl 0 is trivial.

2

Now, we give our proof of Corollary 54: Proof of Corollary 54. By Nielsen-Thurston's classification Theorem and Lemma 57, up to passing to a finite index subgroup of Z k we have that the image of the monodromy map  : Z k ae MCG(S) is generated by reducible elements. Hence, by Lemma 56 there is a non-empty canonical admissible set A of S (S) such that Â(Z k ) preserves A . Since a finite index subgroup of Z k (which is isomorphic to Z k ) acts trivially on A , we can apply Theorem B which concludes the proof. 2

Kähler extension by a surface group whose monodromy map has solvable image

Corollary 59. Let S be a closed surface of genus g Ø 2 and be a Kähler group such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Q / / 1.
If the image of the monodromy map  : Q ae MCG(S) is solvable, then there are finite index subgroups 1 of and Q 1 of Q such that fi 1 (S) is contained in 1 and the restricted short exact sequence

1 / / fi 1 (S) / / 1 / / Q 1 / / 1
splits as a direct product.

According to a result of Birman, Lubotzky and McCarthy (see [9, Theorem B]), every solvable subgroup of the mapping class group of a closed oriented surface of genus g Ø 2 is virtually Abelian. Hence, to prove Corollary 59 it su ces to prove a result analogous to Lemma 57 for the case when the monodromy map  : Q ae MCG(S) has Abelian image. Lemma 60. Let S be a closed surface of genus g Ø 2 and be a Kähler group such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Q / / 1.
If the image of the monodromy map  : Q ae MCG(S) is Abelian, then it does not contain pseudo-Anosov elements.

Proof. The proof is by contradiction. Let us suppose that the image of  contains a pseudo-Anosov element • . As in Lemma 57, up to passing to a finite index subgroup of we may assume that the image of  is the cyclic subgroup generated by • . Let t be an element in such that  ¶ P (t) = • and let 0 be the subgroup of generated by fi 1 (S) and t. The result will follow from exhibiting a homomorphism ae 0 whose restriction to 0 is the identity, since Thurston's hyperbolization Theorem and Carlson and Toledo's Theorem will lead us to a contradiction as in Lemma 57. Now, to construct this homomorphism, observe that since each element in the kernel of  has a unique lift in that centralizes fi 1 (S), there is a subgroup K of isomorphic to the kernel of  that centralizes fi 1 (S). Indeed, K is the centralizer of fi 1 (S) in . Hence, the subgroup [ , K] centralizes fi 1 (S) and since its image under P is contained in the kernel of Â, we get that [ , K] is contained in K and therefore K is a normal subgroup of . Finally, 1.5. Surface groups and Kähler groups since the kernel of  ¶ P is isomorphic to K ◊ fi 1 (S), we obtain that is isomorphic to the semi-direct product (K ◊ fi 1 (S)) o ÈtÍ, and thereby there is a well-defined homomorphism

ae 0 kxt m ' ae xt m ,
where k is in K, x is in fi 1 (S) and m is in Z. To verify that it is a well-defined map we just need to observe that for k 1 x 1 t m 1 and k 2 x 2 t m 2 in we have that

k 1 x 1 t m 1 k 2 x 2 t m 2 = (k 1 t m 1 k 2 t ≠m 1 )(x 1 t m 1 x 2 t ≠m 1 )t m 1 +m 2 ,
where we notice that

t m 1 k 2 t ≠m 1 is in K and t m 1 x 2 t ≠m 1 is in fi 1 (S). 2
The following result is a consequence of a theorem due to Ivanov which states the following. Let S be a closed oriented surface and be an infinite subgroup of MCG(S) that does not preserve any admissible set A of S (S). Then, is either virtually a cyclic group generated by a pseudo-Anosov element, or it contains a free group generated by 2 pseudo-Anosov elements (see [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF]Theorem 2]).

Corollary 61.

Let S be a closed surface of genus g Ø 2 and a a Kähler group such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Q / / 1.
If the image of the monodromy map  : Q ae MCG(S) is infinite, then it contains a free subgroup generated by 2 pseudo-Anosov elements.

Proof. Recall that an extension as above is virtually trivial if and only if the monodromy subgroup of the short exact sequence is finite. Therefore, by Theorem B, Â(Q) does not preserve any admissible set A of S (S). Finally, the result follows from Lemma 60 and Ivanov's result recalled before the statement. 2

More restrictions on Kähler extensions by a surface group

A classical result due to Scott (see [START_REF] Scott | Subgroups of surface groups are almost geometric[END_REF]) allows us to extend Theorem B to any closed curve in S.

Theorem 62 (Scott). Let S be a topological surface and let x be an element of fi 1 (S).

Then, there is a finite covering map S Õ ae S such that x is in fi 1 (S Õ ) and can be represented by a simple closed curve in S Õ .

Lemma 63. Let G be a group and H be a finitely generated normal subgroup of G. If is a finite index subgroup of H, then the normalizer of in G is a finite index subgroup of G.

Proof. Let n be the index of in H. Then G acts on the set of subgroups of H of index n by conjugation. This set is finite and therefore the stabilizer of any of these subgroups gives a finite index subgroup of G. Finally, notice that the stabilizer of is precisely the normalizer of in G and the result follows. 2

Theorem 64. Let be a Kähler group such that there is a short exact sequence

1 / / fi 1 (S) / / P / / Q / / 1,

Chapter 2

Irrational pencils and Betti numbers

We study irrational pencils with isolated critical points on compact aspherical complex manifolds. We prove that if the set of critical points is nonempty, the homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated. This generalizes a result by Dimca, Papadima and Suciu. By considering self-products of the Cartwright-Steger surface, this allows us to build new examples of smooth projective varieties whose fundamental group has a non-finitely generated homology.

Exotic finiteness properties and irrational pencils 2.1.1 Some history

In Section 2.2 we will define the topological finiteness property F n and the homological finiteness property FP n . As we saw in the introduction, Dimca, Papadima and Suciu constructed the first examples of Kähler groups satisfying an exotic finiteness property, i.e. Kähler groups that are of type F n≠1 but not of type FP n . Let us introduce some notations before recalling their results. Notations. Let X be a (connected) compact complex manifold of complex dimension n Ø 2 and S a closed Riemann surface of positive genus. Recall that a surjective holomorphic map with connected fibers f : X ae S is called an irrational pencil. For such a map, we will always denote by the kernel of the induced homomorphism f ú : fi 1 (X) ae fi 1 (S).

The results of Dimca, Papadima and Suciu [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] are the following:

Theorem 65 (Dimca, Papadima and Suciu). If n Ø 3 and if f : X ae S is an irrational pencil with isolated critical points, then the fundamental group of a smooth fiber of f embeds into that of X and coincides with the kernel of the induced homomorphism f ú : fi 1 (X) ae fi 1 (S). Theorem 66 (Dimca, Papadima and Suciu).

Let X = 1 ◊ • • • ◊ n be a direct product
of n Riemann surfaces of genus greater than 1 and let S have genus 1. If f : X ae S is an irrational pencil with isolated critical points then the group H n ( , Q) has infinite dimension.

Lemma 67. Let X and Y be compact complex manifolds and let S be a closed Riemann surface of genus 1. If f : X ◊ Y ae S is a holomorphic map, then there exist holomorphic maps f 1 : X ae S and f 2 : Y ae S such that

f (x, y) = f 1 (x) + f 2 (y).
Definition 81. The canonical augmentation map ' : C 0 ( ' X) ae Z is the map of ZG-modules that sends every 0-cell of ' X to 1.

For each positive integer n, there exists a boundary map C n ( ' X) ae C n≠1 ( ' X) which is a map of ZG-modules. With the canonical augmentation map, this defines a chain complex of ZG-modules

• • • ae C n ( ' X) ae C n≠1 ( ' X) ae • • • ae C 0 ( ' X) ae Z ae 0.
This sequence is called the augmented cellular chain complex of the universal cover of X. We refer to [40, p. 137-139] for a precise definition of the boundary maps

C n ( ' X) ae C n≠1 ( ' X).
Proposition 82. Let G be a group and let X be a classifying space of G. Then the augmented cellular chain complex of the universal covering space of X is a free resolution of Z over ZG.

By the latter result, one can compute the homology groups of G by using the augmented cellular chain complex of the universal covering space of X. Moreover, we have the following result (see Chapter 2 of [START_REF] Brown | Cohomology of groups[END_REF]).

Proposition 83. Let G be group. If X is a classifying space of G which is a CW-complex, then H ú (G) ƒ H ú (X).
We have an equivalent result of the latter proposition for homology groups with coecients in Q.

Finiteness properties

A group G is finitely generated if there exists a free group F of finite rank and a surjective homomorphism F ae G. Equivalently, G is finitely generated if there exists a finite subset S of G such that the smallest subgroup of G containing S is equal to G. If in addition there exists a finite subset R of F whose normal closure (i.e. the smallest normal subgroup of F containing R) is the kernel of the map F ae G, we say that G is finitely presented. These are the classical finiteness properties in group theory. Here we present more finiteness properties of groups.

2.2.3.a Homological

Definition 84. A group G is of type 1. FP, if there exists a projective resolution of Z over ZG such that P i is finitely generated for all i Ø 0 and trivial for all but finitely many i Ø 0.

2. FP n , if there exists a projective resolution of Z over ZG such that P i is finitely generated for all i AE n.

3. FP OE , if there exists a projective resolution of Z over ZG such that P i is finitely generated for all i Ø 0.

As a consequence of a generalization of Schanuel's Lemma (see Chapter V III Section 4 of [START_REF] Brown | Cohomology of groups[END_REF]), we have the following result. Proposition 85. A group G is of type FP n if and only if there exists a free resolution of

Z over ZG • • • ae F i ae • • • ae F 1 ae F 0 ae Z ae 0,
such that F i is of finite rank for all i AE n.

2.2.3.b Topological

Definition 86. A group G is of type 1. F , if there exists a classifying space for G which is a finite CW-complex.

2. F n , if there exists a classifying space for G which is a CW-complex with finite nskeleton.

3. F OE , if there exists a classifying space or G which is a CW-complex of finite type, (i.e., having a finite number of cells of each dimension).

The topological finiteness condition F n and F OE were introduced by Wall [START_REF] Wall | Finiteness conditions for CW-complexes[END_REF]. An important consequence of Proposition 82, is that the topological finiteness condition F n (resp. F OE ) implies the homological finiteness condition FP n (resp. FP OE ). One can verify this as follows: let G be a group of type F n and let X be a classifying space of G with finite n-skeleton. Then, by considering the augmented cellular chain complex of the universal covering space of X, we obtain a free resolution of Z over ZG. Since C i ( ' X) is a free ZGmodule of finite rank for all i AE n (it is generated by the i-cells of X), we conclude that G is of type FP n .

Finally, observe that if G is of type FP n , by Proposition 85, there is a free resolution of

Z over ZG • • • ae F i ae • • • ae F 1 ae F 0 ae Z ae 0,
such that F i is of finite rank for all i AE n. For such a resolution we obtain that (F i ) G is a finitely generated free Abelian group for all i AE n (see Remark 75). Therefore, the kernel of (F i ) G ae (F i≠1 ) G is finitely generated. This implies that H i (G) is finitely generated for all i AE n. We can sum up the above discussion in the following result.

Proposition 87. Let G be a group. Given a group G one can ask whether it satisfies a topological or a homological finiteness property. We refer to [START_REF] Bestvina | Morse theory and finiteness properties of groups[END_REF][START_REF] Bieri | Homological dimension of discrete groups[END_REF][START_REF] Bridson | The subgroups of direct products of surface groups, Dedicated to John Stallings on the occasion of his 65th birthday[END_REF][START_REF] Stallings | A finitely presented group whose 3-dimensional integral homology is not finitely generated[END_REF] for important works on these notions. For a group G, properties F 1 and F 2 are respectively equivalent to being finitely generated and finitely presented. Since the class of finitely generated groups is uncountable (see for instance [39, p. 60-61]) and the class of finitely presented groups is countable, there are many examples of groups which are finitely generated but not finitely presented. It is natural Chapter 2. Irrational pencils and Betti numbers to ask whether one can construct groups of type F n≠1 which are not of type F n . In this context, Stallings [START_REF] Stallings | A finitely presented group whose 3-dimensional integral homology is not finitely generated[END_REF] constructed the first example of a group of type F 2 which is not of type F 3 . This construction was generalized by Bieri [START_REF] Bieri | Normal subgroups in duality groups and in groups of cohomological dimension 2[END_REF] to construct groups which are of type F n≠1 but not of type F n . These groups can be described as follows.

• If G is of type F n , then G is of type FP n • If G is of type FP n , then H i (G)
Example 90 (Stallings-Bieri). Let F 2 be the free group with two generators and let us consider the n-fold direct product F n 2 of F 2 with itself. Then the kernel of the map F n 2 ae Z that sends all generators to 1 is a group of type F n≠1 but not of type F n .

There are not so many constructions of groups which are of type F n≠1 but not of type FP n (or F n ). In this chapter we provide a method to construct examples of groups with these properties using complex geometry.

Remark 91. If X is a CW-complex with finite n-skeleton such that fi k (X) is trivial for all k = 2, . . . , n ≠ 1, then fi 1 (X) is a group of finiteness type F n . This follows from the fact that we can attach cells of dimension greater than n to obtain a space X Õ whose fundamental group is isomorphic to that of X and such that fi k (X Õ ) is trivial for all k Ø 2 (see Example 80). Hence X Õ is an aspherical CW-complex with finite n-skeleton such that fi 1 (X) ƒ fi 1 (X Õ ).

We conclude this section with the following discussion on commensurability that clarifies the fact that the results of Dimca, Papadima and Suciu [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] gave a negative answer to Kollár's question (discussed in the introduction of this text).

Definition 92. Two groups G and H are commensurable if there exist subgroups G 1 < G and H 1 < H of finite index such that G 1 and H 1 are isomorphic.

Two groups G and H are commensurable up to finite kernels if there exists a finite sequence

G = G 1 ae G 2 Ω G 3 ae • • • Ω G n = H,
where each arrow indicates a homomorphism of groups with finite kernel and with image of finite index.

Proposition 93. The property of being of type F n or FP n is invariant by the commensurability relation up to finite kernels.

For a proof of this statement see Corollary 7.2.4 and Theorem 7.2.21 in [START_REF] Geoghegan | Topological methods in group theory[END_REF]Chapter 7] for condition F n , and Proposition 2.7 in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] for condition FP n .

Isolated critical points of holomorphic maps

In this section we recall some definitions and results about holomorphic maps with isolated critical points. Section 2.3.1 can be found with more details in Chapter 14 of [START_REF] Voisin | Théorie de Hodge et géométrie algébrique complexe[END_REF] and Section 2.3.2 in Appendix B of [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF].

Let X be a complex manifold and let f : X ae C be a holomorphic map. has nonzero determinant at x, otherwise it is called degenerate.

The following result known as the holomorphic Morse Lemma characterizes the nondegenerate critical points of a holomorphic map.

Lemma 96. (holomorphic Morse lemma) Let f : X ae C be a holomorphic map and let

x in X be a nondegenerate critical point of f . Then, there exist holomorphic coordinates (z 1 , . . . , z n ) centered at x such that f can be written in these coordinates as

f (z) = f (x) + n ÿ i=1 z 2 i .
For a proof of the latter result see [75, p. 46]. Now, consider the homogeneous polynomial of degree 2

f : C n ≠ae C (2.3.1) (z 1 , . . . , z n ) ' ae n ÿ i=1 z 2 i .
For such a function, 0 oe C n is the only critical point of f . Hence, the central fiber

F 0 = f ≠1 (0)
is the only singular fiber of f and for any t in C ú , the regular fiber F t = f ≠1 (t) is a complex (n ≠ 1)-manifold. Let ' be a positive real number and for any z in C n let us write z = x + iy with x, y in R n . Then the fiber F ' has the form

F ' = {z = x + iy oe C n | |x| 2 = ' + |y| 2 , x‹y},
where x‹y means that x and y are orthogonal for the usual inner product of R n . This fiber contains the real (n ≠ 1)-sphere of radius Ô '

S ' = {z = x + iy oe C n | |x| 2 = ', y = 0},
which contracts to the origin in C n as ' tends to 0.

A similar situation occurs for any fiber F t . We use the following notation for the action of C ú on C n :

t • (z 1 , . . . , z n ) = (tz 1 , . . . , tz n ). Since f (t • (z 1 , . . . , z n )) = t 2 f (z 1 , . . . , z n ),
Chapter 2. Irrational pencils and Betti numbers we get that this action preserves the set of fibers {F t } toeC . In particular the singular fiber F 0 is invariant under this action. From here, we deduce that for all t in C ú , the fibers F t and F |t| are di eomorphic and F t contains the real (n ≠ 1)-sphere

S t = Ô t | Ô t| • S |t| ,
which contracts to the origin as |t| tends to 0. Let T S n≠1 be the tangent space of the unitary (n ≠ 1)-sphere. Then, for any positive real number ', the map Let B be the ball in C n of radius r centered at 0. Then the image of B under the homogeneous polynomial (2.3.1) is exactly the disk of radius r 2 centered at 0 in C that we will denote by D. The following results allow us to understand the topology close to the origin of the singular fiber of such a function. This lemma implies that f ≠1 ( )flB is a manifold with corners whose boundary consists of the union of f ≠1 ( ) fl ˆB and f ≠1 (ˆ ) fl B, which are manifolds with boundary meeting transversally along their boundaries.

F ' ≠ae T S n≠1 z = x +
The compactness of ˆB implies that f ˆB : ˆB ae D is a proper map. Therefore by Ehresmann's fibration Theorem we get that f ≠1 ( ) fl ˆB deformation retracts onto the intersection of a regular fiber of f with the boundary of B More precisely, if t is a complex number contained in \ {0}, then f ≠1 ( ) fl ˆB deformation retracts onto f ≠1 (t) fl ˆB.

The cone over S ' with vertex at 0, is the real n-ball of radius Ô '

B ' = {z = x + iy oe C n | |x| 2 AE ', y = 0}.
Using the action of C ú on the set of fibers of f , one can define the cone over S t with vertex at 0 as the real n-ball of radius

Ô ' Ô t | Ô t| B |t| .
Proposition 99. Let be a disk in C of small radius with respect to the radius of B and let t be a complex number contained in \ {0}. Then f ≠1 ( ) fl B deformation retracts onto (f ≠1 (t) fl B) fi B t . Moreover this retraction by deformation can be chosen to be induced by a retraction by deformation of f ≠1 ( ) fl ˆB onto f ≠1 (t) fl ˆB.

A global version of the last proposition is the following result.

Irrational pencils and fiber products 2.4.1 Fiber product

Let X and Y be topological spaces, f : X ae Y a continuous map and q : ' Y ae Y a covering map of Y . Then the fiber product of f : X ae Y and q : ' Y ae Y is defined as

' X = {(x, ŷ) oe X ◊ ' Y | f (x) = q(ŷ)}.
If we denote by ' f : ' X ae ' Y and ' q : ' X ae X the natural projections, we get that the diagram

' X ' f / / ' q ✏ ✏ ' Y q ✏ ✏ X f / / Y (2.4.1)
commutes. Moreover, ' f : ' X ae ' Y is a continuous map whose fibers are isomorphic to those of f : let ŷ be an element of ' Y , then

' f ≠1 (ŷ) = {(x, ŷ) oe X ◊ ' Y | f (x) = q(ŷ)} = f ≠1 (q(ŷ)) ◊ {ŷ}.
Notice that if the continuois map f is given by an inclusion X Òae Y , then the fiber product ' X is isomorphic to q ≠1 (X). Let x 0 , y 0 and ŷ0 be base points of X, Y and ' Y respectively such that f (x 0 ) = y 0 and q(ŷ 0 ) = y 0 . We will denote by f ú : fi 1 (X, x 0 ) ae fi 1 (Y, y 0 ) and q ú : fi 1 ( ' Y , ŷ0 ) ae fi 1 (Y, y 0 ) the induced maps on fundamental groups of f and q respectively. In general, the fiber product ' X is not connected. The following result gives a condition that guarantees the connectedness of the fiber product.

Proposition 107. If q : ' Y ae Y is the universal covering map of Y , then the induced map on fundamental groups f ú : fi 1 (X, x 0 ) ae fi 1 (Y, y 0 ) is surjective if and only if the fiber product ' X is connected.

Lemma 108. The natural projection ' q : ' X ae X is a covering map and the fundamental group of ' X based at (x 0 , ŷ0 ) is isomorphic to f ≠1 ú (q ú (fi 1 ( ' Y , ŷ0 ))).

Proof. Let x be an element of X and let U be an open neighborhood of f (x) evenly covered by q : ' Y ae Y . We write q ≠1 (U ) = fi ioeI U i , where U i is homeomorphic to U for all i in I. Let V = f ≠1 (U ) and let V i = ' X fl (V ◊ U i ) for all i in I. Then, ' q ≠1 (V ) = fi ioeI V i , where V i is homeomorphic to V for all i in I. Hence V is an open neighborhood of x evenly covered by ' q : ' X ae X and we obtain that ' q : ' X ae X is a covering map. Since q ¶ ' f = f ¶ ' q, we get that

q ú ( ' f ú (fi 1 ( ' X, (x 0 , ŷ0 )))) = f ú (' q ú (fi 1 ( ' X, (x 0 , ŷ0 )))),
which implies that the image of ' q ú is contained in f ≠1 ú (q ú (fi 1 ( ' Y , ŷ0 ))). To prove the other inclusion, let -: [0, 1] ae X be closed curve based at x 0 such that f ú ([-]) is contained in

q ú (fi 1 ( ' Y , ŷ0 )). Now, let " : [0, 1] ae  Y be a lifting of f ¶ -based at ŷ0 . Hence, for all t in [0, 1] we get that q ¶ "(t) = f ¶ -(t). Therefore, (-, ") defines a closed curve in  X such that  q ú ([(-, ")]) = [-], which concludes the proof. 2
Corollary 109. If X and Y are compact complex manifolds and f :

X ae Y is a holomor- phic map, then ' f : ' X ae ' Y is a proper holomorphic map. Proof. To see that ' f is proper, let K be a compact subset of ' Y . Then ' f ≠1 (K) is a closed subset of ' X contained in X ◊ K.
Since the latter space is compact we obtain that ' f ≠1 (K) is compact as well.

2

Remark 110. If X is an aspherical CW-complex or an aspherical smooth manifold, then the fiber product ' X constructed in (2.4.1) is a classifying space of f ≠1 ú (q ú (fi 1 (Y ))). In particular, if q : ' Y ae Y is the universal covering space of Y , we obtain that ' X is a classifying space of ker(f ú ).

Irrational pencils

Let f : X ae S be an irrational pencil, let ' S ae S be the universal covering space of S and let ' X be the fiber product of f : X ae S and ' S ae S. As we saw previously we obtain a commutative diagram

' X ' f / / ✏ ✏ ' S ✏ ✏ X f / / S, (2.4.2)
where ' X ae X is a covering map such that fi 1 ( ' X) is isomorphic to the kernel of f ú : fi 1 (X) ae fi 1 (S) and ' f : ' X ae ' S is a holomorphic proper map. In the following, ' X will be called the universal fiber product of the irrational pencil f : X ae S and ' f : ' X ae ' S a lift of the irrational pencil.

We identify ' S with C or the unit disk of C. If we suppose that f is not a submersion and has isolated critical points, then its critical set Crit(f ) is a nonempty discrete subset of X. Moreover, we can deduce the following.

• The set of critical values of the lift ' f : ' X ae ' S is an infinite discrete subset of ' S: since the diagram (2.4.2) commutes, we get that the set of critical values of ' f is infinite; it is precisely the preimage of the set of critical values of f under the covering map ' S ae S. Finally, we observe that this set is discrete, since it is given by the orbit of a finite number of points (one for each critical value of f ) under the action of the fundamental group of S.

• The set of critical points of ' f is an infinite discrete subset of ' X. The argument is the same as above since the critical set of ' f , denoted by Crit( ' f ), is the preimage of Crit(f ) under the covering map ' X ae X.

Topology of the universal fiber product of an irrational pencil

The following result shows that we can perturb a lift of an irrational pencil with isolated critical points to obtain a C OE map with nondegenerate critical points. It is an application of the results of Section 2.3.

Proposition 111. Let ' f : ' X ae ' S be a lift of an irrational pencil with isolated critical points. Then, there exists a C OE map ' f 0 : ' X ae ' S, C OE -close to ' f , which is holomorphic in a neighborhood of its critical set and such that each critical point of ' f 0 is nondegenerate.

Proof. To construct the map ' f 0 we will perturb ' f in a neighborhood of each degenerate critical point of ' f as follows. Recall that we identified ' S with C or the unit disk of C. For each point q in the critical set Crit( ' f ) of ' f , we pick a neighborhood U q such that the open sets

(U q ) qoeCrit( ' f )
are disjoint. By choosing a holomorphic coordinate system in U q , we can identify U q with an open ball B(0, Á q ) centered at the origin of C n of radius Á q . For each point q in Crit( ' f ), we consider the restriction f : B(0, Á q ) ae ' S. If q is a degenerate critical point of ' f , by Remark 106, we can choose a small enough linear form ¸q on C n such that the map ' f ≠ ¸q : B(0, Á q ) ae ' S µ C still takes values in ' S and has finitely many critical points contained in B(0, Áq 2 ) which are all nondegenerate. The number of its critical points is exactly the Milnor number of the critical point q for the map ' f . If q is a nondegenerate critical point we take ¸q = 0. Under the identification of U q with the open ball B q = B(0, Á q ), we denote by V q the open subset of U q corresponding to B(0, Áq 2 ). Finally, one builds the map ' f 0 by declaring that ' f 0 is equal to ' f ≠ ¸q on V q , to ' f outside the union of the open sets (U q ) qoeCrit( ' f ) and to a deformation between ' f and ' f ≠ ¸q on U q ≠ V q . 2

We would like to point out the following facts about the morsification ' f 0 : ' X ae ' S: • All the critical points of ' f 0 are contained in one of the V q 's. • The construction of ' f 0 : ' X ae ' S can be done in an equivariant way for the action of fi 1 (X), so that the map ' f 0 descends to a map f 0 : X ae S homotopic to f . We will now work with the map ' f 0 instead of ' f . We study the topology of ' X by viewing it as the increasing union of a well-chosen sequence of compact subsets (see [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] for a very similar construction). With that purpose, we are looking for a sequence of closed disks

(D k ) kØ1 of ' S such that • ' S = fi kØ1 D k . • The disk D k is contained in the interior of the disk D k+1 . • No critical value of ' f 0 is contained in the boundary of D k . • D k+1 \ D k contains exactly one critical value of ' f 0 for all k Ø 1. • D 1 contains no critical value.
For such a sequence (D k ) kØ1 , we set

X k := ' f ≠1 0 (D k ).
Hence X 1 retracts onto a smooth fiber of ' f 0 . For k Ø 2, the topology of the space X k is described thanks to the following proposition.

Proposition 112. The space X k+1 has the homotopy type of a space obtained from X k by gluing to it a finite number m k > 0 of n-dimensional cells.

This proposition is well-known (see e.g. Lemma 3.3 in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] for a related statement, although in that lemma the authors work with the original map ' f instead of our perturbation ' f 0 ). Before giving the proof of this result we give a construction of the sequence (D k ) kØ1 . We endow ' S with its classical Riemannian metric of constant sectional curvature and we denote by the set of critical values of ' f 0 . Each pair of points (z, w) in defines the following subsets of ' S. • ¸1(z, w) := the bi-infinite geodesic whose points are equidistant from z and w.

• ¸2(z, w) := the bi-infinite geodesic passing through z and w.

We write

L = € z,woe z" =w ¸1(z, w) fi ¸2(z, w)
and we pick a point z 0 in ' S which is not contained in L. • The point z 0 is a regular value of ' f 0 such that all the points in are at di erent distances from z 0 .

• Any geodesic passing through z 0 contains at most 1 point of .

Hence, if we denote by d(•, •) the distance induced by the Riemannian metric of ' S, we can enumerate the critical values of ' f 0 in such a way that d(z 0 , z

i ) < d(z 0 , z j ) if i < j. Finally, let r 1 = d(z 0 , z 1 ) 2 and r k = d(z 0 , z k ) + d(z 0 , z k+1 ) 2 .
Then the set of disks (D k = D(z 0 , r k )) kØ1 centered at z 0 of radius r k has the desired properties. Notice that if we allow the disks (D k ) kØ1 to have di erent centers, the construction of such a sequence of disks is simpler. Since the restriction of ' f 0 to the preimage of the set of regular values is a locally trivial fibration, X k+1 deformation retracts onto

X k fi ' f ≠1 0 (c([0, 1]) fi D ú ). Let m k > 0 be the number of critical points in the level ' f ≠1 0 (c( 1
)) and let x 1 , . . . , x m k be the corresponding critical points. For each i = 1, . . . , m k , let U i be a small neighborhood of x i such that the sets (U i ) m k i=1 are disjoint. According to Corollary 101, we can fix a regular value t = c(1 ≠ ") (for a small " > 0) of ' f 0 such that ' f ≠1 0 (D ú ) deformation retracts onto the union of the generic fiber ' f ≠1 0 (t) with m k n-balls B 1 , . . . , B k , where the ball B i is contained in U i and it is glued to ' f ≠1 0 (t) along a vanishing sphere

S i of ' f 0 contained in ' f ≠1 0 (t).
By a surface group we mean here the fundamental group of an oriented surface of finite type (open or closed). Hence a surface group is either free or the fundamental group of a closed oriented surface. To prove Proposition 117, we will make use of the following theorem due to Bridson, Howie, Miller and Short [START_REF] Bridson | The subgroups of direct products of surface groups, Dedicated to John Stallings on the occasion of his 65th birthday[END_REF].

Theorem 118. Let F 1 , . . . , F m be surface groups. Let G be a subgroup of the direct product

F 1 ◊ • • • ◊ F m . If G is of type FP m , then G is virtually isomorphic to a direct product of the form H 1 ◊ • • • ◊ H k where k AE m and each H i is a surface group. In particular G is of type FP OE .
Besides this theorem, we will also use the fact that the property of being FP m (for m oe N fi {OE}) is invariant under commensurability (see Proposition 93).

Proof of Proposition 117. The group sits inside the direct product

fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) ◊ • • • ◊ fi 1 (Y ).
(2.5.

3)

The factors of this direct product are the subgroups of the form

{1} ◊ • • • ◊ fi 1 ( j ) ◊ • • • ◊ {1} or {1} ◊ • • • ◊ fi 1 (Y ) ◊ • • • ◊ {1}.
Each factor intersects non-trivially. This implies that contains copies of the group Z a+b . The Gromov hyperbolicity of each factor implies that Z a+b+1 does not embed in . Now suppose that a finite index subgroup 1 of embeds in a direct product of surface groups

F 1 ◊ • • • ◊ F m .
By taking m to be minimal, we may assume that L i = 1 fl F i is non-trivial for i = 1, . . . , m. Otherwise 1 embeds in a direct product of m ≠ 1 surface groups. Since 1 does not contain any non-trivial Abelian normal subgroup, this implies that each F i is non-Abelian, hence hyperbolic. A similar argument as before then shows that 1 contains copies of Z m but no copy of Z m+1 . Hence m = a + b. By Remark 113, the group is of type FP 2b+a≠1 , hence 1 is. Since 2b + a ≠ 1 Ø m = a + b, 1 is of type FP m and Theorem 118 implies that 1 is of type FP OE . This contradicts the fact that 1 is not of type FP a+2b . 2 Finally, we compute, in some cases, the first Betti number of the group = ker(f ú ).

A similar computation appears in [50, §7], which applies to some of the examples built in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF][START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF].

Proposition 119. Assume that a + 2b Ø 3. Assume furthermore that b Ø 2 or that b = 1 and that the map fi 1 ( j ) ae fi 1 (E) is surjective for some j oe {1, . . . , a}. Then the first Betti number of is equal to:

b 1 ( 1 ◊ • • • ◊ a ◊ Y b ) ≠ 2.
Proof. We consider the surjective homomorphism

ae fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) b≠1
obtained by considering the inclusion of in the direct product (2.5.3) and by projecting onto the first a + b ≠ 1 factors. Its kernel N consists of elements of the form

(1, . . . , 1, g) oe fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) ◊ • • • ◊ fi 1 (Y )
where g oe ker(h ú ); it is isomorphic to ker(h ú ). Hence we have the following exact sequence:

0 ae N ae ae fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) b≠1 ae 0.
It induces the following short exact sequence (see [START_REF] Brown | Cohomology of groups[END_REF]VII.6], all homology groups are taken with Z coe cients):

H 1 (N ) ae H 1 ( ) ae H 1 (fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) b≠1 ) ae 0. ( 2
H 2 (Z 2 ) ae H 1 (ker(h ú )) fi 1 (Y ) ae H 1 (Y ) ae H 1 (Z 2 ) ae 0.
Since the map

H 1 (Y ) ¢ Q ae H 1 (Z 2 ) ¢ Q is an isomorphism, we obtain that H 1 (ker(h ú )) fi 1 (Y ) ¢ Q has dimension at most 1. Hence H 1 (N ) ¢ Q has dimension at most 1.
Since is Kähler and hence has even first Betti number, this implies that the first arrow in (2.5.4) has finite image. Hence H 1 ( ) ¢ Q and

H 1 (fi 1 ( 1 ) ◊ • • • ◊ fi 1 ( a ) ◊ fi 1 (Y ) b≠1 ) ¢ Q are isomorphic.

Î

. See [START_REF] Brady | Branched coverings of cubical complexes and subgroups of hyperbolic groups[END_REF][START_REF] Kropholler | Almost hyperbolic groups with almost finitely presented subgroups[END_REF] for related results and motivation. The article [START_REF] Kropholler | Almost hyperbolic groups with almost finitely presented subgroups[END_REF] produces other examples with a smaller bound on the rank of Abelian subgroups. More precisely, for each positive integer n, Kropholler gives in [START_REF] Kropholler | Almost hyperbolic groups with almost finitely presented subgroups[END_REF] an example of a group of type F n≠1 but not of type F n , which does not contain free Abelian groups of rank greater than

' n 3 (
and which is a subgroup of a CAT(0) group.

Irrational pencils on aspherical compact complex surfaces

In this section we give a detailed proof of Theorem 70 due to Kapovich. There are no original results here, this is an expository section.

Remark 126. Let Èx 1 , . . . , x m |r 1 , r 2 , . . .Í be a presentation of fi 1 ( ' X) (not necessarily finite), such that x j corresponds to i ú [" j ] and let r = x j 1 • • • x j d be a relation of such presentation. Then, there exists a positive integer N such that the loop given by the concatenation of

" j 1 • • • " j d (which corresponds to r) is nullhomotopic in X k for all k Ø N .
Lemma 127. If the kernel of the map f ú : fi 1 (X) ae fi 1 (S) is finitely presented, there exists a positive integer N such that for any k Ø N the inclusion map X k Òae ' X induces an isomorphism on fundamental groups.

Proof. Let Èx 1 , . . . , x m |r 1 , . . . , r l Í be a presentation of fi 1 ( ' X), where x j corresponds to i ú [" j ]. Since there are finitely many relations, by Remark 126, for all relation r oe {r 1 , . . . , r l } written as r = x j 1 • • • x j d , the loop given by " j 1 • • • " j d is nullhomotopic in X k for all k Ø N . Now, we fix a positive integer k such that k Ø N and let i : F Òae ' X, i 1 : F Òae X k and i 2 : X k Òae ' X be the inclusion maps. Then, we have that the diagram

F _ i 1 ✏ ✏ Id / / F _ i ✏ ✏ X k i 2
/ / ' X commutes. Hence fi 1 (X k ) is generated by the set {i 1ú [" 1 ], . . . , i 1ú [" m ]} and i 2ú : fi 1 (X k ) ae

fi 1 ( ' X) sends i 1ú [" j ] to i ú [" j ].
We claim that i 2ú is an isomorphism. By construction we have that i 2ú is surjective since F is contained in X k . To prove the injectivity of i 2ú , let F m be the free group with m generators {x 1 , . . . , x m }, and let ◊ : F m ae fi 1 (X k ) be the morphism that sends x j to i 1ú [" j ]. Now, consider the commutative diagram

F m Id / / ◊ ✏ ✏ F m i 2ú ¶◊ ✏ ✏ fi 1 (X k ) i 2ú / / fi 1 ( ' X).
(2.6.1)

Let " : [0, 1] ae F be such that i 1ú (["]) is in the kernel of i 2ú and let x be an element in F m such that ◊(x) = i 1ú (["]). By construction, we get that i 2ú ¶ ◊ : F m ae fi 1 ( ' X) sends x j to i ú ([" j ]). The kernel of i 2ú ¶ ◊ is the subgroup of F m whose elements are of the form jt g ≠1 t ) with g t oe F m and r jt oe {r 1 , . . . , r l }.

(2.6.2)

Hence by the commutativity of diagram (2.6.1) we get that x is of the form (2.6.2). Finally, observe that for any relation r oe {r 1 , . . . , r l } we have that ◊(r) = 1. This follows from the fact that the simple closed curve given by concatenating some of the " Õ j s whose homotopy class represents the relation r is nullhomotopic in X k . Therefore x is in the kernel of ◊, which implies the injectivity of i 2ú . 2

2.6.3.b Foliations induced by the fibers of an irrational pencil

Let {p 1 , . . . , p l } be the set of singular points of F s . Via the choice of some holomorphic coordinates, for each singular point p i , we pick a closed neighborhood B i (Á) of p i that we Chapter 2. Irrational pencils and Betti numbers identify with a closed ball B i (0, Á) centered at the origin of C n , and a closed neighborhood of s that we identify with a closed disk centered at the origin such that f is given in these coordinates by a holomorphic map that vanishes at the origin. Since the singular points of F s are finitely many, there exists a small enough Á 0 such that for all i = 1, . . . , l and for all Á AE Á 0 , the boundary of B i (0, Á) satisfies the conclusion of Milnor's Fibration Theorem.

From now on, we will assume that there is at least one singular point in F s . We fix a Riemannian metric on X. Let Y 0 be a connected component of F s \ fi l i=1 B i (Á 0 ) and let NY 0 ae Y 0 be the associated normal bundle, i.e. NY 0 is orthogonal complement of T Y 0 in T X for the fixed metric on X.

Lemma 128.

There exists an open neighborhood U of the 0-section of the normal bundle NY 0 ae Y 0 and an open neighborhood V of Y 0 such that the exponential map exp : U ae V is a di eomorphism and the intersection of the fibers of f with V are connected.

Proof. Let Á < Á 0 and let Y be the connected component of F s \fi l i=1 B i (Á) containing Y 0 . We denote by U (r) the open r-neighborhood of the 0-section NY ae Y . Since exp : NY ae X is a local di eomorphism in every point of the 0-section of NY ae Y , we get that up to increasing Á (preserving the inequality Á < Á 0 ), there exists a positive number r > 0 such that exp : U (r) ae X is a submersion. Otherwise, there would exist a decreasing sequence of positive numbers (r n ) nØ1 converging to zero and a sequence (x n ) nØ1 in NY of singular points of exp : NY ae X such that x n is contained in U (r n ). If we consider a compact set K such that Y µ K µ F s \ {p 1 , . . . , p l }, we obtain that there exists a point y in the 0-section of NY ae Y such that (x n ) converges to y, a contradiction to the existence of a neighborhood of y where exp is a di eomorphism. By a similar result, we can also assume that exp : U (r) ae X is injective. Hence, if we denote by V (r) the image of U (r) under exp we obtain that exp : U (r) ae V (r) is a di eomorphism. Finally, since the restriction of f to the open set V (r) is continuous, we can find an open subset V of V (r) such that the fibers of F intersecting V are connected.

2

Lemma 129. There is a neighborhood U of Y 0 such that if a generic fiber F of f intersects U , then F fl U is a connected finite covering space of Y 0 .

Proof. Let U and V be open neighborhoods of the 0-section of NY 0 ae Y 0 and of Y 0 respectively, such that the conclusion of Lemma 128 holds. Let y be a base point in Y 0 and let D be a holomorphic disk centered in y, contained in V and transverse to the fibers of f . We can identify D to the unit disk of C in such a way that the restriction of f to D corresponds to the function z ' ae z k , where k is the multiplicity of the irreducible component of F s containing Y 0 . Let us consider the foliation of V given by the fibers of f . Hence, the holonomy associated to this foliation is given by a morphism fl : fi 1 (Y 0 , y) ae G, where G is a subgroup of the germ of di eomorphisms of D, whose elements are of the form z ' ae ›z, where › is a k-th root of unity. Let q : ( ' Y , ŷ) ae (Y 0 , y) be the finite covering map such that fi 1 ( ' Y , ŷ) is isomorphic to the kernel of fl, and let N ' Y ae ' Y be the pullback bundle induced by q, i.e., we have the commutative diagram

N ' Y ' q ✏ ✏ / / ' Y q ✏ ✏ NY 0 / / Y 0 .
2.6. Irrational pencils on aspherical compact complex surfaces

75

The foliation in V given by the fibers of f induces a foliation F in U , which induces a foliation ' F in ' U := ' q ≠1 (U ) (via the covering ' q). Let L and ' L be the leafs in F and ' F given by the 0-sections of NY 0 ae Y 0 and N ' Y ae ' Y respectively. Since ' q is a covering map, we get that for every point ŷ in ' L there is a holomorphic disk transverse to the foliation. We deduce that the holonomy of L is precisely given by fl and the germinal holonomy group of ' L is trivial. By Reeb Stability Theorem (see Remark 125) we obtain that there is a neighborhood of ' L that is a union of leaves that are homeomorphic to ' L. Then, the result follows from the fact that ' U ae U is a finite covering map and that exp : U ae V is a di eomorphism. 2

2.6.3.c Irrational pencils on aspherical compact Kähler surfaces

Suppose now that X is an aspherical compact Kähler surface and let f : X ae S be an irrational pencil with a singular value s. We resume with the notation of Section 2.6.3.b.

Lemma 130. If Y 0 is a connected component of F s \ fi l i=1 B i (Á 0 ). Then the map fi 1 (Y 0 ) ae fi 1 (X) induced by the inclusion Y 0 Òae X has infinite image.

Proof. For every irreducible component Z of F s , let Z ú ae Z be its desingularization map (see [START_REF] Gri | Principles of algebraic geometry, Pure and Applied Mathematics[END_REF] p. 498). Then, the map fi 1 (Z ú ) ae fi 1 (X) induced by the composition Z ú ae Z Òae X, has infinite image. Otherwise, if we denote by  X ae X the universal covering space of X and by Ê Z ú the fiber product of Z ú ae X and  X ae X, we would obtain a finite covering map Ê Z ú 0 ae Z ú , where Ê Z ú 0 is a connected component of Ê Z ú (by Lemma 108 fi 1 ( Ê Z ú 0 ) ƒ ker(fi 1 (Z ú ) ae fi 1 (X))). Since the map Z ú ae X is a nonconstant holomorphic function, we would obtain that there is a nonconstant holomorphic map from the closed Riemann surface Ê Z ú 0 to  X, which contradicts the asphericity of X. Finally, suppose that Y 0 is contained in the irreducible component Z of F s and let denote the singular points of Z. Hence, there exists a retraction of Z \ onto Y 0 . Therefore, the composition of the inclusions Y 0 Òae Z \ Òae Z ú induces a surjective map on fundamental groups and the result follows directly.

2

Remark 131. In the proof of the latter lemma, we used the fact that the universal covering space of an aspherical compact Kähler surface cannot contain the image of a closed Riemann surface under a nonconstant holomorphic map. This follows from the fact that for a Kähler surface X, the integral of the Kähler form over any analytic compact curve is always positive, which cannot occur if H 2 (X) is trivial.

A direct consequence of Lemmas 129 and 130 is the following result.

Corollary 132. There exists a neighborhood of F s such that for any regular fiber F of f :

X ae S contained in such a neighborhood and any connected component Y of F \fi l i=1 B i (Á 0 ), the map on fundamental groups induced by the inclusion Y Òae X has infinite image.

Proof of Kapovich's Theorem

In this section we give the proof of Theorem 70 for the general case.

Let X be an aspherical compact Kähler surface and let f : X ae S be an irrational pencil whose singular fibers are of multiplicity one. We resume with the construction of Section 2.4.3. Let us assume by contradiction that f is not a submersion and that the kernel of f ú : fi 1 (X) ae fi 1 (S) is finitely presented. By Lemma 127, we know that there Lemma 134. The map H 1 (  F ) ae H 1 (V ) induced by the inclusion  F Òae V has non-trivial kernel.

Proof. Let C be a connected component of p ≠1 (F fl B i (Á 0 )) for some i = 1, . . . , l (which is homeomorphic to F fl B i (Á 0 )). By Theorem 122, we know that C has the homotopy type of a CW-complex of dimension 1, and by Theorem 123, C is not simply connected. Hence, C is a compact surface with non-empty boundary and it is not a disk. Now, let H 1 (C) ae H 1 (  F ) be the map on homology groups induced by the inclusion C Òae  F . Lemma 134 follows from the following two facts (a) The image of H 1 (C) in H 1 (  F ) is contained in the kernel of the map H 1 (  F ) ae H 1 (V ).

(b) The map H 1 (C) ae H 1 (  F ) is non-trivial. Notice that (a) follows directly from the fact that C is contained in the ball B i (Á 0 ). From now on, to prove (b) we will be working with the surface  F as our total space. If the surface C has a handle, (b) follows immediately since there exists an element in H 1 (C) which has non-trivial image in H 1 (F ) in this case (which is stronger than having non-trivial image in

H 1 ( Â F )).
If C is a planar surface the last argument does not work. Using the Mayer-Vietoris sequence for the sets (or rather small enough neighborhoods of these sets) C and ( Â F \ C), we get the exact sequence

H 1 (ˆC) (iú,≠jú) / / H 1 (C) ü H 1 ( Â F \ C) kú+lú / / H 1 ( Â F ), (2.6.5)
where, i, j, k, l are the respective inclusions.

Claim 135. All the connected components of  F \ C are non-compact. We now conclude the proof by using Claim 135. If ˆC has n connected components {a 1 , . . . , a n }, then H 1 (ˆC) is a free Abelian group generated by the a i 's with the unique relation n ÿ i=1 i ú a i = 0.

Let W be a connected component of  F \ C. Up to reordering the set {a 1 , . . . , a n }, we can suppose that {a 1 , . . . , a l } are the common boundary components of C and W for some l AE n. Let us assume by contradiction that the map k ú : H 1 (C) ae H 1 (  F ) is trivial. Then (i ú a 1 , 0) lies in ker(k ú + j ú ) = Im(i ú , ≠j ú ), and by the exactness of 2.6.5 there is a non-zero element a in H 1 (ˆC) such that (i ú a, ≠j ú a) = (i ú a 1 , 0). Thus a ≠ a 1 lies in ker(i ú ), which is generated by

a 1 + • • • + a n , i.e. a = a 1 + ⁄(a 1 + • • • + a n ). Since H 1 ( Â F \ C) = H 1 (W ) ü H 1 ( Â F \ (C fi W ))
and j ú a = 0, we deduce that j ú (a 1 + ⁄(a 1 + • • • + a l )) = 0. Finally, since W is noncompact, {j ú a 1 , . . . , j ú a l } form a linearly independent set and thereby ⁄ = ≠1 and l = 1. A contradiction to the fact that a is non-zero.

Résumé

Dans cette thèse nous nous sommes intéressés à l'étude des sous-groupes normaux de type fini des groupes kählériens. Nous étudions les groupes de type fini munis d'une action sur un arbre qui admettent un plongement dans un groupe kählérien comme sous-groupes normaux et dont l'action sur l'arbre peut être virtuellement étendue au groupe kählérien. L'un des ingrédients principaux de cette étude est un résultat de Gromov et Schoen sur les groupes kählériens agissant sur un arbre. Nous donnons également de nouveaux exemples de groupes kählériens qui apparaissent comme des sous-groupes normaux de groupes kählériens précédemment connus. Les nouveaux exemples qui apparaissent de cette manière sont liés aux propriétés de finitude dans la théorie des groupes. Notre principal outil pour construire ces exemples est l'étude des pinceaux irrationnels avec des points critiques isolés sur des variétés complexes compactes asphériques.

Résumé en anglais

In this thesis, we focus on the study of finitely generated normal subgroups of Kähler groups. We present some restrictions on finitely generated groups that can occur as normal subgroups of Kähler groups. We study finitely generated groups acting on a tree that admit an embedding into a Kähler group as a normal subgroup, and whose action on the tree can be virtually extended to the Kähler group. One of the main ingredients of this study is a result of Gromov and Schoen about Kähler groups acting on trees. We also give new examples of Kähler groups which occur as normal subgroups of previously known Kähler groups. The new examples that occur in this way are related to finiteness properties in group theory. Our main tool to construct these examples is the study of irrational pencils with isolated critical points on compact aspherical complex manifolds.

Prénom NOM TITRE de la thèse !

  1 and Theorems B, C, D and E are proved in Section 1.4.2. Let us summarize which results about extensions of group actions are used in each proof. • To prove Theorem B, we use the results we establish in Section 1.3.4. • To prove Theorem C we can use either Karras, Pietrowski and Solitar's work (Theorem 40) or a special case of it that we reprove here (Theorem 28 in Section 1.3.

Chapter 1 .

 1 Kähler groups acting on trees Now, suppose that there exists a common subgroup C of A and B and let i A : C Òae A i B : C Òae B be the respective inclusions. Let N denote the normal closure in A ú B of the set of elements {i A (c)i ≠1 B (c) | c oe C}. Then, the amalgamated product of A and B along C denoted by A ú C B is defined as the quotient A ú B/N . Given presentations of A and B as in (1.2.1), we get that the amalgamated product A ú C B is given by

1 .Figure 1 . 2 . 1 :Figure 1 . 2 . 2 :

 1121122 Figure 1.2.1: Bass-Serre tree of an amalgamated product A ú C B have that for any element x of G one can associate the edge xC of T to its pair of vertices {xt ≠1 A, xA}. Therefore, the edges of its Bass-Serre tree T joined to the vertex xA is given by {xaC | a oe A} fi {xatC | a oe A}, (see Figure 1.2.2). The action of G on its Bass-Serre tree T has the following properties: • It is minimal, i.e. there is no G-invariant proper subtree of T . To see this, let T Õ be a G-invariant subtree of T . If G decomposes as an amalgamated product A ú C B and xA is a vertex of T Õ , then by the transitivity of the action of G on G/A, we obtain that G/A is contained in T Õ . Then for all b in B, the path between A and bA is contained in T Õ . In particular, the vertex B and the edge C are contained in T Õ . By the transitivity of the

Figure 1 . 2 . 3 :

 123 Figure 1.2.3: Cutting along a simple closed curve

Figure 1

 1 Figure 1.2.4: Surfaces S 1 and S 2

Figure 1 . 2 . 7 :

 127 Figure 1.2.7: intersection of S 1 and S 2

Figure 1 . 2 . 8 :

 128 Figure 1.2.8: Connected components of H 2 \ fiE separated by an element of E

4 )

 4 l x ' ae xC sends the edge l x in the dual tree (V, E) to the edge xC in the Bass-Serre tree T which by Theorem 6 joins the vertices xA and xB. Therefore, the bijections (1.2.3) and (1.2.4) define an isomorphism between such trees.

2 . 6 )

 26 l x ' ae xC sends the edge l x in the dual tree (V, E) to the edge xC in the Bass-Serre tree T which by Theorem 6 joins the vertices xt ≠1A and xA. We conclude that the bijections (1.2.5) and (1.2.6) define an isomorphism between such trees.

Figure 1 . 2 . 9 :

 129 Figure 1.2.9: Dual tree embedded in H 2

  the stabilizer of v and x • v are equal. From this, we obtain that G v fixes v and x • v, and thereby the path [v, x • v]. Finally, we obtain that

2 1. 3 . 1 .Definition 19 .

 23119 b Induced isometries of the Bass-Serre tree We define the subgroup Aut T (G) of Aut(G) by the following property: an automorphism Ï : G ae G is an element of Aut T (G) if for every edge e = (v 1 , v 2 ) of T there is an element x in G such that

2 Definition 22 .Lemma 23 .

 22223 which implies that Ï is given by the action of x on T . An isometry Ï : T ae T induced by an automorphism Ï of G that satisfies (1.3.1) is called G-compatible. An extension of the induced action of Inn(G) on T to a subgroup of Aut(G) is called G-compatible if every automorphism of defines a Gcompatible isometry of T . Suppose that the stabilizer of any vertex of T is its own normalizer. Then the induced action of Inn(G) on T extends to a G-compatible action of Aut T (G) on T .

x 2 Proposition 27 .

 227 = ab, for some a oe A and some b oe B, which implies the result. The induced action of Inn(G) on T extends to T . Proof. By Lemma 20 and Lemma 23 it su ces to show that T is contained in Aut T (G). Let Ï be an automorphism of G contained in T and let x, y be elements in G such that Ï(A) = xAx ≠1 and Ï(B) = yBy ≠1 .

Lemma 98 .

 98 Let be a disk in C of small radius with respect to the radius of B. Then the restriction of f to the boundary of B f ˆB : ˆB ae D is a submersion along f ≠1 ( ) fl ˆB.

Chapter 2 .

 2 Irrational pencils and Betti numbersIf f is defined in a complex manifold and x is an isolated critical point of f , by choosing a holomorphic coordinate system in a neighborhood U of x, we can identify U with an open ball B centered at the origin of C n . Under this identification 0 is an isolated critical point of f B : B ae C with Milnor number equal to µ Ø 1. The following remarks are an interpretation of Lemmas 102 and 103.Remark 105. The critical point x is nondegenerate if and only if µ is equal to 1.Remark 106. If we perturb f B : B ae C by subtracting a linear map ¸1z 1 + • • • + ¸nz n from f B , which is a regular value of df : B ae (C n ) ú su ciently close to the origin, then 0 splits up into µ nondegenerate critical points contained in B.

Proof of Proposition 112 .Figure 2 . 4 . 1 :

 112241 Figure 2.4.1: The disks D k and D ú

2

 2 Proof of Claim 135. Let W be a connected component of F \C. Recall that C is a connected component of p ≠1 (F fl B i (Á 0 )) which indeed is homeomorphic to F fl B i (Á 0 ). Now, let Y be a connected component of F \ fi l i=1 B i (Á 0) and let  Y be a connected component of p ≠1 (Y ) contained in W . Hence, we get the commutative diagram

  On renvoie à la Section 1.3.1.b pour la définition d'action G-compatible. La première application de ce résultat est la suivante. Soient S une surface fermée de genre g Ø 2 et un groupe kählérien tel qu'il existe une suite exacte courte

	Théorème B. 1	/ / fi 1 (S)	/ /	/ / Q	/ / 1.

/ 1 P / / P ( 1 ) / / 1 est un produit direct. Supposons que l'action par conjugaison de sur fi 1 (S) préserve la classe de conjugaison d'une courbe fermée simple dans S. Alors, il existe un sous-groupe d'indice fini 1 de contenant fi 1 (S), tel que la suite exacte courte restreinte 1 / / fi 1 (S) / / 1 / / Q 1 / / 1 est un produit direct (où Q 1 est l'image de 1 dans Q). En particulier, la monodromie Q ae Out(fi 1 (S)) est finie.

  1 fl S 2 ) ae fi 1 (S 2 ) induced by the inclusions S 1 fl S 2 Òae S 1 and S 1 fl S 2 Òae S 2 . Let • 1 : [0, 1] ae S 1 be the path in S 1 obtained from • after cutting S along " and let ‡ be the loop based at •

	1.2. Groups acting on trees
	S
	⌧
	S 2	S 1
	⌧
	Figure 1.2.5: Surfaces S, S 1 and S 2

[START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF]

(1) defined by the boundary component of S 1 which is not homotopic to ". If S is a closed oriented surface of genus g, we obtain that S 1 is a surface of genus g ≠ 1 without two disks removed. Hence, one can obtain S 1 from a 4g ≠ 4 polygon without two disks removed. Up to changing the orientation of ‡, we may assume that the (oriented) boundary of the 4g ≠ 4 polygon deformation retracts onto the concatenation "• 1 ‡• 1 , where ", ‡ and •1 denote the inverse curves of ", ‡ and • 1 respectively (see Figure

1

.2.6). Hence, if we denote by c 1 the homotopy class of " and by c 2 the homotopy class of • 1 ‡• 1 , we obtain that fi 1 (S, x 0 ) has the following presentation

  is finitely generated for all i AE n. Proposition 87 implies that if G is of type FP n , then H i (G, Q) has finite dimension for all i AE n. If G is a finitely presented group, the finiteness conditions F n (resp. F OE ) and FP n (resp. FP OE ) are equivalent.

	Remark 88. Proposition 89. See Chapter 8 Section 7 of [16] for a proof of Proposition 89.

1 Vanishing sphere of a nondegenerate critical point

  Definition 94. A point x in X is called a critical point of f if the di erential map df x : T x X ae C is identically zero, otherwise it is called a regular point. A critical point x of f is called isolated if there exists a neighborhood U of x such that x is the only critical point of f contained in U . The preimage F t = f ≠1 (t) of a point t in C, is called a singular fiber if it contains a critical point of f , otherwise it is called a regular fiber.Let x in X be a critical point of a holomorphic map f : X ae C and let (z 1 , . . . , z n ) be a holomorphic coordinate system in a neighborhood of x. The Hessian of f at x is defined as the C-bilinear form on T x X such that Let f : X ae C be a holomorphic map. A critical point x of f is called nondegenerate if the Hessian of f at x is a nondegenerate quadratic form, i.e. if the Hessian matrix

	2.3. Isolated critical points of holomorphic maps	59
	2.3.Hess x f	A	ẑi	|x,	ẑj	B |x	=	ˆzi ˆzj ˆ2f	(x),
	which is well defined since df x is zero.
	Definition 95. 1 ˆ2f 2
	ˆzi ˆzj	i,j

  iy ' ae ' to S n≠1 , seen as the zero section of T S n≠1 . Hence, the homology class of S ' is a generator of H n≠1 (F ' , Z) and in general the homology class of S t is a generator of H n≠1 (F t , Z) . The homology class of S t in H n≠1 (F t , Z) is called the vanishing sphere of the homogeneous polynomial (2.3.1).

	3	4
	x
	|x|	, y
	is a di eomorphism that sends S Definition 97.

  .5.4) Here H 1 (N ) is the group of coinvariants of H 1 (N ) for the -action. It is isomorphic to the quotient of N by the group [N, ] generated by commutators of elements of and of N . Note that if x = (1, . . . , 1, g) oe N and y = (y 1 , . . . , y a , h 1 , . . . , h b ) oe then

	xyx ≠1	y ≠1 = (1, . . . , 1, gh b g ≠1	h ≠1 b ).

Hence when we identify N with ker(h ú ), [N, ] is identified with the group [ker(h ú ), fi 1 (Y )] (we are using here that b Ø 2 or that one of the fi 1 ( j ) surjects onto fi 1 (E)). In particular the groups H 1 (N ) and H 1 (ker(h ú )) fi 1 (Y ) are isomorphic. Now the short exact sequence 0 ae ker(h ú ) ae fi 1 (Y ) ae Z 2 ae 0 induces the short exact sequence (see

[START_REF] Brown | Cohomology of groups[END_REF] VII.6

] again):

Une question importante est de savoir si ces deux classes de groupes coïncident (voir[START_REF] Claudon | The fundamental group of compact Kähler threefolds[END_REF]).

An important question is whether these two classes of groups coincide (see[START_REF] Claudon | The fundamental group of compact Kähler threefolds[END_REF]).

Remerciements

Remerciements

where π c m 1 1 , . . . , c m k k ∫ is the normal closure of {c m 1 1 , . . . , c m k k } in fi 1 (S \ ). The following lemma is a well-known result. A more general statement of this result can be found in [START_REF] Catanese | Some remarks on the universal cover of an open K3 surface[END_REF].

Lemma 46. Let X be a compact complex manifold, S be a closed Riemann surface and f : X ae S be a surjective holomorphic map whose generic fiber F is connected. If we denote by ÿ : F Òae X the inclusion map and by N the image of the induced homomorphism on fundamental groups ÿ ú : fi 1 (F ) ae fi 1 (X), we obtain a short exact sequence

Since F is compact we get that N is finitely generated. The kernel of the map fl :

fi orb 1 ( ) ae fi 1 (S) is isomorphic to ker(f ú )/N and we have the commutative diagram

fi orb 1 ( ) fl : :

The result of Gromov and Schoen is the following.

Theorem 47 (Gromov-Schoen). Let X be a compact Kähler manifold whose fundamental group acts on a tree which is not isomorphic to a line nor a point. Suppose that the action is minimal with no fixed points on the boundary. Then there is a surjective holomorphic map with connected fibers from X to a closed hyperbolic orbifold inducing the short exact sequence

such that the restriction of the action to N is trivial.

Proof of Theorem A. Let 0 be the finite index subgroup of satisfying the hypothesis of the theorem and let 0 be the preimage of 0 under the morphism ae Aut(G). Then, 0 is a finite index subgroup of containing G such that the action of G on T can be extended to 0 . By Theorem 47 there is a short exact sequence

where the restriction of the action of 0 on T to the subgroup N is trivial and fi orb 1 ( ) is virtually isomorphic to a surface group. Now, let ⁄ : G ae fi orb 1 ( ) be the morphism given by the restriction of to G. This morphism is injective since N fl G is trivial. The latter assertion follows from the faithfulness of the action of G on T . We claim that the subgroup 1 = ≠1 (⁄(G)) is the subgroup we are looking for. First of all, since for a normal subgroup of fi orb 1 ( ) being finitely generated is equivalent to having finite index, we get that ⁄(G) has finite index in fi orb 1 ( ). This implies that 1 is a finite index subgroup of 0 (and thereby of ) and that G is virtually isomorphic to a surface group. To end the proof, it su ces to observe that the morphism

x ' ae (⁄ ≠1 ¶ (x), P (x))

Surface groups and Kähler groups

In this section we use Theorem B to study Kähler extensions of solvable groups by surface groups. First, we focus on the simpler case of Kähler extensions of Abelian groups by surface groups, which gives an alternative proof of Bregman and Zhang's result. At the end of this section we present a slightly more general statement of Theorem B.

Kähler extension of an Abelian group by a surface group

Corollary 54 (Bregman-Zhang). Let S be a closed surface of genus g Ø 2, be a Kähler group and k be a positive integer such that there is a short exact sequence

Then there is a finite index subgroup 1 of containing fi 1 (S) such that the restricted short exact sequence

splits as a direct product.

Note that k must be even since 1 is a Kähler group. Before proving Corollary 54, we will recall some definitions and facts about the mapping class group of a closed oriented surface S (these can be found with more details in [START_REF] Birman | Abelian and solvable subgroups of the mapping class groups[END_REF]). Let S (S) denote the set of isotopy classes of simple closed curves in S. If • is an element of the mapping class group of S and A is a subset of S (S), we denote by

where • (-) denotes the isotopy class of t(a) for t oe • and a oe -. A subset A of S (S) is said admissible if there is a set of disjoint simple closed curves that represent all the isotopy classes of A . Notice that an admissible subset of S (S) must be finite. An element • of the mapping class group of S is said to be 1. reducible: if there is a non-empty admissible set A such that • (A ) = A , 2. pseudo-Anosov: if for any isotopy classin S (S) and for every nonzero integer n,

• n (-) is di erent from -(this is one definition of a pseudo-Anosov mapping class among many others).

For a closed oriented surface S, the Nielsen-Thurston classification Theorem states the following (see [START_REF] Farb | A primer on mapping class groups[END_REF] Chapter 13).

Theorem 55 (Nielsen-Thurston). Every element of the mapping class group of S is either of finite order, reducible or pseudo-Anosov. Furthermore, pseudo-Anosov mapping classes are neither periodic nor reducible.

To prove Corollary 54 we will need the Nielsen-Thurston classification Theorem and two lemmas. The first lemma is due to Birman, Lubotzky and McCarthy (see Lemma 3.1 in [START_REF] Birman | Abelian and solvable subgroups of the mapping class groups[END_REF]).

with S a closed surface of genus g Ø 2. If the conjugation action of on fi 1 (S) preserves the conjugacy class of a non-trivial element of fi 1 (S), then there is a finite index subgroup 1 of containing a finite index subgroup of fi 1 (S) which is normal in 1 such that the extension

splits as a direct product.

Proof. Let " be the closed curve in S whose conjugacy class is preserved by the action of . We may assume that " is not simple. By Theorem 62, there is a finite covering map S Õ ae S, such that " lifts to a simple closed curve in S Õ . Let be the fundamental group of S Õ and let Õ be the normalizer of in . By Lemma 63, Õ is a finite index subgroup of and therefore Õ is a Kähler group. Hence, the short exact sequence

satisfies the hypothesis of Theorem B, which implies the existence of a subgroup 1 of Õ of finite index (and therefore a finite index subgroup of ) such that the exact sequence

splits as a direct product.

Chapter 2. Irrational pencils and Betti numbers

Proof. We view S as the quotient of C by a lattice , so that tangent vectors to S can be identified with complex numbers. Given a point y in Y and a tangent vector v in T y Y , we have that the holomorphic map

x ' ae df (x,y) (v, 0) is constant. This follows from the compactness of X. If now we fix a point x 0 in X, the group structure on S allows us to define the holomorphic map

We deduce from the above argument that this map only depends on x and therefore f (x, y) = f 1 (x) + f (x 0 , y), which implies the result.

2

Remark 68. Lemma 67 implies that any irrational pencil f : X ae S as in Theorem 66 is the sum of holomorphic maps f i : i ae S, i.e.

If all the f i 's are nonconstant and n Ø 2, f has connected fibers if and only if it is fi 1surjective, see Lemma 2.1 in [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF]. The set of critical points of f is the product of the critical sets of the f i 's. Hence f has isolated critical points if and only if all the f i 's are nonconstant.

Let '

X ae X the covering space of X such that fi 1 ( ' X) ƒ . Under the hypothesis of Theorem 65, the authors proved that the inclusion of a smooth fiber of f in ' X induces isomorphisms on the homotopy groups of dimension 0, . . . , n ≠ 2 (see Lemma 3.3 in [31]). Hence, if f : X ae S is an irrational pencil as in Theorem 66, we get that the group is of type F n≠1 and by Theorem 66 it cannot be of type FP n as the group H n ( , Q) has infinite dimension (see Proposition 87).

Finiteness properties for arbitrary pencils

Let f : X ae S be an irrational pencil with dim C X = n Ø 2. As before, let denote the kernel of the morphism f ú : fi 1 (X) ae fi 1 (S) and let ' X ae X be the covering space such that fi 1 ( ' X) ƒ . We assume that the critical points of f are isolated and that f is not a submersion; its critical set is then a nonempty finite set. Our main results are the following:

If X is aspherical, the space ' X is a classifying space of the group (see Remark 110). Hence Theorem G follows from Theorem F. The main ingredient of the proof of Theorem F is the study of the topology of the covering space ' X (see Section 2.4.3). This method, which can be seen as a complex analog of Bestvina and Brady's work (see [START_REF] Bestvina | Morse theory and finiteness properties of groups[END_REF]) appears already in the articles [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF][START_REF] Kapovich | On normal subgroups in the fundamental groups of complex surfaces[END_REF].

When n Ø 3, the group is isomorphic to the fundamental group of a generic fiber of f , thanks to Theorem 65. Hence, when n Ø 3 and X is aspherical and Kähler (resp. projective), is a Kähler (resp. projective) group of type F n≠1 but not of type FP n .

2.2. Homology of a group, classifying spaces and finiteness properties 53

Related results

2.1.3.a Direct products of Riemann surfaces

In the special case when X is a product of Riemann surfaces and S has genus 1, Theorem G reduces to Theorem 66 above. Dimca, Papadima and Suciu's proof was based on the notion of characteristic varieties. However, the proof we will present in Section 2.4.4 is more direct and applies in full generality: X can be any aspherical complex manifold and the genus of S need not be equal to 1. On the other hand the article [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] also studies the finiteness properties of arbitrary normal coabelian subgroups of direct products of fundamental groups of closed surfaces, not necessarily coming from irrational pencils.

If X is a product of Riemann surfaces, the fact that is not of type FP n can be deduced from the work of Bridson, Howie, Miller and Short [START_REF] Bridson | The subgroups of direct products of surface groups, Dedicated to John Stallings on the occasion of his 65th birthday[END_REF]. See [START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF][START_REF] Isenrich | Kähler groups and subdirect products of surface groups[END_REF] for further results which rely on properties of subgroups of direct products of surface groups.

2.1.3.b Irrational pencils with nondegenerate critical points

Under similar assumptions as in Theorem G, Biswas, Mj and Pancholi proved the following result (see Theorem 7.3 in [START_REF] Biswas | Homotopical height[END_REF]), whose conclusion is weaker than the one of Theorem G.

Theorem 69 (Biswas, Mj and Pancholi). Let f : X ae S be an irrational pencil which is not a submersion and whose critical points are isolated. If X is an aspherical complex manifold of dimension greater or equal than 3, then the fundamental group of the generic fiber of f is not of type FP.

See Section 2.2.3 for the definition of the homological finiteness property FP.

2.1.3.c Aspherical compact Kähler surfaces

We recall the result of Kapovich on irrational pencils of aspherical compact Kähler surfaces stated in the introduction (see [START_REF] Kapovich | On normal subgroups in the fundamental groups of complex surfaces[END_REF]).

Theorem 70. Let X be an aspherical compact Kähler surface, S a compact Riemann surface of positive genus and f : X ae S an irrational pencil whose singular fibers have multiplicity one. Then f is a submersion or the kernel of the induced map on fundamental groups is not finitely presented.

In the case of maps with isolated critical points, Theorem G gives a slight strengthening of Kapovich's result since a finitely presented group must have finitely generated second homology group. A detailed proof of Theorem 70 will be given in Section 2.6. We will see as well in this section that the hypothesis on singular fibers implies that the kernel of

Homology of a group, classifying spaces and finiteness properties 2.2.1 Homology of a group

Here we summarize some definitions and results about the homology of a group. See [START_REF] Brown | Cohomology of groups[END_REF] for an introduction to this subject.

Chapter 2. Irrational pencils and Betti numbers Definition 71. Let G be a group. The integral group ring ZG is defined as the free Z-module generated by the elements of G.

An element of ZG is uniquely expressed as

where n(g) is an integer and n(g) = 0 for all but finitely many g in G.

Definition 72. A ZG-module is an Abelian group A together with an action of G on A.

Given two ZG-modules A and B, a homomorphism of ZG-modules is a homomorphism of Abelian groups Ï : A ae B such that for all g in G and for all a in A, Ï(g

Example 73. Let G be a group. The trivial action of G on Z induces a structure on Z of ZG-module.

A free ZG-module is a ZG-module of the form ü ioeI ZG. We will also use the notion of projective ZG-module, we refer the reader to [16, p. 21] for the definition.

Definition 74.

Let G be a group. A projective (resp. free) resolution of Z over ZG is an exact sequence of ZG-modules

such that P i is a projective (resp. free) ZG-module for every i Ø 0.

Let G be a group, A be a ZG-module and let us consider the subgroup of A generated by the elements of the form g • a ≠ a, where g is an element of G and a is an element of A. The quotient of A by this subgroup is called the group of co-invariants of A and it is denoted by A G . Since the group of co-invariants of A is obtained from A by identifying each element of A with all the elements of its G-orbit, we obtain that A G is a ZG-module endowed with a trivial G-action.

Remark 75.

If A is a free ZG-module with basis {a i } ioeI , and if we denote by āi the image of a i in A G , then A G is a free Abelian group with basis { āi } ioeI . Definition 76. Let G be a group and let

be a projective resolution of Z over ZG. The n-th homology group of G is defined as

Every ZG-module has a projective resolution. Then, one can always define the homology groups of G. Moreover, these groups do not depend on the choice of the projective resolution (see for instance Chapter 1 of [START_REF] Brown | Cohomology of groups[END_REF]). If M is a ZG-module, one can define the homology groups of G with coe cients in M in a similar way as before, by considering a projective resolution of M over ZG. Then, H n (G) is the n-th homology group of G with coe cients in Z. In this chapter we will work with the homology groups of a group G with coe cients in Q, where the trivial action of G on Q induces a structure of ZG-module on Q.

Classifying space

Definition 77. A classifying space of a discrete group G (or a K(G, 1) space) is a topological space X such that 1. X is path connected.

2. The fundamental group of X is isomorphic to G.

3. The universal covering space of X is contractible.

The last condition implies that X is aspherical, i.e. that fi k (X) is trivial for k Ø 2. In the context of CW-complexes (see [START_REF] Hatcher | Algebraic topology[END_REF] for an introduction to this subject), being aspherical and having a contractible universal covering space are equivalent conditions. This is a consequence of a theorem of Whitehead. Combined with a theorem of Hurewicz it implies that for a CW-complex, being aspherical is equivalent to having a universal covering space with trivial integral homology. See [START_REF] Hatcher | Algebraic topology[END_REF] for all the details of these statements. Throughout this text we will work with smooth manifolds and CW-complexes, and we will use these equivalences without recalling them.

Example 78. Since the universal covering space of

The following example shows that given a group G, one can always choose a classifying space of G which is a CW-complex. In this case the homotopy type of such a space is uniquely defined.

Example 80. Given a presentation of a group G = Èg 1 , g 2 , . . . |r 1 , r 2 , . . .Í one can construct a classifying space of G which is a CW-complex as follows. The 1-skeleton X 1 is given by a wedge-sum of circles, one for each generator g i . Hence, each generator g i of G defines a loop " i in X 1 . Now, each relation r i = g i 1 g i 2 • • • g i k defines a cellular map S 1 ae X 1 whose image is the concatenation of the loops " i 1 , " i 2 , . . . , " i k . By attaching a 2-cell to X 1 for each of this cellular maps, one obtains a CW-complex X 2 of dimension 2 whose fundamental group is isomorphic to G. Now, one can attach a 3-cell to X 2 for each generator of fi 2 (X 2 ) to obtain a CW-complex X 3 of dimension 3, whose fundamental group is isomorphic to G and such that fi 2 (X 3 ) is trivial. One can iterate this process as follows. Once the CWcomplex X k of dimension k is constructed, one can attach a (k + 1)-cell to X k for each generator of fi k (X k ) to obtain a CW-complex X k+1 of dimension k + 1 whose fundamental group is isomorphic to G and whose homotopy groups of dimension 2, . . . , k are trivial. The limit space obtained in this way is the desired space.

Let G be a group and let X be a classifying space for G which is a CW-complex. The universal covering space ' X ae X of X inherits a CW-structure and the action of G on ' X freely permutes the cells of ' X. In particular, G acts on the set of n-cells of ' X. Therefore, the free Abelian group generated by the set of n-cells of ' X denoted by C n ( ' X) is a free ZG-module with one basis element for each G-orbit of n-cells.

Therefore, one can define the ZG-module C n ( ' X) as the free Abelian group generated by the set of n-cells of ' X. The action of G on this set extends to a Z-linear action of G on C n ( ' X). By construction, one obtains that the set of n-cells of X is a basis of C n ( ' X) as a ZG-module.

Theorem 100. Let X be a complex n-manifold, and f : X ae C a proper holomorphic map. Suppose that there exists a point x in X such that f : X \ {x} ae C is a submersion and x is a nondegenerate critical point of f . If t oe C is a regular value of f su ciently close to f (x), then X deformation retracts onto the union of the fiber F t = f ≠1 (t) with a real n-ball glued to F t along a vanishing sphere of f contained in F t .

The proof of Theorem 100 can be found in [START_REF] Voisin | Théorie de Hodge et géométrie algébrique complexe[END_REF] and it essentially an application of the holomorphic Morse Lemma and Proposition 99. Indeed, by Proposition 99, the n-ball glued to F t is given by the cone over the vanishing sphere of f contained in F t with vertex at x. We can deduce from this the following result.

Corollary 101. Under the hypothesis of Theorem 100, given a neighborhood U of the nondegenerate critical point x, we can choose a regular value t su ciently close to f (x), so that X deformation retracts onto the union of the fiber F t = f ≠1 (t) with a real n-ball contained in U , glued to F t along a vanishing sphere of f contained in F t .

Milnor number and degenerate critical points

Let g 1 , . . . , g n : (C n , 0) ae (C, 0) be holomorphic map-germs and let us write g = (g 1 , . . . , g n ). Let S be a small enough sphere centered at 0 and suppose that 0 is the only solution of

in the ball bounded by S. The multiplicity of g at 0 is defined as the degree of the mapping

We recall some important results on this notion. We refer to [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF] for a detailed introduction to this subject and for the proofs of these results.

Lemma 102. Suppose that 0 is an isolated solution of (2.3.2). If the Jacobian matrix

has nonzero determinant at 0, then the multiplicity of 0 is equal to 1, otherwise it is strictly greater than 1.

Lemma 103. Let 0 be a an isolated solution of (2.3.2) of multiplicity µ and let B be a ball in C n centered at 0 which does not contain any other solution of (2.3.2). Then, for almost all points ¸in C n su ciently close to the origin, the equation g(z) = ¸has precisely µ di erent isolated solutions of multiplicity 1 within B.

Remark 104. More precisely, the latter lemma holds if ¸is a regular value of g, close enough to the origin. Sard's Theorem ensures that there are plenty of regular values.

Suppose that 0 is an isolated critical point of a holomorphic map-germ f : (C n , 0) ae (C, 0). The Milnor number µ of 0 is defined as the multiplicity at 0 of g =

each sphere S i can be identified to a sphere S ú i µ ' f ≠1 0 (c(0)) in such a way that the (S ú i ) 1AEiAEm k are disjoint. Hence X k+1 retracts onto the space obtained from X k by gluing a ball to each of the spheres S ú i . This proves the result. 2

Growth of the n-th Betti number

Here we give a proof of Theorem F Observation. Proposition 112 implies that the sequence b n≠1

Let k 0 be a large enough integer such that the sequence

is constant. We will now prove that the sequence (b n (X k )) kØk 0 is strictly increasing with k and that each map

induced by the inclusion X k Òae X k+1 is injective for k Ø k 0 . This immediately implies Theorem F since the vector space H n ( ' X, Q) is the direct limit of the H n (X k , Q)'s. We use the notation from the proof of Proposition 112. Let k Ø k 0 . We know that X k+1 has the homotopy type of a space W k+1 obtained by gluing a ball to each of the spheres

We write:

where each B j is homeomorphic to an n-dimensional ball with B j fl B l = ÿ for l " = j and X k fl B j is equal to the boundary of B j (or to the sphere S ú j depending on whether one views it inside X k or B j ). Since the inclusion of X k into X k+1 induces an isomorphism on (n ≠ 1)-dimensional homology groups, the same occurs for each inclusion X k Òae W k+1 . We now apply the Mayer-Vietoris exact sequence to the decomposition of W k+1 given in (2.4.4).

We obtain (all homology groups being with Q coe cients):

(2.4.5)

Since the groups

) are zero we obtain:

(2.4.6) The last arrow on the right hand side being an isomorphism, this implies that the following sequence is exact:

(2.4.7)

2.5. New Kähler groups with exotic finiteness properties 67 This implies that each inclusion X k Òae W k+1 (and hence the inclusion X k Òae X k+1 ) induces an injective map on H n (•, Q) and that b n (X k ) = b n (X k+1 ) + m k . Note that m k > 0. This is the desired result and concludes the proof of Theorem F.

Remark 113. Assume that X is aspherical. Then the space ' X is a classifying space of the kernel of f ú : fi 1 (X) ae fi 1 (S). As we said in the introduction, Dimca, Papadima and Suciu proved in [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF] that the inclusion of the smooth generic fiber of f in ' X induces isomorphisms on the homotopy groups of dimension 0, . . . , n ≠ 2 (see Lemma 3.3 and Corollary 5.8 in [31]). This essentially follows from Proposition 112. This implies that the kernel of f ú :

New Kähler groups with exotic finiteness properties

In this section we will construct new examples of Kähler groups with exotic finiteness properties using the Cartwright-Steger surface. We recall the definition of the complex hyperbolic space to introduce this complex surface.

Complex hyperbolic space

Let us denote by È• , •Í : C n+1 ◊ C n+1 ae C the product given by

Observe that the action of C ú on C n+1 given by the scalar multiplication, preserves the set

Definition 114. The complex hyperbolic space of dimension n, denoted by H n C , is defined as the subset of CP n given by

C , then z n+1 " = 0, otherwise Èz , zÍ would be non-negative. Moreover, the usual coordinate chart (U n+1 , Ï n+1 ) of CP n given by the homogeneous coordinate

Under this identification we can endow H n

C with a Kähler metric as follows. Let z be a point in the unit ball B of C n and let v, w oe T z B. We write

where (u, v)

The metric defined in (2.5.1) is called the hyperbolic metric.

Definition 115. The unitary group U(n, 1) of signature (n, 1) is defined as the subgroup of Gl(n + 1, C) that preserves È• , •Í.

Chapter 2. Irrational pencils and Betti numbers

The projective unitary group PU(n, 1) which is defined as the quotient of U(n, 1) by its center, acts on H n C by holomorphic isometries. Furthermore, we have the following result. Proposition 116. The group of holomorphic isometries of H n C is PU(n, 1). Hence for any torsion-free discrete subgroup of PU(n, 1), the quotient H n C / is a Kähler manifold.

Examples built from self-products of the Cartwright-Steger surface

The Cartwright-Steger surface is a smooth compact complex surface which is a quotient of

C by a discrete subgroup of PU(2, 1). It is characterized (up to changing the sign of the complex structure) by the fact that its Euler characteristic is equal to 3 and its first Betti number is equal to 2. It was discovered in [START_REF] Cartwright | Enumeration of the 50 fake projective planes[END_REF] in the context of the classification of fake projective planes. It was further studied by several authors, see for instance [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF][START_REF] Stover | On general type surfaces with q = 1 and c 2 = 3p g[END_REF][START_REF] Vidussi | The slope of surfaces with Albanese dimension one[END_REF].

We now denote by Y the Cartwright-Steger surface and by h : Y ae E its Albanese map, whose target is an elliptic curve since b 1 (Y ) = 2. Cartwright, Koziarz and Yeung [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF] have proved that the map h has isolated critical points and Koziarz and Yeung later proved that these critical points are nondegenerate [START_REF] Koziarz | Stability of the Albanese fibration on the Cartwright-Steger surface[END_REF] (see also Rito [START_REF] Rito | Surfaces with canonical map of maximum degree[END_REF]). We can thus consider the product Y b of Y with itself b times and the map

(

As we said in the introduction, this provides natural examples to which one can apply Theorems F and G. If we denote by < PU(2, 1) the fundamental group of the Cartwright-Steger surface (we refer the reader to [START_REF] Cartwright | On the Cartwright-Steger surface[END_REF] for a detailed description of this lattice), we obtain from Theorem G together with the latter construction the following result:

Theorem H. The direct product of b copies of contains a coabelian normal subgroup N which is of type FP 2b≠1 but satisfies that H 2b (N, Q) has infinite dimension.

The group N appearing above is the kernel of the morphism on fundamental groups induced by the map (2.5.2). Recall that by Remark 113, we have that N is of type FP 2b≠1 .

Besides considering the products Y ◊ • • • ◊ Y , one can also build more examples by combining the construction by Dimca, Papadima and Suciu and our construction. We fix a family of ramified covers p i : i ae E of the elliptic curve E (1 AE i AE a), where each i has negative Euler characteristic. We then consider the map

(where there are b Ø 1 copies of Y ) which is the sum of the p i 's and of the map h on each copy of Y . All the results until the end of this section also apply when a = 0, i.e. when one studies the map

The map f has a finite non-empty set of critical points and connected fibers. This last point follows from the fact that h : Y ae E has connected fibers. We denote by the kernel of the map induced by f on fundamental groups. Theorem 65 and Theorem G imply that the Q-vector space H a+2b ( , Q) has infinite dimension and that is projective if and only if 2b + a Ø 3. The following proposition shows that the group is of a di erent nature compared to the examples from [START_REF] Dimca | Non-finiteness properties of fundamental groups of smooth projective varieties[END_REF][START_REF] Isenrich | Branched covers of elliptic curves and Kähler groups with exotic finiteness properties[END_REF].

Proposition 117. No finite index subgroup of embeds in a direct product of surface groups.

Milnor's Fibration Theorem

Let f : C n ae C be a nonconstant holomorphic map that vanishes at the origin and let S Á be the real (2n ≠ 1)-sphere of radius Á centered at the origin. We write

Assume that 0 is a critical point of f (not necessarily isolated). Then, Milnor's Fibration Theorem states the following.

Theorem 121. There exists Á 0 such that for all Á AE Á 0 , the space S Á \ K is a smooth fiber bundle over S 1 with projection mapping

The following results give a description of the topology of the fibers of Milnor's Fibration Theorem. The first one is a result of Milnor (see [START_REF] Milnor | Singular points of complex hypersurfaces[END_REF]) and the second one of A'Campo (see [START_REF] Campo | Le nombre de Lefschetz d'une monodromie[END_REF]) Theorem 122. The fiber of Milnor's Fibration Theorem is parallelizable and has the homotopy type of a CW-complex of dimension n ≠ 1.

Theorem 123. If the fiber of Milnor's Fibration Theorem has the homotopy type of a point, then 0 is regular point of f .

Reeb Stability Theorem

We recall some facts on the holonomy of foliations that we will need later.

Let X be a smooth manifold endowed with a foliation F and let L be a leaf of F. Given a simple closed curve " in L, the holonomy of the foliation describes the behavior of the foliation in a neighborhood of ". Given a base point x in L and a transverse space D to the foliation containing x, the holonomy of the foliation is given by a surjective map

where G is a subgroup of the germ of di eomorphisms of D (see [START_REF] Candel | Foliations. I[END_REF] for the details on this subject). The following result is a version of Reeb Stability Theorem (see §2.4 of [START_REF] Candel | Foliations. I[END_REF]) Theorem 124. Let (X, F) be a foliated manifold. If there is a compact leaf L in F with trivial germinal holonomy group, then there exists a neighborhood of L in X that is a union of leaves that are homeomorphic to L.

Remark 125. If we suppose L to be contained in a compact subset of X instead of being compact, the conclusion of Theorem 124 holds. In the proof of Theorem 124 the compactness of L guarantees that there is a finite number of plaques covering L which can be done as well if L is contained in a compact subset of X. The rest of the proof is not a ected by this change of hypothesis.

Let f : X ae S be a holomorphic map with connected fibers from a complex manifold of dimension n to a Riemann surface. If s is a singular value of f such that the critical points of the singular fiber F s = f ≠1 (s) are non-isolated, then F s decomposes as the union of a finite number of irreducible components Z 1 , . . . , Z k . Given a smooth point p in an irreducible component Z j , there exist local coordinates (z 1 , . . . , z n ) centered at p such that f is locally given by the map (z 1 , . . . , z n ) ' ae z

The integer m j does not depend on the choice of the smooth point p in Z j and it is called the multiplicity of the irreducible component Z j . The multiplicity m(s) of the singular fiber f ≠1 (s) is defined as the greatest common divisor of m 1 , . . . , m k .

2.6.3.a Irrational pencils with singular fibers of multiplicity one

Let ' X be the universal fiber product of an irrational pencil f : X ae S with singular fibers of multiplicity one and let ' f : ' X ae ' S be a lift of f (see Section 2.4.2). As we saw in Section 108, ' f and f have the same fibers and ' f is locally defined as f (see Corollary 109). Therefore, ' f : ' X ae ' S has connected fibers and all the singular fibers have multiplicity 1. As in Section 2.4.3, let (D k ) kØ1 be a sequence of closed disks of ' S such that

For such a sequence (D k ) kØ1 , we set

Let F be a regular fiber whose image is contained in D 1 . In Chapter 1 Section 1.4.1, we saw that a surjective holomorphic map with connected fibers from a compact complex manifold to a closed Riemann surface induces an orbifold structure on the Riemann surface. The latter assertion holds as well for a proper holomorphic map with connected fibers from a non-compact complex manifold to a Riemann surface. In this case, the orbifold structure is given by a discrete set of marked points on the Riemann surface that might be infinite. Hence, the lift ' f : ' X ae ' S induces an orbifold structure on ' S and we obtain an exact sequence

Since the singular fibers of ' f have multiplicity one, we get that fi orb 1 ( ) = fi 1 ( ' S), which is trivial since ' S is simply connected. Therefore the map fi 1 (F ) ae fi 1 ( ' X) induced by the inclusion i : F Òae ' X, is surjective. The compactness of F implies that fi 1 (F ) is finitely generated. We fix some loops " 1 , . . . , " m : [0, 1] ae F with the same base point such that the set of homotopy classes {[" 1 ], . . . , [" m ]} generates fi 1 (F ). Hence, the set {i ú [" 1 ], . . . , i ú [" m ]} generates fi 1 ( ' X). Notice that F is contained in X k for all positive integer k and the same argument as before shows that fi 1 (F ) generates fi 1 (X k ).

Chapter 2. Irrational pencils and Betti numbers exists a positive integer k 0 such that for all k Ø k 0 , the inclusion map X k Òae ' X induces an isomorphism on fundamental groups. We fix k Ø k 0 .

As in the proof of Proposition 112, let c : [0, 1] ae Int(D k+1 ) be an embedded arc going from a boundary point c(0) of D k to the unique critical value contained in D k+1 ≠ D k . We assume that c(t) / oe D k for t > 0 and let D ú be a small disk centered at c(1) and contained in Int(D k+1 \ D k ). We may assume as well that f ≠1 (D ú ) is contained in a neighborhood of F s satisfying the conclusion of Corollary 132.

We denote by p : Â X ae X the universal covering space of X and we write,

The desired contradiction to prove Theorem 70 will follow from studying the following fragment of the Mayer-Vietoris sequence associated to U and V .

(2.6.3)

Let ' X be the universal fiber product of f . Since p : Â X ae X is the universal covering space of X and ' q : ' X ae X is a covering space of X, there exists a covering map p 1 : Â X ae ' X such that the diagram

commutes and p = ' q ¶ p 1 . We obtain from this that • U fl V deformation retracts onto  F := p ≠1 (F ), where F is a regular fiber of f , whose image is contained in D ú .

• U deformation retracts onto  X k := p ≠1 1 (X k ). •  X k+1 := p ≠1 1 (X k+1 ) deformation retracts onto U fi V . By Proposition 107, we get that  F is a connected surface contained in  X, since the inclusion F Òae X induces a surjective map on fundamental groups. The asphericity of X implies that  F is non-compact (see Remark 131). Therefore, H 2 (  F ) is trivial. We next observe that Â

X k and  X k+1 are simply connected, since the inclusions X k , X k+1 Òae ' X induce isomorphisms on fundamental groups. Hence (2.6.3) can be simplified to the exact sequence

(2.6.4)

Lemma 133. The map H 2 (  X k ) ae H 2 (  X k+1 ) induced by the inclusion  X k Òae  X k+1 is injective and H 2 (  X k+1 ) is non-trivial.

The injectivity of H 2 ( Â X k ) ae H 2 ( Â X k+1 ) follows directly from the exactness of (2.6.4). Then Lemma 133 follows from the following result (by the exactness of (2.6.4)).

Chapter 2. Irrational pencils and Betti numbers By Corollary 132, the map fi 1 (Y ) ae fi 1 (X) has infinite image, and therefore p ú (fi 1 (  Y )) is an infinite index subgroup of fi 1 (Y ). This implies that  Y ae Y is an infinite covering map. We deduce from this that  Y is non-compact and thus, so is W . 2 Now, we can give the proof of Theorem 70

Proof of Theorem 70. On the one hand, by Lemma 133, for large enough k, the map

) induced by the inclusion  X k Òae  X k+1 is injective. Since H 2 (  X) is the direct limit of H 2 (  X k ) kØ1 , we get that for large enough k, the map H 2 (  X k ) ae H 2 (  X) is injective. From this and the asphericity of X, we deduce that for large enough k, H 2 (  X k ) is trivial.

On the other hand, by Lemma 133, one can find large enough k such that H 2 ( Â X k ) is non-trivial, which is the desired contradiction.