Jacques Chirac

Quand est venu le moment d'écrire ces remerciements qui formeront la partie de cette thèse qui sera sans doute la plus lue, je me suis maintes fois demandé comment être original si ce n'est en écrivant que je me suis posé la question de l'originalité dans les remerciements des thèses. C'est une des ouvertures de recherche de ce manuscrit et je suis prêt à m'y investir : donnez moi un financement, trois ans et je vous promets d'y réfléchir.

De manière classique donc mais ô combien sincère, je souhaite d'abord adresser très chaleureusement ma gratitude à mon directeur de thèse, Olivier Guichard. Je lui suis redevable de tout ce qu'il m'a apporté, tant sur le plan professionnel qu'humain, m'aidant à me relever de chaque chute mathématique et morale, avec une patience et une abnégation sans failles. J'ai été chanceux d'avoir fait cette thèse sous ta direction inspirante. Infiniment merci Olivier.

Je souhaite ensuite exprimer ma reconnaissance vive à mes rapporteurs Bill Goldman et Florent Schaffhauser pour l'honneur qu'ils m'ont fait d'avoir accepté d'écrire un rapport. Un immense merci à mes examinateurs Julien Marché et Maxime Wolff, votre intérêt pour ces travaux, les discussions que nous avons eues et vos conseils précieux furent d'une grande aide à cette thèse. Je souhaite témoigner une gratitude particulière à Charles Frances pour avoir non seulement accepté d'être examinateur de ma thèse, mais aussi pour tout ce qu'il m'a apporté comme enseignant, collègue ainsi que pour ses sympathie et bienveillance constantes.

Cette thèse aura été l'occasion de connaître les joies des collaborations. En ça, je remercie grandement Gianluca Faraco, Arnaud Maret, Suzanne Schlich et Clément Guérin pour toutes nos discussions autour de thèmes passionnants qui ne me quitteront pas et pour leurs amitiés qui me laisseront un souvenir indéfectible.

Que seraient ces remerciements si je n'adressais pas un mot privilégié à Thibault Lorscheider. Je ne suis pas toujours certain de réellement mériter sa loyauté, mais qu'importe mon éligibilité à celle-ci sa présence lors de ces sept dernières années a été salutaire. Merci pour tout l'ami.

Je tiens également à remercier Renan Laureti pour son soutien et son amitié indefectible. Je souhaite émettre une pensée affectueuse à mes frères mathématiques. A mes aînés Clément, suscité, et Stéphane Marséglia pour leurs conseils toujours bienveillants et avisés. Un immense merci à l'actuel benjamin Clarence Kineider pour sa bonne humeur et les bons moments partagés.

Je dois évidemment une salutation à mes cobureaux Francisco Nicolas, Pierre-Alexandre Arlove, Djibril Gueye, Valdo Tatitscheff, Martin Mion-Mouton, Guillaume Ferrière et Thomas Saigre. Ils ont tous été les soleils de mes journées au bureau 114.

Ces trois ans et demi auront été pavés de rencontres formidables au sein des doctorants de l'IRMA. Aussi je remercie Philippe, Laura, Victoria, Florian, Clément et Adam pour ces bribes de bonheur pur que m'ont offert nos cafés, repas, soirée et autres réjouissances. Votre existence aura été essentielle à cette thèse. Plus modestement

Introduction en français

Soient Γ un groupe de type fini et G un groupe de Lie semi-simple. L'espace des représentations Hom(Γ, G) est l'ensemble de tous les homomorphismes Γ → G que nous munissons de la topologie compact-ouverte. Le groupe G agit par conjugaison sur Hom(Γ, G). En effet, pour g ∈ G et ρ : π 1 Σ → G, nous définissons la représentation g •ρ comme étant le morphisme γ → gρ(γ)g -1 . Le quotient de Hom(Γ, G) par l'action de G n'est généralement pas séparé. La variété des caractères est le quotient Hausdorff Hom(Γ, G)//G et sera notée par X (Γ, G). Dans le cas plus général où G est un groupe algébrique complexe, nous pourrions définir la variété des caractères comme le quotient GIT de l'espace des représentations par l'action de G. Cependant, si G est un groupe réductif, le quotient GIT et le quotient Hausdorff coïncident. Dans cette thèse, G est supposé être un groupe de Lie réel semi-simple, cette question n'intervient pas. La variété des caractères contient des points singuliers : Définition 1. Une représentation est régulière si son centralisateur, qui est le stabilisateur pour l'action par conjugaison de G, est discret.

Soit Rep(Γ, G) le sous-ensemble de X (Γ, G) formé des classes de conjugaison de représentations régulières. C'est un sous-espace dense, ouvert et lisse. Dans Goldman 1, lorsque Γ est le groupe fondamental d'une surface Σ orientée, compacte, connexe et de genre g ≥ 2, Goldman construit une structure symplectique sur Rep(π 1 Σ, G) issue de la forme de Killing sur G. La forme symplectique de Goldman induit une mesure de Radon λ sur Rep(π 1 Σ, G). Cette mesure s'étend naturellement en une mesure de Radon sur X (π 1 Σ, G), que nous noterons encore λ, telle que Rep(π 1 Σ, G) est de mesure pleine pour λ. Cette mesure a de plus le bon goût d'être dans la classe de Lebesgue de X (π 1 Σ, G).

La variété des caractères encodent des structures géométriques et est l'espace des modules d'objets géométriques :

Exemple 1. Supposons que Γ est un groupe de surface et que G = PSL 2 (R), c'est à dire que G est le groupe d'isométrie du demi-plan de Poincaré. D'après les travaux de Goldman, le second point du théorème 5 précisément, la variété des caractères X (π 1 Σ, PSL 2 (R)) contient alors deux composantes connexes formées de classes de conjugaison d'holonomies de structures hyperboliques sur Σ. En particulier, ces composantes sont toutes deux homéomorphes à l'espace de Teichmüller de Σ.

Exemple 2. Supposons que Γ est le groupe fondamental d'une surface de Riemann Σ et que G = SU(n). Le théorème de Narasimhan-Seshadri assure l'existence d'une bijection entre Rep(π 1 Σ, SU(n)) et l'espace des modules de fibrés vectoriels holomorphes de rang n, semi-stables et de degré 0 sur Σ, voir Donaldson 2018.

Ces deux exemples suggèrent la richesse géométrique de l'étude des variétés de caractères.

Dans une direction plus dynamique, le groupe des automorphismes de Γ agit par pré-composition sur l'espace des représentations. Précisément, pour ψ ∈ Aut(Γ) et ρ ∈ Hom(Γ, G), nous définissons ψ • ρ comme la représentation γ → ρ(ψ -1 (γ)). De plus, l'action du groupe des automorphismes intérieurs de Γ est contenue dans l'action par conjugaison de G sur Hom(Γ, G). En effet, soit ψ x : γ → xγx -1 un automorphisme intérieur et ρ ∈ Hom(Γ, G), alors ψ x • ρ : γ → ρ(x -1 )ρ(γ)ρ(x) est conjuguée à ρ. Nous obtenons ainsi une action de Out(Γ), défini comme le quotient Aut(Γ) Inn(Γ) , sur la variété des caractères via la formule

[ψ] • [ρ] = [ψ • ρ].
Cette action préserve la mesure de Goldman et implique une dynamique mesurable sur la variété des caractères. L'objet de cette thèse est d'étudier les propriétés dynamiques de cette action.

Dans les travaux que cette thèse présente, Γ est le groupe fondamental d'une surface fermée, connexe et compacte Σ. Le théorème de Dehn-Nielsen-Baer, voir Farb and Margalit 2011, nous assure alors de l'existence d'un isomorphisme entre Out(Σ) et le groupe modulaire de Σ, que nous noterons par Mod(Σ). Ce groupe est le groupe des difféomorphismes de Σ à isotopie près. Nous noterons par Mod + (Σ) le sous-groupe des classes d'isotopies de difféomorphismes directs de Σ. Ce point de vue permet une description topologique de Out(Σ) et nous procure une approche géométrique de son action. Introduisons certains sous-groupes particuliers du groupe modulaire :

• Le groupe de Torelli est le noyau de l'action de Mod + (Σ) sur le premier groupe d'homologie de Σ, H 1 (Σ, Z).

• Le premier sous-groupes de Johnson est le sous-groupe de Mod + (Σ) engendré par les twists de Dehn le long de courbes séparantes.

Remarquons que le premier sous-groupe de Johnson est inclus dans le groupe de Torelli.

Cas compact

Une action d'un groupe H sur un espace mesuré (X, µ), dont nous supposons la H-invariance de µ, est ergodique si tout sous-ensemble mesurable et H-invariant de X est de mesure nulle ou a un complémentaire de mesure nulle (nous dirons que l'ensemble est de mesure pleine si cette condition est vérifiée). En particulier, si H est un sous-groupe de H et si H agit sur un espace mesuré en préservant la mesure, l'ergodicité de l'action de H implique l'ergodicité de l'action de H. L'exemple classique d'une action ergodique est celui d'une translation irrationnelle du tore T n = R n /Z n . Le groupe engendré par le vecteur (a 1 , . . . , a n ) ∈ R n agit par translation sur T n et préserve sa mesure de Haar. L'action de (a 1 , . . . , a n ) sur T n est ergodique par rapport à la mesure de Haar si et seulement si les réels a 1 , . . . , a n , 1 sont linéairement indépendants sur Q.

Théorème 1. (Goldman, Pickrell-Xia) Soit G un groupe de Lie compact et soit Γ = π 1 Σ un groupe de surface. Le groupe modulaire Mod + (Σ) agit ergodiquemment sur les composantes connexes de la variété des caractères X (π 1 Σ, G) par rapport à la mesure symplectique de Goldman.

Ce théorème a été démontré pour la première fois par Goldman dans Goldman 1 lorsque G est localement isomorphe à SU(2) puis Pickrell et Xia ont traité le cas général dans Pickrell-Xia 1. Certaines questions peuvent être posées à partir de l'ergodicité, en particulier la possibilité d'avoir des propriétés plus précises : trouver des sous-groupes propres de Mod + (Σ) qui agissent ergodiquement sur les variétés de caractères.

Funar et Marché ont prouvé dans Funar-Marche le résultat suivant :

Théorème 2. (Funar-Marché) Le premier sous-groupe de Johnson agit ergodiquement sur la variété de caractères X (π 1 Σ, SU(2)) par rapport à la mesure de Goldman.

En particulier :

Théorème 3. Le sous-groupe de Torelli agit ergodiquement sur la variété de caractères X (π 1 Σ, SU(2)) par rapport à la mesure de Goldman.

La preuve utilise des développements de Taylor des fonctions de traces associées aux courbes séparantes en la représentation triviale.

Nous proposons une nouvelle preuve du théorème 3 et généralisons celle-ci au cas des groupes de Lie semi-simples et compacts : Théorème A. Soient une surface fermée, compacte, connexe et orientée Σ de genre g ≥ 3 et G un groupe de Lie connexe, semi-simple et compact. Le groupe de Torelli agit ergodiquement sur les composantes connexes de la variété de caractères X (π 1 Σ, G) par rapport à la mesure de Goldman.

Pour prouver le théorème A, nous montrons que toute fonction mesurable f : X (π 1 Σ, G) → R qui est invariante par l'action du groupe de Torelli est presque partout invariante par le groupe modulaire. Pour ce faire, nous introduisons pour chaque paire de courbes co-bordantes, un sousensemble de X (π 1 Σ, G) pour lequel nous prouvons qu'il est de mesure pleine. Sur cet ensemble, chaque fonction invariante (sous l'action du groupe de Torelli) est presque partout invariante sous l'action des twists de Dehn des courbes de la paire. Il suffit donc de restreindre une fonction qui est invariante par le groupe de Torelli à une intersection dénombrable de ces sous-ensembles pour qu'elle soit presque partout invariante par le groupe modulaire.

En corollaire, quitte à remplacer G par un produit fini

G × • • • × G et identifier X (π 1 Σ, G k ) à X (π 1 Σ, G) k , nous obtenons :
Théorème B. Soient Σ et G respectivement une surface et un groupe vérifiant l'hypothèse du théorème A. Alors, pour tout k ≥ 1, le groupe de Torelli Tor(Σ) agit ergodiquement sur chaque composante connexe du produit X (Γ, G) k . En particulier, sur les composantes de X (π 1 Σ, G), l'action du groupe de Torelli est faiblement mélangeante.

Cas de PSL 2 (R)

Soit Σ une surface connexe, orientée et fermée de genre g ≥ 2. Son groupe fondamental admet la présentation finie : se projette sur l'identité. On en déduit que g i=1 [ ρ(a i ), ρ(b i )] est un élément de π 1 PSL 2 (R) ∼ = Z. Cet entier ne dépend que de ρ et en aucun cas du choix des relevés. Nous le noterons donc eu(ρ). L'application eu : Hom π 1 Σ, PSL 2 (R) → Z est de plus invariante par l'action de PSL 2 (R) par conjugaison et descend donc comme une application eu : X → Z. Dans Milnor 1957/58, Milnor montre 2. Une représentation ρ : π 1 Σ → PSL 2 (R) est maximale si et seulement si c'est l'holonomie d'une structure hyperbolique sur Σ.

Le deuxième point de ce théorème montre que X ±|χ(Σ)| est homéomorphe à l'espace de Teichmüller de Σ.

Action du groupe modulaire pour les surfaces fermées

Il est bien connu que l'action du groupe modulaire sur l'espace de Teichmüller de Σ est proprement discontinue : pour tout sous-ensemble compact K de l'espace de Teichmüller de Σ, l'ensemble

[φ] ∈ Out(π 1 Σ) | [φ] • K ∩ K = ∅ est fini.
Le comportement de cette action sur les composantes non-maximales est diamétralement différent. L'existence d'éléments γ ∈ π 1 Σ qui sont envoyés sur des isométries non-hyperboliques permet de s'approcher des arguments utilisés dans le cas compact, voir Goldman and Xia 2011. Dans W. [START_REF] Goldman | MAPPING CLASS GROUP DYNAMICS ON SURFACE GROUP REPRE-SENTATIONS[END_REF], Goldman a conjecturé : Conjecture 1. (Conjecture 3.1, W. Goldman 2006) Soit k ∈ Z∩ χ(Σ), -χ(Σ) . Le groupe modulaire agit ergodiquement sur X k par rapport à la mesure λ.

En reliant cette conjecture à la condition de Bowditch, voir Bowditch 1998, et En genre 2 et pour la classe d'Euler 0, la composante X 0 se décompose en deux sous-espaces disjoints X 0 -et X 0 + de mesures non nulles et sur lesquels le groupe modulaire agit ergodiquement par rapport à λ, [START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF][START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF][START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF] Cas des variétés de caractères relatives dans PSL 2 (R) Soit Σ une surface connexe, orientée et compacte de genre g ≥ 1 et à n > 0 composantes de bord, telle que χ( Σ) = 2 -2g -n < 0. Le groupe fondamental π 1 Σ de Σ est le groupe libre :

a 1 , b 1 , . . . , a g , b g , c 1 , . . . c n | g i=1 [a i , b i ]

Une représentation ρ

: π 1 Σ → PSL 2 (R)
(R) et C := (C 1 , . . . , C n ), on définit l'espace Hom Σ, PSL 2 (R), C des représentations ρ : π 1 Σ → PSL 2 (R) telles que pour tout i ∈ {1, . . . , n}, ρ(c i ) ∈ C i .
Le groupe PSL 2 (R) agit sur l'espace Hom Σ, PSL 2 (R), C par conjugaison. Définition 3. La variété de caractères relative associée à C, notée X ( Σ, C), est le quotient Hausdorff

Hom Σ, PSL 2 (R), C //PSL 2 (R).
Notons Rep( Σ, C) le sous-espace des classes de représentations Zariski-denses de X ( Σ, C). Cet ensemble est une variété lisse de dimension 6g -6 + 2n, voir Mondello 2017. Le groupe modulaire pur de Σ, noté PMod( Σ) est le groupe des classes d'isotopies de difféomorphismes de Σ qui agissent trivialement sur ∂ Σ. Il agit naturellement sur les variétés de caractères relatives de manière similaire à celle décrite pour l'action du groupe modulaire dans le cas des surfaces fermées et préserve les composantes connexes de la variété des caractères relative ainsi que la mesure de Goldman.

Définition 4. Soit ρ : π 1 Σ → PSL 2 (R) et fixons une relevé ρ : π 1 Σ → PSL 2 (R).

Résultats

Nous démontrons un analogue du théorème 6 de Marché-Wolff pour les surfaces à bord : soient La preuve de ce théorème utilise essentiellement la même stratégie que la preuve du théorème 6.6 de Marché and Wolff 2016.

C 1 , . . . , C n des classes de conjugaison d'isométries elliptiques et soit e = ±(k -||{r}|| 1 ) avec k ∈ Z ∩ ||{r}|| 1 , -χ( Σ) . Alors le sous-espace N H e ( Σ,
Nous conjecturons que le résultat dynamique du théorème 6 peut être renforcé par l'existence d'un sous-groupe propre agissant ergodiquement sur N H k .

Conjecture A. Si g ≥ 3 et k est un entier tel que |k| ≤ 2g -5, alors le groupe Tor(Σ) agit ergodiquement sur les sous-espaces N H k par rapport à λ.

Ce résultat serait une version non-compacte du théorème A. Nous developpons une stratégie en section ?? et expliquons en quoi notre proposition est reliée à un analogue de la conjecture de Goldman.

Dynamique topologique

L'exemple de l'action des translations sur les tores nous donne une intuition dynamique des orbites. Dans le cas du cercle, on voit facilement que l'action d'une rotation d'angle irrationnel n'a que des orbites denses. Ce phénomène chaotique peut être vu en dimension supérieure et est un résultat général provenant de l'ergodicité : Proposition 1. Soit H un groupe topologique agissant sur un espace borélien (X, µ) et qui préserve la mesure µ. Supposons que chaque ouvert non-vide a une mesure non nulle et que la topologie de X est engendrée par une base dénombrable d'ouverts. Si l'action de H sur (X, µ) est ergodique, alors presque toutes les orbites sont denses.

Malheureusement

Lien avec le théorème d'approximation de Kronecker

Le résultat dynamique fourni par le théorème D trouve une application en théorie géométrique des nombres. Un théorème important de cette théorie est le théorème de Kronecker concernant l'approximation diophantienne inhomogène.

Théorème E. Pour g ≥ 1. Soit b (i) = b (i) 1 , . . . , b (i) 2g , avec i = 1, . . . , n, des vecteurs de R 2g tel que b (1) , . . . , b (n) , πe 1 , . . . , πe 2g sont linéairement indépendants sur Q dans l'espace vectoriel R 2g . Soit A ∈ M n, 2g; R) une matrice réelle et ε un nombre positif. Alors il existe un élément K ∈ Sp 2g (Z) tel que A -B K < Cε mod 2π.
(1)

Introduction

Let Γ be a finitely-generated group and G be a semi-simple Lie group. The representation space Hom(Γ, G) is the set of all homomorphisms Γ → G we endow with the compact-open topology.

The group G acts by conjugation on the set Hom(Γ, G). Precisely, for g ∈ G and ρ : π 1 Σ → G, we define by g • ρ the representation γ → gρ(γ)g -1 . The quotient of Hom(Γ, G) by the action of G, is generally not Hausdorff. The character variety is the Hausdorff quotient Hom(Γ, G)//G and will be denoted by X (Γ, G). In the more general case when G is a complex algebraic group, we could define the character variety as the GIT quotient of the representation space by G. However, if G is reductive the GIT quotient and the Hausdorff quotient coincide. In this thesis, G is assumed to be a semi-simple real Lie group. The character variety may be singular at some points.

Definition 1. A representation is regular if its centralizer, which is its stabilizer under the conjugation action of G, is discrete.

Let Rep(Γ, G) be the subset of X (Γ, G) of classes of regular representations. It is a dense and open subset of the character variety. This subspace is moreover smooth. In Goldman 1, when Γ is the fundamental group of a connected, compact and oriented surface Σ of genus g ≥ 2, Goldman constructs a symplectic structure on Rep(Γ, G) which comes from the Killing form on G. The Goldman symplectic form gives rise to a Radon measure λ on Rep(Γ, G), which we extend naturally to a Radon measure on X (Γ, G), we will denote by λ too, and such that Rep(Γ, G) has full measure. This measure lies in the Lebesgue class of X (Γ, G).

Geometrically, character varieties often encode structures and are moduli spaces of geometric objects. The two following examples explain this geometric point of view.

Example 1. Assume that Γ = π 1 Σ for a closed, connected and oriented surface Σ of genus g ≥ 2 and that G = PSL 2 (R). Goldman works and precisely second point of Theorem 5 imply that the character variety X (π 1 Σ, PSL 2 (R)) contains two connected components of classes of representations which are holonomies of hyperbolic structures on Σ. In particular, these components are both isomorphic to the Teichmüller space of Σ.

Example 2. Assume that Γ is the fundamental group of a Riemann surface Σ and that G = SU(n). Then, by Narasimhan-Seshadri theorem, the character variety X (π 1 Σ, SU(n)) is in bijection with the moduli space of holomorphic vector bundles of rank n, semi-stable and of degree 0 on Σ, see Donaldson 2018.

These two examples explain the geometric richness of the study of character varieties and the reasons of this point of view.

The automorphisms group of Γ acts by pre-composition on the representation space: precisely, for ψ ∈ Aut(Γ) and ρ ∈ Hom(Γ, G), we define ψ • ρ as the representation γ → ρ(ψ -1 (γ)). Moreover the action of the group of inner automorphisms of Γ is contained in the action by conjugation of G on Hom(Γ, G) in the following sense: let ψ x : γ → xγx -1 be an inner automorphism and ρ ∈ Hom(Γ, G), then ψ x • ρ : γ → ρ(x -1 )ρ(γ)ρ(x) which is in the conjugacy class of ρ. We hence have an action of the group Out(Γ), which is the quotient Aut(Γ) Inn(Γ) , on the character variety via the formula

[ψ] • [ρ] = [ψ • ρ].
This action preserves the Goldman measure and define a measurable dynamic on the character varieties.

In a lot of case we treat in these works, Γ is the fundamental group of a closed, connected and oriented surface Σ. The Dehn-Nielsen-Baer theorem ensures of an isomorphism between Out(Σ) and the mapping class group of Σ, we will denote by Mod + (Σ), see Farb and Margalit 2011. This group is the quotient of the group of positive diffeomorphisms of Σ by the equivalence relation of isotopy. This point of view allows to describe the group Out(Σ) well and to have a geometric approach of this action.

We introduce briefly two remarkable subgroups of the mapping class group:

• The Torelli subgroup of Mod + (Σ) is the kernel of the action of Mod + (Σ) on the homological subspace H 1 (Σ, Z).

• The first Johnson subgroups is the subgroup of Mod + (Σ) generated by Dehn twists along separating and simple curves.

Denoting by tw c ∈ Mod + (Σ) the Dehn twist along c, the Torelli group is generated by the mapping classes of the form tw c tw -1 c , for two cohomologuous curves c and c and by the Dehn twists along separating simples curves.

Compact case

An action of a group H on a measurable space (X, µ), where we assume the measure µ to be H-invariant, is ergodic if every measurable and H-invariant subset of X has null measure or is of full measure, this means that its complement has null measure. In particular, if H is a subgroup of H and if H acts on a measurable set, preserving the measure, then, if the action of H is ergodic, the action of H is ergodic. The most classical example of an ergodic action is the case of the n-torus

T n = R n /Z n .
The group generated by a vector (a 1 , . . . , a n ) ∈ R n acts by translation of T n and preserve the Haar measure. The action of (a 1 , . . . , a n ) on T n is ergodic with respect to the Haar measure if and only if the real numbers a 1 , . . . , a n , 1 are linearly independent over Q.

Theorem 1. (Goldman, Pickrell-Xia) Let G be a compact Lie group and let Γ = π 1 Σ be the fundamental group of a closed, connected and oriented surface Σ. Then the modular group Mod + (Σ) acts ergodically on the character variety X (π 1 Σ, G) with respect to the Goldman measure.

This theorem was first proved by Goldman in Goldman 1 when G is locally isomorphic to SU(2) and Pickrell-Xia treated the general case in Pickrell-Xia 1. Some questions can be asked from the ergodicity, especially the possibility to have stronger properties: find proper subgroups of Mod + (Σ) which acts ergodically on character varieties.

Funar and Marché proved in Funar-Marche the following:

Theorem 2. (Funar-Marché) The first Johnson subgroup act ergodically on the character variety X (π 1 Σ, SU(2)) with respect to the Goldman measure.

From this theorem and the inclusion of the first Johnson subgroup in the Torelli subgroup, we can prove directly the following corollary:

Theorem 3. The Torelli subgroup acts ergodically on the character variety X (π 1 Σ, SU(2)) with respect to the Goldman measure.

The proof use Taylor series at the trivial representation of traces functions associated to separating curves.

We propose a new proof of theorem 3 and generalize this one for semi-simple and compact Lie groups:

Theorem A. Let Σ be a closed, compact, connected and oriented surface of genus g ≥ 3 and let G be a connected, semi-simple and compact Lie group. Then the Torelli group acts ergodically on each connected component of the character variety X (π 1 Σ, G) w.r.t Goldman measure.

To prove Theorem A, we show that any measurable function f : X (π 1 Σ, G) → R which is invariant by the Torelli subgroup is almost everywhere invariant by the mapping class group. To do that we introduce for each bounding pair, a subset of X (π 1 Σ, G) for which we prove that it has full measure. On this full measure set, every invariant function (under the Torelli group action) is almost-everywhere invariant under the action of the Dehn twists of the curves of the pair. It is so sufficient to restrict a function which is invariant by Torelli group to an intersection of enough of these subsets to be mapping class group invariant.

As corollary, replacing G by a finite product

G × • • • × G and identifying X (π 1 Σ, G k ) with X (π 1 Σ, G) k , we obtain: Theorem B.
Let Σ and G be a surface and a group verifying the hypothesis of theorem A. Then, for all k ≥ 1, the Torelli group Tor(Σ) acts ergodically on each connected component of the product

X (π 1 Σ, G) k .
In particular on these components, the action of the Torelli group is weakly mixing.

Non-compact Lie group case

Let Σ be a closed surface of genus g ≥ 2. Its fundamental group is finitely generated and admits the classical presentation The character variety has singular points but contains a dense and open subset of regular points. Precisely, a representation ρ : π 1 Σ → PSL 2 (R) is Zariski-dense if its image is a Zariski-dense subgroup of PSL 2 (R). Let Hom ZD Σ, PSL 2 (R) be the subset of Hom Σ, PSL 2 (R) of Zariski-dense representations and let Rep be its projection in the character variety. In W. M. Goldman 1984, Goldman proved that Rep is a smooth manifold of dimension 6g -6. Goldman constructed a symplectic form on Rep, we will denote by ω. This symplectic structure gives rise to a Radon measure, which will be denoted by λ. We extend this Radon measure to a Radon measure X , we will denote again by λ, such that:

π 1 Σ ∼ = a 1 , b 1 , . . . , a g , b g | g i=1 [a i , b i ] = 1 .
λ X \ Rep = 0.

Connected components for the closed case

In this paragraph, we introduce the Euler class. It is an invariant associated to each representation and which is known to classify the connected components of X .

Let ρ : π 1 Σ → PSL 2 (R) be a representation. For each generator x ∈ {a 1 , b 1 , . . . , a g , b g }, we choose a lift ρ(x) in the universal cover PSL 2 (R) of PSL 2 (R). Hence the product

g i=1 [ ρ(a i ), ρ(b i )]
projects on the identity. We hence deduce that g i=1 [ ρ(a i ), ρ(b i )] lies in π 1 PSL 2 (R) ∼ = Z. This integer only depends on ρ and not on the choice of the lifts. We hence will denote it by eu(ρ). The map eu : Hom Σ, PSL 2 (R) → Z is moreover invariant by the action of PSL 2 (R) by conjugation and thus descends to a map eu : X → Z.

In Milnor 1957/58, Milnor shows that this invariant is bounded: 1. For all k ∈ Z∩ χ(Σ), -χ(Σ) , the subspace eu -1 (k) is non-empty and the connected components of X are exactly the eu -1 (k) for k ∈ Z ∩ χ(Σ), -χ(Σ) . We will denote them by X k and by

Rep k their intersections with Rep.

2. A representation ρ : π 1 Σ → PSL 2 (R) is maximal if and only if it is the holonomy of a hyperbolic structure on Σ.

The second point of Theorem 5 implies that X ±|χ(Σ)| is homeomorphic to the Teichmüller space of Σ.

Action of the modular group in the closed surface case

The modular group of Σ is the quotient group of positive diffeomorphisms of Σ up to isotopy. Dehn-Nielsen-Baer Theorem (Theorem 8.1, [START_REF] Farb | A primer on mapping class groups[END_REF] gives an isomorphism between the modular group of the surface Σ and the outer automorphisms group:

Out(π 1 Σ) := Aut(π 1 Σ) Inn(π 1 Σ) .
It is well known that the action of the modular group on the Teichmüller space of Σ is properly discontinuous: for all compact subset K of the Teichmüller space of Σ, the set

[φ] ∈ Out(π 1 Σ) | [φ] • K ∩ K = ∅ is finite.
The behavior of this action on non-maximal components seems completely opposite. In W. Goldman 2006, Goldman conjectured:

Conjecture 1. (Conjecture 3.1, W. Goldman 2006) Let k ∈ Z ∩ χ(Σ), -χ(Σ)
. Then the modular group acts ergodically on X k with respect to the measure λ.

Related to the Bowditch condition Bowditch 1998 and in the way of this conjecture, Marché and Wolff introduced in Marché and Wolff 2016, for k ∈ Z ∩ χ(Σ), -χ(Σ) , the subspace

N H k := [ρ] ∈ Rep k | ∃γ ∈ π 1 Σ simple , tr ρ(γ) ∈ [-2, 2] .
They proved the following: Theorem 6. (Theorem 1.5 and Theorem 1.6, Marché and Wolff 2016) 1. Assume Σ has genus 2. Then the action Mod + (Σ) on X ±1 is ergodic with respect to λ.

2. Assume g ≥ 3 and k is an integer such that |k| < |χ(Σ)|, then the action of Mod + (Σ) on N H k is ergodic with respect to λ.

In genus 2 and for Euler class 0, the component X 0 decomposes in two disjoint subspaces X 0 and X 0 + of non-zero measure and on which the modular group acts ergodically with respect to λ, see [START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF][START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF][START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF][START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF] Relative PSL 2 (R)-character varieties Let Σ be a connected, oriented and compact surface of genus g ≥ 1 and with n > 0 punctures, such that χ( Σ) = 2 -2g -n < 0. Fix c 1 , . . . , c n be curves on Σ which are homotopic to a circle around each puncture.

Figure 4

The fundamental group π 1 Σ is the free group:

a 1 , b 1 , . . . , a g , b g , c 1 , . . . c n | g i=1 [a i , b i ] n j=1 c j = 1 .
The representation space Hom Σ, PSL 2 (R) is the space of morphisms ρ : π 1 Σ → PSL 2 (R). The group PSL 2 (R) acts on the representation space by conjugation.

Definition 2. The character variety X ( Σ) is the Hausdorff quotient

Hom Σ, PSL 2 (R) //PSL 2 (R). A representation ρ : π 1 Σ → PSL 2 (R) is called Zariski-dense if its image is Zariski-dense in PSL 2 (R).
The subspace Rep( Σ) of classes of Zariski-dense representations is a dense, open and full measure subset of X ( Σ). This space is partitioned by behavior on the boundary components. Precisely we impose conditions on the boundary: for C 1 , . . . , C n be n conjugacy classes in PSL 2 (R) and

C := (C 1 , . . . , C n ), we define the space Hom Σ, PSL 2 (R), C of representations ρ : π 1 Σ → PSL 2 (R) such that for all i ∈ {1, . . . , n}, ρ(c i ) ∈ C i .
The groupe PSL 2 (R) acts on the space Hom Σ, PSL 2 (R), C by conjugation.

Definition 3. The relative character variety associated with C, denoted by X ( Σ, C), is the Hausdorff quotient

Hom Σ, PSL 2 (R), C //PSL 2 (R).
Let us denote by Rep( Σ, C) the subspace of classes of Zariski-dense representations of X ( Σ, C). This set is a smooth manifold of dimension 6g -6 + 2n.

For each C i ∈ PSL 2 (R), we choose a lift C i ∈ PSL 2 (R), we denote by r i the translation number of C i and by r the vector (r 1 , . . . , r n ). We define, for the vector r ∈ R n , the vector {r} of fractional parts of the coordinates of r. This means that {r} = {r 1 }, . . . , {r n } with for each i ∈ {1, . . . , n},

{r i } = r i -r i . Definition 4. Let ρ : π 1 Σ → PSL 2 (R) and fix a lift ρ : π 1 Σ → PSL 2 (R).
For each i ∈ {1, . . . , n}, let r i be the translation number of any lift ρ(c i ). The Euler number of ρ is the number eu(ρ) defined by

- n i=1 r i .
The Euler number of a representation does not depend of the choices of the lifts and is constant on each conjugacy class of it. We thus define the Euler number of a class of representations [ρ] by the Euler number of a representative of this class. This gives a map eu : X ( Σ) → R which induces the restriction map

eu C : X ( Σ, C) → R such that eu C ([ρ]) + {r} 1 ∈ Z
with {r} be the vector of fractionnal parts of the r i . The preimage of eu -1 C (e), for e ∈ R, will be denoted by X e ( Σ, C). In Mondello 2017, Mondello described the topology of X ( Σ, C).

Theorem 7. (Theorem 2.20,Mondello 2017) 1. The image of eu : X ( Σ) → R is the interval χ( Σ), -χ( Σ) . If eu(ρ) = -χ( Σ), then all the ρ(c i ) are hyperbolic and ρ is the holonomy of a hyperbolic structure on Σ with geodesic boundary components.

2. Assume e > 0. Then the space X e ( Σ, C) is non-empty if and only if e + {r} 1 + s 0 + s -∈ Z ∩ 0, -χ( Σ) , where s 0 is the cardinal of {i ∈ {1, . . . , n} | C i = id} and s -is the cardinal of {i ∈ {1, . . . , n}|C i negative unipotent} where a matrix is negative unipotent if it is conjugated to the matrix 1 -1 0 1 . In this case, the space X e ( Σ, C) is connected and smooth.

This work will always consider conjugacy classes of elliptic isometries. In this situation the number s 0 and s -are both zero.

The pure mapping class group of Σ, denoted by PMod( Σ), is the group of isotopy classes of diffeomorphisms of Σ whose action on ∂ Σ is trivial. It acts naturally on the relative character varieties with an action which is similar than the one of mapping class group on character varieties in closed case and preserves the connected components of relative character varieties and Goldman measure. We prove the theorem:

Statements

Fix C = (C 1 , . . . , C n ) with each C i be a conjugacy class of an elliptic isometry of angle υ i in PSL 2 (R). The number {r} 1 = n i=1 |υ i -υ i |
Theorem C. For e = ±(k -{r} 1 ) with k ∈ {r} 1 , -χ( Σ) ∩ Z, the pure mapping class group acts ergodically on N H e ( Σ, C).
The proof of Theorem C is an adaptation to the case of surfaces with boundaries of the proof of Theorem 1.6 of Marché and Wolff 2016.

We conjecture the dynamic of the modular group can be ensured, stating that a proper subgroup acts ergodically for the case of closed surfaces:

Conjecture A. Let Σ be a closed surface of genus g ≥ 3. If k is an integer such that |k| ≤ 2g -5, then the group Tor(Σ) acts ergodically on the subspaces N H k .

The section ?? is devoted to explain a strategy and a large part of proofs in the way of this conjecture. A positive answer to a question which is analogue to the Goldman conjecture, see Conjecture ??, could prove Conjecture A.

Topological dynamic

The example of translations on torus gives us a dynamical intuition of the orbits. In the case of the circle, we easily see that the action of a rotation of irrational angle has only dense orbits. This chaotic phenomena can be seen in higher dimension and is a general result coming from ergodicity: Proposition 1. Let H be a topological group acting on an Borelian space (X, µ) and which preserves the measure µ. Assume that each non-empty and open subset has non-zero measure and that the topology of X is countably generated. If the action of H on (X, µ) is ergodic, then almost every orbit is dense.

Unfortunately, this fact is a probabilistical result and no description of point of X for which the orbit is dense is given in general.

In W. Goldman 2006, Goldman ask the following:

Question 1. Could we find a condition on a representation ρ : π 1 Σ → SU(2) to have that its

Mod + (Σ)-orbit is dense ?

In Previte and Xia 2000 and in Previte and Xia 2002, Previte-Xia gives a necessary and sufficient condition:

Theorem 8. A representation ρ : π 1 Σ → SU(2) has a dense Mod + (Σ)-orbit if and only if ρ has a dense image in SU(2).
We start to prove a similar statement with the Abelian case. A compact, Abelian and connected Lie group is isomorphic to a n-dimensional torus. Using Ratner theorem, see Morris 2005, we prove:

Theorem D. A representation ρ : π 1 Σ → T n has a dense Mod + (Σ)-orbit if and only if ρ has a dense image in T n .
We conjecture such a result is true when G = SU(3) and Σ is a one-holded torus.

Connection with the Kronecker's Approximation Theorem

The dynamical result provided by Theorem D finds an application on the theory of geometry of numbers. An important theorem in this topic is the Kronecker's theorem concerning inhomogeneous Diophantine approximation, see section 4 below for the precise statement. n) , πe 1 , . . . , πe 2g are linearly independent over Q in the vector space R m . Let A ∈ M n, 2g; R) be a real matrix and let ε be a positive number. Then there is an element

Theorem E. Let g ≥ 1. Let b (i) = b (i) 1 , . . . , b (i) 2g , with i = 1, . . . , n, be vectors of R 2g such that b (1) , . . . , b (
K ∈ Sp 2g (Z) such that A -B K < Cε mod 2π. ( 2
)
where C is a constant depending only on 2g and n, {e 1 , . . . , e 2g } is the canonical basis of R 2g and the norm is any norm on M n, 2g; R).

Background

Let Σ be a connected, oriented closed surface of genus g ≥ 2

The Torelli group

For more details on this part, see Farb and Margalit 2011. The group Mod + (Σ) is generated by the 3g -1 Dehn twists tw a 1 , tw b 1 . . . , tw ag , tw bg , tw d 1 , . . . , tw d g-1 where, for i = 1, . . . , g, the curves a i , b i on Σ are given by the presentation of the fundamental group of Σ :

π 1 Σ = a 1 , b 1 , . . . , a g , b g | [a 1 , b 1 ] • • • [a g , b g ] = 1
and, for j = 1, . . . , g -1, the curves d j are the products a -1 j a j+1 . See Figure 2.

The first homology group H 1 (Σ, Z) ∼ = Z 2g is freely generated by the curves [a 1 ], [b 1 ], . . . , [a g ], [b g ]
and is a lattice in the real homology group H 1 (Σ, R).

Definition 1.1. The Torelli group Tor(Σ) is the kernel of the action of the mapping class group on

H 1 (Σ, Z).
An explicit generating set of the Torelli group is the set

{tw c , tw a tw -1 b | c separating, a, b cohomologous}.

Borel cross sections

Let X be a topological set on which a topological group H acts. The set X is endowed with its Borelian σ-algebra and a Borelian measure µ. A function X 1 → X 2 between two measured sets is called bimeasurable if it is measurable, invertible and has a measurable inverse. If such a function exists, we then say that X 1 and X 2 are bimeasurable. The quotient set X/H carries the quotient topology such that the canonical projection X → X/H is continuous. A Borel cross section is a subset S ⊂ X which intersects exactly once the orbit H.x, for every element x ∈ X. The following theorem is proven by Edward Effros in Effros 1965. For our purposes we shall need the following partial statement:

Theorem 1.2. (Theorem 2.1, Effros 1965) If X is a separable, complete, metrizable and locally compact topological space and if H is a topological group acting continuously on X, then the following conditions are equivalent :

• Every orbit H.x is locally closed.

• Every orbit H.x is locally compact.

• There exists a Borel cross section S for the orbits of H in X.

A key corollary of this result is given by James Bondar in Bondar 1976: [START_REF] Bondar | Borel cross-sections and maximal invariants[END_REF]) If X and H verify the hypothesis of the previous theorem and if furthermore the action of H on X is free and one of the conditions of the previous statement holds, then the Borel cross section S is bimeasurable to the quotient space X/H and there exists a Borelian measure µ S on S such that for every measurable function f : X → R one has:

Theorem 1.3. (Theorem 2,
X f dµ = S H f (h.s)dµ H (h) dµ S (s),
where µ H is the Haar-measure on H.

The mapping class group action on the SU(2)-characters

In the compact case and in rank 1 case we consider, the Lie algebra of SU( 2) is the Lie algebra su(2) of traceless skew-Hermitian complex 2 × 2-matrices. Let f : SU(2) → -2, 2 denote the trace function. Its variation function F is defined as the unique function

F : SU(2) → su(2) such that, for all x ∈ SU(2) and X ∈ su(2), d dt |t=0 f (x • exp(tX)) = F (x), X .
Goldman proved, in W. M. Goldman 1986, that:

F (x) = x - tr(x) 2 id.
Following Goldman and Xia 2011, we let:

ζ t : SU(2) → SU(2) defined as the map sending x ∈ SU(2) on exp(t • F (x)) For x ∈ SU(2), the map t → ζ t (x) is a one-parameter subgroup of SU(2).

Separating curve

If a curve α on Σ is separating, then Σ \ α can be written as the disjoint union Σ 1 Σ 2 and the fundamental group π 1 Σ is the amalgamated product

π 1 Σ 1 * α π 1 Σ 2 .
The data of two representations ρ 1 :

π 1 Σ 1 → SU(2) and ρ 2 : π 1 Σ 2 → SU(2), such that ρ 1 (α) = ρ 2 (α),
allows us to construct by amalgamation a unique representation ρ :

π 1 Σ → SU(2) defined by ρ |π 1 Σ 1 = ρ 1 and ρ |π 1 Σ 2 = ρ 2 . Let ξ t α : X (π 1 Σ, SU(2)) → X (π 1 Σ, SU(2)) be the twist flow defined, for γ ∈ π 1 Σ, by ξ t α ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ 1 ) ζ t (ρ(α))ρ(γ)ζ -t (ρ(α)) if γ ∈ π 1 (Σ 2 ) .
This flow is well defined since ζ t (ρ(α)) is the exponential of a polynomial in ρ(α) and thus commutes with ρ(α).

Non-separating curve

If a curve α on Σ is non-separating, the fundamental group π 1 Σ is the HNN-extension:

π 1 (Σ|α) * β βα -β -1 (α + ) -1
where α ± represent the boundary components of Σ \ α. Hence the data of a representation

ρ 0 : π 1 (Σ|α) → SU(2)
and a matrix B ∈ SU(2) such that

Bρ 0 (α -)B -1 = ρ 0 (α + ) defines a unique representation ρ : π 1 Σ → SU(2) such that ρ |π 1 (Σ|α) = ρ 0 and ρ(β) = B. Let ξ t α : X (π 1 Σ, SU(2)) → X (π 1 Σ, SU(2)) be the twist flow defined, for γ ∈ π 1 Σ, by ξ t α ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ|α) ζ t (ρ(α))ρ(β) if γ = β .
This flow is well defined since the relation

ξ t α ρ(β)ξ t α ρ(α -)ξ t α ρ(β) -1 = ξ t α ρ(α + )
is satisfied.

The Dehn twists and the flows

For a simple and closed curve α, consider the trace function

f α : X (π 1 Σ, SU(2)) → [-2, 2
] associated with the curve α, that is:

f α ([ρ]) = tr(ρ(α)).
In W. M. Goldman 1986, Goldman proved that the flow ξ α is the Hamiltonian flow of the trace function f α , that means:

df α X = ω G d dt |t=0 ξ t α , X for all X ∈ T X (π 1 Σ, SU(2))
and where ω G is the symplectic form constructed by Goldman in W. M. Goldman 1984.

If x ∈ SU(2), then there exists g ∈ SU(2) such that x = g e iθ 0 0 e -iθ g -1 with θ = cos -1 f (x)

2

.

We then compute F (x) = x -f (x) 2 id which we can write:

F (x) = g i sin(θ) 0 0 -i sin(θ) g -1 ,
and so, by definition

ζ t (x) = g e it sin(θ) 0 0 e -it sin(θ) g -1 .
In particular, if x = ±id, then for all t ∈ R we have ζ t (x) = id and for x = ±id, we have the equality

ζ t (x) = id, if and only if t ∈ 2π sin(θ) Z.
Thus we state the following (see [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]:

Lemma 1.4. If x = ±id, then x belongs to the one-parameter subgroup {ζ t (x)} t∈R and more precisely,

x = ζ s(x) (x) for: s(x) = θ sin(θ)
.

For x ∈ SU(2) such that θ / ∈ πQ, the subgroup x is dense in the circle {ζ t (x)|t ∈ R} ∼ = S 1 and acts ergodically on it with respect to the Lebesgue measure.

Furthermore, the Dehn twist tw α acts on ρ via the relation

tw α .ρ = ξ s(ρ(α)) α ρ. Thus, if cos -1 f (ρ(α)) 2 × 1
π is irrational, then the orbit tw α .ρ is dense in the circle defined by the orbit {ξ t α ρ} t∈R . The Hamiltonian flow gives an action of the circle U α := S 1 on the character variety and a free action on the subspace consisting of classes of representations [ρ] such that ρ(α) = ±id.

1.4 Action of the modular group on PSL 2 (R)-character varieties and

Hamiltonian flows

Similarly to the previous part, the Lie algebra sl 2 (R) of PSL 2 (R) is the space of traceless real 2 × 2 matrices. Let exp = sl 2 (R) → PSL 2 (R) be the exponential map. The Lie algebra sl 2 (R) is endowed with its Killing form we will denote by K( . , . ). Let f : PSL 2 (R) → R be a smooth function. Then there exists a unique map F : PSL 2 (R) → sl 2 (R), called the variation function associated to f , such that for all g ∈ PSL 2 (R) and all X ∈ sl 2 (R),

d dt|t=0 f (g • exp(tX)) = K(F (g), X).
In W. M. Goldman 1986, Goldman computed that the variation function of the trace function f := tr is given, for all g ∈ PSL 2 (R), by the formula F (g) = g -tr(g) 2 id. We introduce the flow

ζ t : PSL 2 (R) → PSL 2 (R) g → exp(tF (g)) .
Let g ∈ PSL 2 (R) be an elliptic isometry and let θ its angle. We assume sin(θ) = 0. The isometry

g has the form g = h cos(θ) -sin(θ) sin(θ) cos(θ) h -1 for some h ∈ PSL 2 (R).
We directly compute that

F (g) = h 0 -sin(θ) sin(θ) 0 h -1
and hence that

ζ t (g) = h cos(t sin(θ)) -sin(t sin(θ)) sin(t sin(θ)) cos(t sin(θ)) h -1 .
In particular, ζ t (g) = id if and only if t ∈ 2π sin θ Z and the matrix g is contained in the flow {ζ t (g)} t∈R . More precisely g = ζ s(g) (g) for s(g) = θ sin θ and moreover: Lemma 1.5. If g is an elliptic isometry of infinite order then the subgroup g is dense in the circle

{ζ t (g) | t ∈ R} ∼ = S 1
and acts ergodically on it.

Non-separating curve

If a curve α is non-separating, the fundamental group π 1 Σ is the HNN-extension:

π 1 (Σ \ α) β / βα -β -1 = α +
where α ± are the boundary components of Σ \ α. Hence the data of a representation

ρ 0 : π 1 (Σ \ α) → PSL 2 (R) and a matrix B ∈ PSL 2 (R) such that Bρ 0 (α -)B -1 = ρ 0 (α + ) allows to construct a representation ρ : π 1 Σ → PSL 2 (R) defined by ρ |π 1 (Σ\α) = ρ 0 and ρ(β) = B.
We define the flow ξ t α : Rep k → Rep k by:

ξ t α ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ \ α) ζ t (ρ(α))ρ(β) if γ = β .
This flow is well defined since the relation

ξ t α ρ(β)ξ t α ρ(α -)ξ t α ρ(β) -1 = ξ t α ρ(α + ).
is satisfied.

Separating curve

If α is separating, then Σ \ α is the disjoint union Σ 1 Σ 2 and its fundamental group Γ is the amalgamated product:

π 1 Σ 1 * α π 1 Σ 2
Similarly to the HNN-extension, the data of two representations ρ 1 : π 1 Σ 1 → PSL 2 (R) and ρ 2 :

π 1 Σ 2 → PSL 2 (R) such that ρ 1 (∂Σ 1 ) = ρ 2 (∂Σ 2 )
allows to construct the representation ρ :

π 1 Σ → PSL 2 (R) defined by ρ |∂Σ 1 = ρ 1 and ρ |∂Σ 2 = ρ 2 .
We let the twist flow ξ t α : Rep k → Rep k by :

ξ t α ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ 1 ) ζ t (ρ(α))ρ(γ)ζ -t (ρ(α)) if γ ∈ π 1 (Σ 2 ) .
This flow is well defined because ζ t (ρ(α)) commutes with ρ(α) as the exponential of a polynomial in ρ(α).

In W. M. Goldman 1986, Goldman proved that the flow {ξ t α } we defined is the Hamiltonian flow of the trace function

f α : ρ → tr ρ(α) .
Theses flows defines action of the circle U α := S 1 on the subspace of classes of representations which send α on an elliptic element. A corollary of Lemma 1.5 is the following: Lemma 1.6. Let ρ : π 1 Σ → PSL 2 (R) and let α be a simple closed curve such that ρ(α) is elliptic of infinite order. Then the action of T α on the circle U α • [ρ] is ergodic with respect to the Lebesgue measure.

Remark 1.7. If α 1 , . . . , α are simple curves and if we assume there are pairewise disjoint, then the flows ξ t 1 α 1 , . . . , ξ t α commute. Their action on ρ give a topological torus orbit

U α 1 × • • • × U α • ρ.

Torelli group action on compact character varieties

In this section, we prove Theorem A.

2.1 Ergodicity for the case of SU(2)

The strategy in proving the ergodicity of the Torelli group action is to find a full measure subset of the character variety on which every measurable and Tor(Σ)-invariant function is almost everywhere invariant by the mapping class group action.

Ergodicity of translation actions

In this subsection, we generalize Lemma 1.4. We remark that if two simple curves c 1 and c 2 are disjoint, then the associated flows ξ t c 1 and ξ s c 2 commute. Hence, under this assumption, the actions of these flows on a representation ρ give a topological torus orbit {ξ t c 1 .ξ s c 2 ρ} t,s∈R obtained by the action of U c 1 × U c 2 on the characters which are not ±id evaluated in c 1 , c 2 .

Lemma 2.1. Let [ρ] be a class of representations in X (π 1 Σ, SU(2)) and suppose that there exist simple closed curves c 1 , . . . , c on Σ which are pairwise disjoint and such that

θ 1 = cos -1 f (ρ(c 1 )) 2 , . . . , θ = cos -1 f (ρ(c )) 2 , π
are linearly independent over Q. Then the action of h = tw c 1 . . . tw c on the orbit

U c 1 × • • • × U c • [ρ]
is ergodic with respect to the Lebesgue measure on this torus orbit.

In particular, every orbit for the action of h is dense in the topological torus

U c 1 × • • • × U c • [ρ].
Remark 2.2. With the notation of the Lemma 2.1 and if [ρ] satisfies the hypothesis of this lemma, then for all i = 1, . . . , , the matrix ρ(c i ) is not ±id and the torus orbit

U c 1 × • • • × U c • [ρ]
is the torus obtained as the quotient:

R /Λ where Λ is the lattice 2π sin θ 1 Z⊕• • •⊕ 2π sin θ Z. By definition of the torus R /Λ, the action of U c 1 ו • •×U c on the character variety is free.
The following is a classical result needed for proving Lemma 2.1 and for which a proof may be find in [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF] We denote by T the torus R/2πZ. we precise that T is isomorphic to a maximal torus of SU(2).

Lemma 2.3. Let t = (t 1 , . . . , t ) ∈ R such that t 1 , . . . , t , π are linearly independent over Q and let f t : T → T be the translation of vector t. Then the action of f t on T is ergodic with respect to the Lebesgue measure.

Proof of the lemma 2.1. The Dehn twists tw c k act as a translation of the torus orbit

U c 1 ו • •×U c •[ρ]. The orbit h Z • [ρ] is the orbit ξ t 1 c 1 . . . , ξ t c • [ρ]
with for all k ∈ {1, . . . , }:

t k = θ k 2 sin(θ k )
.

For such a t k , the action of ξ t k c k is given by the multiplication by a matrix conjugated to

e iθ k 0 0 e -iθ k .
We deduce from this that the action of

h on U c 1 × • • • × U c • [ρ]
is given by the translation of the vector (θ 1 , . . . , θ ). Then the lemma 2.3 shows that this action is ergodic with respect to the Lebesgue measure since π, θ 1 , . . . , θ are linearly independent over Q.

A full measure set

A multicurve m is the union of a finite number of simple, closed and pairwise disjoint curves. Let us denote by MC(Σ) the set of multicurves m whose curves are simple, closed and non-separating and such that m is the boundary of a subsurface of Σ. By MC 0 (Σ) we will denote its subset of bounding pair of Σ.

Definition 2.4. Let m = c 1 ∪ • • • ∪ c ∈ MC(Σ) be a mutlicurve. We say that a class [ρ] ∈ X (π 1 Σ, SU(2)) satisfies the condition (M m ) if the real numbers : π, θ 1 = cos -1 tr(ρ(c 1 )) 2 , . . . , θ = cos -1 tr(ρ(c )) 2
are linearly independent over Q.

Following the previous definition, for m ∈ MC 0 (Σ), we now consider the set :

M m (π 1 Σ, SU(2)) = {[ρ] ∈ X (π 1 Σ, SU(2)) | [ρ] statisfies the condition (M m )}
The aim of this section is to prove the proposition :

Proposition 2.5. For m ∈ MC 0 (Σ), the set M m (π 1 Σ, SU(2)) has full measure in X (π 1 Σ, SU(2)).
Before to prove this proposition, we note that for a curve γ, the angle of ρ(γ), expressed by the formula

cos -1 f (ρ(γ))
2 , defines a function θ γ : X (π 1 Σ, SU(2)) → S 1 . To simplify notation, as previously, for a multicurve m = c 1 ∪c 2 ∈ MC 0 (Σ), we will denote by θ 1 and θ 2 the functions θ c 1 and θ c 2 . Moreover, c 1 ∪c 2 separate Σ in two subsurface Σ 1 and Σ 2 of respective genus g 1 and g 2 and with two boundary components. Let Σ 1 and Σ 2 be the two subsurfaces of Σ 1 and Σ 2 which have genus g 1 and g 2 and with one boundary component which for a pair of pant with c 1 and c 2 . See next picture.

Figure 6

Let N m be the subset of X (π 1 Σ, SU(2)) of conjugacy classes of representations ρ such that the centralizers Z SU(2) ρ |π 1 Σ 1 and Z SU(2) ρ |π 1 Σ 2 are discrete in SU(2). This subspace has full measure because if one of the centralizers is non discrete, the image of at least one restriction is contained in a parabolic subgroup. Hence the complement of N m is a finite union of submanifolds and hence has measure zero.

The complement of M m (π 1 Σ, SU(2)) is the set:

(q 0 ,q 1 ,q 2 )∈Z 3 \{0} [ρ] ∈ X (π 1 Σ, SU(2)) | q 1 θ 1 (ρ) + q 2 θ 2 (ρ) = q 0 π .
Let q denote the vector (q 1 , q 2 ). If q 0 = 0 and q = 0, the relation q 1 θ 1 (ρ) + q 2 θ 2 (ρ) = q 0 π is empty. Thus, we only have to consider the case q = 0. The proposition will be hence proved if for every (q 0 , q 1 , q 2 ) ∈ Z 3 \{0} with q = 0 Z 2 , the set:

[ρ] ∈ X (π 1 Σ, SU(2)) | q 1 θ 1 (ρ) + q 2 θ 2 (ρ) =ψ m,q ([ρ]) = q 0 π = ψ -1 m,q (q 0 π) has null measure. Fix (q 0 , q 1 , q 2 ) ∈ Z 3 \{0} with q = 0 Z 2 .
Lemma 2.6. The map ψ m,q is a submersion on every class of representation [ρ] ∈ N m .

By Goldman work W. M. Goldman 1984 part 3.7, the map E 1 : SU(2) 2g 1 +1 → SU(2) 2 defined by :

E 1 (A 1 , . . . , B g 1 , C) = g 1 j=1 [A j , B j ], C
is a submersion at the point {A 1 , . . . , B g 1 } if the centralizer Z SU(2) {A 1 , . . . , B g 1 } is discrete. For the same reasons, the map E 2 : SU(2) 2g 2 +1 → SU(2) 2 defined by:

E 2 (A 1 , . . . , B g 2 , C) = g 2 j=1 [A j , B j ], C is a submersion at the point {A 1 , . . . , B g 2 } if the centralizer Z SU(2) {A 1 , . . . , B g 2 } is discrete.
Hence the evaluation Ev Σ 1 : Hom π 1 Σ 1 , SU(2) → SU(2) 2 defined by:

Ev Σ 1 (ρ) = (ρ(c 1 ), ρ(c 2 ))
is a submersion at every ρ such that the centralizer

Z SU(2) ρ |π 1 Σ 1 is discrete since Ev Σ 1 is the com- position of E 1 with the diffeomorphism: SU(2) 2 → SU(2) 2 (A, B) → (B -1 A, B)
.

Since the trace function tr : SU(2) → -2, 2 is a submersion at each matrix A = ±id, the application

(f c 1 , f c 2 ) : Hom π 1 Σ, SU(2) → -2, 2 × -2, 2 is a submersion at each representation ρ such that both Z SU(2) ρ |π 1 Σ 1 and Z SU(2) ρ |π 1 Σ 2 are discrete. In particular, under this condition, the linear form d [ρ] f c 1 and d [ρ] f c 2 are non-colinear.
Proof of Lemma 2.6. Let ρ such that ρ(c 1 ) and ρ(c 2 ) are not equal to ±id. We then have that both sin(θ 1 (ρ)) and sin(θ 2 (ρ)) are non-zero. We hence compute that:

d [ρ] ψ m,q = -q 1 sin(θ 1 (ρ)) d [ρ] f c 1 + -q 2 sin(θ 2 (ρ)) d [ρ] f c 2 . If [ρ] ψ m,q = 0, then we deduce that d [ρ] f c 1 and d [ρ] f c 2 are colinear. But it is impossible if [ρ] ∈ N m . Then ψ m,q is a submersion at each point of N m .

Proof of the ergodicity

In order to prove Theorem A for SU(2), we will consider a measurable function which is Tor(Σ)invariant and prove that, up to a restrict this function to a full measure subset, it is invariant under the action of the Dehn twists which generate the mapping class group.

Let F : X (π 1 Σ, SU(2)) → R be a measurable function and assume that F is Tor(Σ)-invariant. Let

x ∈ {a 1 , b 1 , . . . , a g , b g , d 1 , . . . , d g-1 } be a curve of the generating set of the mapping class group, fix a bounding pair m x = x ∪ c 2 in MC 0 (Σ) and denote by h the product of Dehn twists tw x • tw c 2 . The set

M mx (π 1 Σ, SU(2))
has full measure by Proposition 2.5. As the orbits U x × U c 2 • [ρ] are tori and so are compact in X (π 1 Σ, SU(2)), Theorem 1.2 ensures the existence of a Borel cross section of X (π 1 Σ, SU(2)) for the action of U x × U c 2 = T 3 . We will denote this section by S. Since T 2 and X (π 1 Σ, SU(2)) verify the assumption of Theorem 1.3, then the section S is bimeasurable to the quotient

X (π 1 Σ, SU(2))/T 2
and the measure λ decomposes, for all function f : X (π 1 Σ, SU(2)) → R + , by the formula :

X (π 1 Σ,SU(2)) f dλ = S T 2 f (t.s)dµ T 2 (t) dµ S (s)
where µ T 2 is the Haar measure on the tori T 2 given by Theorem 1.3 and µ S is a measure on S given by the same theorem.

The function F induces a measurable function F : S × T 2 → R defined by : We then deduce that on the space

F (s, t) = F (t • s). Fix [ρ] ∈ M mx (π 1 Σ, SU(2)), denote by [ρ] its projection in S and let F |{[ρ]}×T 2 : T 2 → R. Such a function is measurable and invariant by the action of h since F is. Since [ρ] ∈ M mx (π 1 Σ, SU(2)), the action of the translation h on T 2 is ergodic with respect to µ T 2 . It implies that F |{[ρ]}×T 2 is
x∈{a 1 ,b 1 ,...,ag,bg,d 1 ,...,d g-1 } M mx (π 1 Σ, SU(2)),
which has full measure in X (π 1 Σ, SU(2)), the function F is almost everywhere invariant by the Dehn twists tw x , for all x ∈ {a 1 , . . . , b g , d 1 , . . . , d g-1 }. It implies that F is almost everywhere invariant under the action of Mod(Σ), which is known to be ergodic from Theorem 1. Hence F is almost everywhere constant and this proves the ergodicity of the Torelli group action on X (π 1 Σ, SU(2)).

Ergodicity for G = SU(n)

This part is devoted to the proof of Theorem A in the more general case G = SU(n), for n ≥ 2.

We apply the same strategy as in the previous section, adapting the tools developed there. Let us introduce the notation we are going to use.

Torus actions on

M α-reg (π 1 Σ, SU(n))
For a matrix A ∈ SU(n) with distinct eigenvalues, denote by λ 1 (A), . . . , λ n (A) its eigenvalues and θ 1 (A), . . . , θ n (A) their arguments which can be expressed up to the sign as

cos -1 λ i (A) + λ i (A) -1 2 with the normalisation 0 ≤ θ 1 (A) ≤ • • • ≤ θ n (A) < 2π.
The group SU(n) has rank n -1 and every maximal torus is conjugate to the group :

      e iθ 1 0 . . . 0 0 e iθ 2 . . . 0 . . . . . . . . . . . . 0 0 . . . e iθn       , (θ 1 , . . . , θ n ) ∈ 0, 2π and n k=1 e iθ k = 1 which is isomorphic to T n-1 . A matrix A ∈ SU(n) is called regular if its eigenvalues are simple. Similarly, for a curve α, a character [ρ] ∈ X (π 1 Σ, SU(n)) is called α-regular if the matrix ρ(α)
is regular. We will denote by M α-reg (π 1 Σ, SU(n)) the subsets of α-regular SU(n)-characters. For any curve α, the subset

M α-reg (π 1 Σ, SU(n)) is an open subset of X (π 1 Σ, SU(n))
and has full measure. We will define actions of T n-1 on the character variety X (π 1 Σ, SU(n)). Let z = (z 1 , ..., z n-1 ) ∈ T n-1 and h z be the associated diagonal matrix :

diag z 1 , . . . , z n-1 , 1 z 1 • • • z n-1 ∈ SU(n).
Let A ∈ SU(n) be a regular matrix. There exists a unique decomposition [e 1 ] ⊕ • • • ⊕ [e n ] of C n in lines such that:

Ae i = λ i (A)e i ,
for all i ∈ {1, . . . , n}.

Let α be a simple and closed curve and let [ρ] ∈ M α-reg (π 1 Σ, SU(n)). With the same notation as in part 1.3 and with respect to the unique basis of

C n in which ρ(α) = h (λ 1 (ρ(α)),•••λ n-1 (ρ(α)))
, we define for z ∈ T n-1 and for α non-separating, the representation z • ρ by:

z • ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ \ α) h z ρ(β) if γ = β .
In the case when α is separating, the representation z • ρ by:

z • ρ(γ) = ρ(γ) if γ ∈ π 1 (Σ 1 ) h z ρ(γ)h -1 z if γ ∈ π 1 (Σ 2 ) .
Therefore, the action of the torus U α := T n-1 on the subspace of α-regular characters depends on α. Moreover the action of Dehn twists along α on M α-reg (π 1 Σ, SU(n)) is given by:

tw α • [ρ] = λ(ρ(α)) • [ρ],
where λ(ρ(α)) = (λ 1 (ρ(α)), . . . , λ n-1 (ρ(α))). A direct computation shows that if α and β are disjoint curves, then the actions of the tori U α and

U β on M α-reg (π 1 Σ, SU(n)) ∩ M β-reg (π 1 Σ, SU(n)) commute.
This hence defines an action of U α × U β on the space of α-regular and β-regular characters. We thus obtain an analogue of Lemma 2.1:

Lemma 2.7. Let [ρ] ∈ X (π 1 Σ, SU(n))
and suppose that there exist c 1 , . . . , c pairwise disjoints, simple and closed curves on Σ such that:

θ 1 (ρ(c 1 )), . . . , θ n-1 (ρ(c 1 )), . . . , θ 1 (ρ(c )), . . . , θ n-1 (ρ(c )), π are linearly independent over Q. Then the action of h = tw c 1 • • • tw c on the orbit U c 1 × • • • × U c • [ρ]
is ergodic with respect to the Lebesgue measure on this torus orbit.

Proof. As in the proof of Lemma 2.1, the action of the Dehn twists tw c i is an action by translation on the torus. The flows commute on the character variety because the curves are disjoint and the orbit is given by the formula

h k • [ρ] = (λ 1 (ρ(c 1 )) k , . . . λ n-1 (ρ(c )) k , . . . , λ 1 (ρ(c )) k , . . . λ n-1 (ρ(c )) k ) • [ρ].
Hence the condition on the θ i (ρ(c k )) implies the desired ergodicity by the lemma 2.3.

A full measure set and the ergodicity

In this subsection, we define a full measure subspace of the character variety with the conditions of the previous lemma and conclude with the ergodicity of the Torelli group action.

Definition 2.8. Let m = c 1 ∪ • • • ∪ c be a multicurve of simple closed and non-separating curves.

We say that a class of a representation

[ρ] ∈ X (π 1 Σ, SU(n)) satisfies the condition (M m ) if θ 1 (ρ(c 1 )) . . . , θ n-1 (ρ(c 1 )), . . . , θ 1 (ρ(c )), . . . θ n-1 (ρ(c )), π are linearly independent over Q. Remark 2.9. If a class of a representation [ρ] satisfies the condition (M m ) for some m = c 1 ∪ • • • ∪ c ∈ MC(Σ) then [ρ] is c i -regular for all i ∈ {1, . . . , }. To simplify notation, for a multicurve m = c 1 ∪ • • • ∪ c , we will denote by M m-reg (π 1 Σ, SU(n)) the intersection i=1 M c i -reg (π 1 Σ, SU(n))
which has full measure.

We define, for m ∈ MC 0 (Σ), the set:

M m (π 1 Σ, SU(n)) = [ρ] ∈ X (π 1 Σ, SU(n)) | [ρ] statisfies the condition (M m ) .
Note that Remark 2.9 assures that

M m (π 1 Σ, SU(n)) is contained in M m-reg (π 1 Σ, SU(n)).
Proposition 2.10. For all m = c 1 ∪ c 2 ∈ MC 0 (Σ), the set M m (π 1 Σ, SU(n)) has full measure in the character variety.

A previously, c 1 ∪ c 2 separate Σ in two subsurface Σ 1 and Σ 2 of respective genus g 1 and g 2 and with two boundary components. Let Σ 1 and Σ 2 be the two subsurfaces of Σ 1 and Σ 2 which have genus g 1 and g 2 and with one boundary component which for a pair of pant with c 1 and c 2 . See Figure 6.

We can write its complement as the set:

q=(q 1 1 ,...,q 1 n-1 ,...,q 2 1 ,...q 2 n-1 )∈Z 2(n-1) \{0},q 0 ∈Z [ρ] ∈ X (π 1 Σ, SU(n)) | 2 k=1 n-1 i=1 q k i θ i (ρ(c k )) = q 0 π .
We will show next that each set in the previous union is a codimension 1 submanifold and hence this union has null measure.

Let ψ m,q : M m-reg (π 1 Σ, SU(n)) → R be defined by the formula :

ψ m,q ([ρ]) = 2 k=1 n-1 i=1 q k i θ i (ρ(c k )).
For

π k i : [ρ] → λ i (ρ(c k )) + λ i (ρ(c k )) -1 , we may write θ i (ρ(c k )) = cos -1 ( π k i ([ρ])
2 ) up to the sign. The differential of ψ m,q is the computed to be

d [ρ] ψ m,q = 2 k=1 n-1 i=1 - q k i 2 sin(θ i (ρ(c k ))) d [ρ] π k i .
Let N m be the subset of

X (π 1 Σ, SU(n)) of conjugacy classes of representations ρ such that the centralizers Z SU(n) ρ |π 1 Σ 1 and Z SU(n) ρ |π 1 Σ 2 are discrete in SU(n).
By the same reason we explained for the case of SU(2), we have:

Lemma 2.11. The subset N m has full measure in X (π 1 Σ, SU(n))

We will show the next Lemma as for SU(2).

Lemma 2.12. The map ψ m,q is a submersion on every class of representation [ρ] ∈ N m .

By Goldman work W. M. Goldman 1984 part 3.7, the map E 1 : SU(n) 2g 1 +1 → SU(n) 2 defined by :

E 1 (A 1 , . . . , B g 1 , C) = g 1 j=1 [A j , B j ], C
is a submersion at the point {A 1 , . . . , B g 1 } if the centralizer Z SU(n) {A 1 , . . . , B g 1 } is discrete. For the same reasons, the map E 2 : SU(n) 2g 2 +1 → SU(n) 2 defined by:

E 2 (A 1 , . . . , B g 2 , C) = g 2 j=1 [A j , B j ], C is a submersion at the point {A 1 , . . . , B g 2 } if the centralizer Z SU(n) {A 1 , . . . , B g 2 } is discrete.
Hence the evaluation Ev Σ 1 : Hom π 1 Σ 1 , SU(n) → SU(n) 2 defined by:

Ev Σ 1 (ρ) = (ρ(c 1 ), ρ(c 2 ))
is a submersion at every ρ such that the centralizer

Z SU(n) ρ |π 1 Σ 1 is discrete since Ev Σ 1 is the com- position of E 1 with the diffeomorphism: SU(n) 2 → SU(n) 2 (A, B) → (B -1 A, B) .
Since the functions λ i + λ -1 i : SU(n) → -2, 2 are submersions at each regular matrix A, the application (π k i ) k=1,2 ; i=1,...,n-1 : Hom

π 1 Σ, SU(n) → -2, 2 2(n-1) is a submersion at each representation ρ such that both Z SU(n) ρ |π 1 Σ 1 and Z SU(n) ρ |π 1 Σ 2 are discrete.
In particular, for the first q k 0 i 0 = 0, there exists a vector field X such that d

[ρ] π k i X = δ k 0 i 0 .
For such a vector fiel:

d [ρ] ψ m,q X = -q k 0 i 0 sin(θ i 0 (ρ(c k 0 ))) d [ρ] π k 0 i 0 X = 0.
This concludes the proof of Lemma 2.12 and thus of Proposition 2.10. The proof of the ergodicity uses the same arguments as in case of SU(2).

Let F : X (π 1 Σ, SU(n)) → R be a measurable and Tor(Σ)-invariant function. For each curve x in the set {a 1 , b 1 , . . . , a g , b g , d 1 , . . . , d g-1 }, we fix a bounding pair m x = x ∪ c 2 .

Replacing the torus T 2 we used for the case SU(2) by the torus T 2(n-1) , we conclude using the same methods that on the space

x∈{a 1 ,...,bg,d 1 ,...,d g-1 } M mx (π 1 Σ, SU(n)),
which has full measure from Proposition 2.10, the function F is almost everywhere invariant by the Dehn twists tw x for all x ∈ {a 1 , b 1 , . . . , a g , b g , d 1 , . . . , d g-1 }. This implies that it is almost everywhere invariant by the mapping class group. Theorem 1 now shows that F is constant on a full measure subset of X (π 1 Σ, SU(n)). This proves the ergodicity of the Torelli group action on X (π 1 Σ, SU(n)).

Ergodicity for semi-simple, connected and compact Lie groups

In this section we generalize the proofs of the ergodicity of the Torelli group on character varieties with values in a semi-simple, connected and compact Lie group. We will use the same strategy as for compact Lie groups SU(n) but need to replace the tools we used by their appropriate analogues in a more general sense.

Preliminaries on compact Lie group theory

Let G be a semi-simple, connected and compact Lie group with Lie algebra g. A maximal torus is a connected and abelian subgroup of G which is maximal in these properties. Such a subgroup exists, so fix T < G be a maximal torus and let t be its Lie algebra. It is an abelian subalgebra of g. The subgroup T is isomorphic to an r-dimensional torus T r and its Lie algebra t is isomorphic to the commutative Lie algebra R r . We will so use the existence of coordinates on t via this isomorphism. Precisely, for i ∈ {1, . . . , r} and t ∈ T , we denote by λ i (t) the projection on the i-th factor of t ∈ T ∼ = T r . It is a well known fact that every element of G is contained in a maximal torus. We have however the more precise result (see Bröcker and tom Dieck 1995 for more details):

Theorem 2.13. Every k ∈ G is conjugated to an element of T . Moreover, all the maximal tori are conjugated and hence are isomorphic to T r .

Remark that a maximal torus can contain two conjugated elements. The integer r is called the rank of the group G. The Weyl group associated to T is the group N G (T )/T , where the subgroup

N G (T ) is the normalizer of T in G.
Proposition 2.14. [START_REF] Bourbaki | Groupes et algèbres de Lie[END_REF] The Weyl group associated to a maximal torus T < G is finite.

A weight of T is a real and irreducible representation. Let ω be a weight of T and σ : G → Aut(V ) be a representation. The sum of all invariant subspaces of σ| T isomorphic to ω is called the weight space associated to ω of σ. Define, for n = (n 1 , . . . , n r ) ∈ Z r , the linear form:

Θ * n : t → R (x 1 , . . . , x r ) → n 1 x 1 + • • • + n r x r ,
where we use the coordinates on t given by the isomorphism t ∼ = R r coming from

T ∼ = R/Z r .
It is then well known that the weights of T are either the trivial one-dimensional representation or the representations Θ n : T ∼ = T r → SO 2 (R), for n = (n 1 , . . . , n r ) ∈ Z r \{0}, defined by :

Θ n ([x 1 , . . . , x r ]) = cos(2πΘ * n (x 1 , . . . , x r )) -sin(2πΘ * n (x 1 , . . . , x r )) sin(2πΘ * n (x 1 , . . . , x r )) cos(2πΘ * n (x 1 , . . . , x r ))
.

Definition 2.15. A linear form α ∈ t * is a root of G if there exists n = (n 1 , . . . , n r ) ∈ Z r \{0} such that α = Θ * n and if the weight space of the adjoint representation Ad : G → GL(g) associated to Θ n is non-trivial.

We denote by ∆ the set of roots of G. Since G is a semi-simple Lie group, the Killing form ., . is a scalar product and the subspace t < g becomes a Euclidean space. Using the induced isomorphism t ∼ = t * , we can see ∆ as a subset of t and for α ∈ ∆, we define the reflection

r α : β → β - 2 β, α α, α α.
It is well known that the Weyl group associated to T is isomorphic to the subgroup of GL(t):

r α |α ∈ ∆ .
The alcoves of t are the connected components of t\ α∈∆,n∈N ker(r α -nid).

The Weyl group acts simply transitively on the images of the alcoves in T .

Definition 2.16. An element k ∈ G is called regular, if it is contained in a unique maximal torus.

Let M be the image by the exponential map t → T of an alcove of t. Such M will be called an alcove of T and let k ∈ G be a regular element. There exists a unique class g k of G/Z G (k), with

Z G (k) the centralizer of k in G, such that g k kg -1 k ∈ M.
Example 2.17. For G = SU(n) and T the set of diagonal matrices, the roots of G are given by λ i -λ k for i, k ∈ {1, . . . , n} such that i = k and where the λ i are the eigenvalues. The Weyl group is then the symmetric group S n .

Density of some orbits

Let G be a semi-simple, connected and compact Lie group of rank r, let T be a maximal torus and

M be an alcove of T . Let α ∈ π 1 Σ be a simple curve. A character [ρ] ∈ X (π 1 Σ, G) is called α-regular if ρ(α) is regular. Then there exists a unique class g ρ(α) in the quotient G/Z G (ρ(α)), such that g ρ(α) ρ(α)g -1 ρ(α) ∈ M.
The set of α-regular characters is an open subset of X (π 1 Σ, G) and has full measure. The maximal torus T acts on the space Hom α-reg (π 1 Σ, G) of α-regular representations via the action, defined as follows: if α is non-separating, for t ∈ T and ρ ∈ Hom α-reg (π 1 Σ, G),

t • ρ(γ) = g ρ(α) ρ(γ)g -1 ρ(α) if γ ∈ π 1 (Σ|α) tg ρ(α) ρ(β)g -1 ρ(α) if γ = β
and if α is separating:

t • ρ(γ) = g ρ(α) ρ(γ)g -1 ρ(α) if γ ∈ π 1 (Σ 1 ) tg ρ(α) ρ(γ)g -1 ρ(α) t -1 if γ ∈ π 1 (Σ 2 ).
. Since this action commutes with the conjugation action of G on the representations, we thus defined an action of the maximal torus U α := T on the α-regular characters, which only depends on the curve α. As for the case of SU(n), we have an expression of the tw α action on the subspace of α-regular characters, via the formula

tw α • [ρ] = (g ρ(α) ρ(α)g -1 ρ(α) ) • [ρ].
We then note the important fact that for two disjoint curves α and β, the actions of the maximal tori

U α and U β on M α-reg (π 1 Σ, G) ∩ M β-reg (π 1 Σ, G) commute.
It hence implies an action of the product U α × U β on the previous intersection. We define the map t α :

Hom α-reg (π 1 Σ, G) → M by t α (ρ) = g ρ(α) ρ(α)g -1 ρ(α) .
For every i ∈ {1, . . . , r}, the projection λ i induces a function on the space Hom α-reg (π 1 Σ, G) which we will denote by λ i,α . If we conjugate ρ by g ∈ G, we obtain by the uniqueness of g ρ(α) up to the centralizer of ρ(α), that there exists z ∈ Z G (ρ(α)) such that

g gρ(α)g -1 g = g ρ(α) z
and hence we obtain that t α is invariant under the conjugation action of G on the representation variety Hom α-reg (π 1 Σ, G) and descends to a map

M α-reg (π 1 Σ, G) → M,
which we will denote by t α again. An element k ∈ G is said to be generic if g k kg -1 k is dense in T .

Claim 2.18. An element k ∈ G is generic if and only if for all non-trivial character χ :

T → S 1 , χ(g k kg -1 k ) = 1.
Proof. Let φ : T → T r be an isomorphism which identifies T with the r-torus. Then the map χ(φ -1 ) is a non-trivial character of the torus T r . Since a character of T r is induced by a linear form of R r which maps Z r on Z, it as the form (x 1 , . . . , x r ) → n 1 x 1 + • • • + n r x r with the n i ∈ Z. An element k ∈ T is generic if and only if φ(k) is generic, that means if it generates a dense subgroup in T r . We then have that φ(k) is generic, if and only if χ(φ -1 )(φ(k)) = 1.

In particular, a generic element of k is regular.

Lemma 2.19. Let ρ : π 1 Σ → G be a representation and α be a simple closed curve such that ρ(α) is generic. Then the orbit

tw α • [ρ] is dense in the torus orbit U α • [ρ].
Moreover, the action of tw α on U α • [ρ] is ergodic with respect to the Lebesgue measure.

Definition 2.20.

Let m = c 1 ∪ • • • ∪ c be a multicurve on Σ. A class of a representation [ρ] ∈ X (π 1 Σ, G
) is called m-regular, if the elements ρ(c 1 ), . . . , ρ(c ) are regular and we denote by M m-reg (π 1 Σ, G) the set all of m-regular characters. Precisely, we set

M m-reg (π 1 Σ, G) := i=1 M c i -reg (π 1 Σ, G).
Since the curves c 1 , . . . , c are disjoint, the actions of the tori U c 1 , . . . , U c on M m-reg (π 1 Σ, G) commute. In this general setting and similarly to Lemmas 2.1 and 2.7, we state: Lemma 2.21. Let [ρ] ∈ X (π 1 Σ, G) and suppose that there exist c 1 , . . . , c pairwise disjoint, simple and closed curves of Σ such that for all non-trivial characters χ : T → S 1 :

χ(t c 1 ([ρ]), . . . , t c ([ρ])) = 1. Then, if we set h := tw c 1 • • • tw c , the action of h on U c 1 × • • • × U c • [ρ]
is ergodic with respect to the Lebesgue measure.

To simplify notation we shall denote by T the product

U c 1 × • • • × U c .
Proof of the lemma 2.21. Let φ be the isomorphism T ∼ = T r and let χ : T → S 1 be a non-trivial character. Then the composition χ • (φ -1 , . . . , φ -1 ) is a non-trivial character of T r , which we may identify with R r /Z r . The action of h is then given by the translation of the vector (θ 1 (ρ(c 1 )), . . . , θ r (ρ(c 1 )), . . . , θ 1 (ρ(c )), . . . , θ r (ρ(c ))), where θ i (ρ(c k )) is the argument of λ i (ρ(c k )). Then, by Lemma 2.3, the action of h is ergodic on the torus orbit

U c 1 × • • • × U c • [ρ]
with respect to the Lebesgue measure if and only if θ 1 (ρ(c 1 )), . . . , θ r (ρ(c 1 )), . . . , θ 1 (ρ(c )), . . . , θ r (ρ(c )) and 1 are linearly independent over Q. As a character of a torus is given by a linear form of R r with integer coefficients, this condition is equivalent to the fact that for all non-trivial characters χ of T r , χ (θ 1 (ρ(c 1 )), . . . , θ r (ρ(c 1 )), . . . , θ 1 (ρ(c )), . . . , θ r (ρ(c ))) = 1.

The proof of the lemma is then complete.

Proof of the ergodicity

We will adapt the previous proofs of ergodicity with the condition of Lemma 2.21.

Definition 2.22. A class of a representations [ρ] ∈ X (π 1 Σ, G) satisfies the condition (M m ), if for all non-trivial characters χ : T → S 1 , χ(t c 1 ([ρ]), . . . , t c ([ρ])) = 1.
We thus introduce

M m (π 1 Σ, G) = [ρ] ∈ X (π 1 Σ, G) | [ρ] satisfies condition (M m ) . Remark 2.23. The set M m (π 1 Σ, G) is contained in M m-reg (π 1 Σ, G).
We hence prove the following: Proposition 2.24. For all m = c 1 ∪ c 2 ∈ MC 0 (Σ), the space M m (π 1 Σ, G) has full measure in the character variety.

As in the previous cases, we prove that the set M m (π 1 Σ, G) is the complement of a countable union of submanifolds of codimension 1. The strategy we use is the same as in Propositions 2.5 and 2.10. For m = c 1 ∪ c 2 ∪ c 3 ∈ MC 0 (Σ), we write the complement of the set of characters which satisfy condition (M m ) as the union:

χ:T 2 →S 1 non-trivial character [ρ] ∈ X (π 1 Σ, G) | χ(t c 1 ([ρ]), t c 2 ([ρ])) = 1 .
We will hence prove that for all non-trivial characters χ : T 2 → S 1 , the set

[ρ] ∈ X (π 1 Σ, G) | χ(t c 1 ([ρ]), t c 2 ([ρ])) = 1
has null measure, being the preimage of 1 by the map ψ χ,m := χ(t c 1 (•), t c 2 (•)) on M m-reg (π 1 Σ, G) which we will show to be a submersion. This is the goal of the following lemmas.

Let N m be the subset of X (π 1 Σ, G) of conjugacy classes of representations ρ such that the centralizers

Z G ρ |π 1 Σ 1 and Z G ρ |π 1 Σ 2 are discrete in G.
With the same explanation than in the case of SU(2), we have the following fact:

Lemma 2.25. The subset N m has full measure in X (π 1 Σ, G)

We will hence prove that N m ∩ M m (π 1 Σ, G) has full measure.

Lemma 2.26. For all non-trivial characters χ : T 2 → S 1 , the map ψ χ,m is a submersion at each point of N m .

Proof. It suffices, for

[ρ] ∈ X (π 1 Σ, G), to find a vector X ∈ T [ρ] X (π 1 Σ, G) such that d [ρ] ψ χ,m X = 0. Write dψ χ,m X = dχ(dt c 1 X, dt c 2 X)
where

d [ρ] t c k X is dφ -1 φ(tc k ([ρ])) d [ρ] λ 1,c k X, . . . , d [ρ] λ r,c k X ,
for φ be the isomorphism T ∼ = T r we use in the proof of Lemma 2.21. With the same notations as the cases of SU(n), by Goldman work W. M. Goldman 1984 part 3.7, the map E 1 : G 2g 1 +1 → G 2 defined by :

E 1 (A 1 , . . . , B g 1 , C) = g 1 j=1 [A j , B j ], C
is a submersion at the point {A 1 , . . . , B g 1 } if the centralizer Z G {A 1 , . . . , B g 1 } is discrete. For the same reasons, the map E 2 : G 2g 2 +1 → G 2 defined by:

E 2 (A 1 , . . . , B g 2 , C) = g 2 j=1 [A j , B j ], C is a submersion at the point {A 1 , . . . , B g 2 } if the centralizer Z G {A 1 , . . . , B g 2 } is discrete.
Hence the evaluation Ev Σ 1 : Hom π 1 Σ 1 , G → G 2 defined by:

Ev Σ 1 (ρ) = (ρ(c 1 ), ρ(c 2 ))
is a submersion at every ρ such that the centralizer

Z G ρ |π 1 Σ 1 is discrete since Ev Σ 1 is the composition of E 1 with the diffeomorphism: G 2 → G 2 (A, B) → (B -1 A, B) .
Since the functions λ i : M → R are submersions at each regular matrix point, the application (λ i,c k ) k=1,2 ; i=1,...,r : Hom π 1 Σ, G → R 2r are submersions at each representation ρ such that both Z G ρ |π 1 Σ 1 and Z G ρ |π 1 Σ 2 are discrete. In particular, for the first indexes k 0 and i 0 such that dχe i 0 , with (e i ) i is the k 0 -th copy of R r , there exists a vector field X such that dt c k X = δ k k 0 e i 0 . This allows to conclude that ψ χ,m is a submersion.

The set of characters T 2 → S 1 is countable because such a character is given by a linear form χ : R 2r → R, such that χ(Z 2r ) ⊂ Z. Hence there is a countable number of possibilities to obtain characters of the 2r-torus, looking the image by χ of the canonical basis. Since the complement of M m (π 1 Σ, G) is a countable union of codimension 1 submanifolds and hence a countable union of null measure sets, we conclude to the statement of Proposition 2.24.

The proof of the ergodicity is completed using the same arguments as in the previous cases. We thus prove that all Tor(Σ)-invariant and measurable functions X (π 1 Σ, G) → R can be restricted to a full measure set on which it will be invariant under the generators of the mapping class group and then Theorem 1 allows to conclude that such a function is almost everywhere constant, that is the Torelli group action on X (π 1 Σ, G) is ergodic. Therefore, we derived the Theorem A.

Modular action on relative PSL 2 (R)-character varieties

This section is devoted to the proof of Theorem C. We adapt the strategy and the tools which was used by [START_REF] Marché | Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group[END_REF] We denote by EI e ( Σ, C) the subspace of N H e ( Σ, C) of classes of representations which admit a simple, closed and non-separating curve γ ⊂ Σ which is not homotopic to a component of ∂ Σ and such that the isometry ρ(γ) is elliptic with infinite order.

The next results are steps in order to prove Theorem C. The proofs of these proposition will be found in the next pages. Let U e ( Σ, C) be the open subset of classes of representations [ρ] for which there is N = 6g -6 + 2n curves γ 1 , . . . , γ N ∈ π 1 Σ such that for all i ∈ {1, . . . , N } the isometry ρ(γ i ) is elliptic and:

df γ i | i = 1, . . . , N = T * [ρ] Rep k ( Σ, C Proposition 3.3. The space EI e ( Σ, C) is contained in U e ( Σ, C).
Assuming the Propositions 3.1, 3.2 and 3.3, we prove the ergodicity of the pure mapping class group:

Proposition 3.4. Let f : U e ( Σ, C) → R be a measurable and Mod( Σ)-invariant. Then each [ρ] ∈ U e ( Σ, C) is contained in a neighborhood V [ρ] such that the restriction of f to V [ρ] is almost everywhere constant.
The proof is essentially the same than the one of Proposition 6.5 of Marché and Wolff 2016. Proof. Let [ρ] ∈ U e ( Σ, C) and let γ 1 , . . . γ N be the curves given by the definition of U e ( Σ, C). Let us denote by φ i its Hamiltonian flow of the vector field X γ i . This flow is 2π-periodic and the action of tw γ i is given by:

tw γ i • [ϕ] = φ θγ i i • [ϕ].
We deduce that for every

[ϕ] such that ϕ(γ i ) elliptic, if θ γ i ([ϕ]) / ∈ πQ, an invariant function f : U e ( Σ, C) → R satisfies the equality f (φ θ i • [ϕ]) = f ([ϕ]), for almost-every θ ∈ S 1 . Let V [ρ]
be an open neighborhood such that every [ϕ] ∈ V [ρ] sends the γ i on elliptic isometries and satisfies the condition:

X γ 1 , . . . , X γ N = T [ϕ] Rep k ( Σ, C . Up to reducing V [ρ]
, we can assume that the flows φ i act transitively on V [ρ] and such that f |V [ρ] is almost-everywhere constant on almost of their orbits. We hence deduce that f |V [ρ] is almost-everywhere constant.

The ergodicity of PMod( Σ) on N H e ( Σ, C) comes from Proposition 3.4 and the fact that U e ( Σ, C) has full measure in N H e ( Σ, C) and is connected since it is open and since it contains the full measure and connected subset EI e ( Σ, C).

Condition on e to have EI e ( Σ, C) = ∅

We already now from Mondello 2017 that the set X e ( Σ, C) is non-empty if and only if e = k -{r} 1 with k ∈ Z ∩ {r} 1 , -χ( Σ) . Consider Σ a connected, compact and oriented surface of genus g -1 and with n + 2 boundary components, denoted by c 1 , . . . , c n , d 1 , d 2 . Let A ⊂ PSL 2 (R) be a conjugacy class of an elliptic isometry of angle υ A / ∈ Z ∪ πQ and hence of translation number υ A π . Denote by C = (C 1 , . . . , C n , A, A -1 ) and by Rep(Σ , C ) the relative character variety associated to Σ and C . Since

| υ A π -υ A π | + | -υ A π --υ A π | = 1 and χ( Σ) = χ(Σ ), Theorem 7 shows that for e ∈ R >0 , the pre-image Rep e (Σ , C ) is non-empty if and only if e = k -1 -{r} 1 with k -1 ∈ Z ∩ 0, -χ( Σ) . In other words if and only if e = k -{r} 1 with k ∈ Z ∩ 0, -χ( Σ) .
The assumption e > 0 imposes k > {r} 1 . Under this condition, Rep e (Σ , C ) is connected and smooth.

The fundamental group π 1 Σ is the HNN-extension

[π 1 Σ * b ]/ bd 1 b -1 = d 2 .
Proof. We can easily find for all t ∈ 0, 1 , an isometry

D 1 (t) ∈ PSL 2 (R) such that A(t)D 1 (t)
is hyperbolic and such that the map t → D 1 (t) is continuous. We hence choose D 2 (t) to be the isometry D 1 (t) -1 A(t) -1 . Hence we define the expected path A : 0, 1 → Hom(π 1 P, PSL 2 (R)) by the data

A(t)(c) = A(t), A(t)(d 1 ) = D 1 (t) and A(t)(d 2 ) = D 2 (t).
As it is state previously, fixing a representation ρ of the pair of pant P with Euler class ±1 and with ρ(d) = A(0) corresponds to fix D 1 , D 2 two hyperbolic isometries such that A(0)D 1 D 2 = id. Hence we the same reasoning than the proof of Lemma 3.7, fixing D 1 (0) = D 1 , we can extend the path A in a path of representations A as expected in the lemma with the condition A(0) = ρ.

Lemma 3.8. Let S n+1 be a n + 1-sphere with as boundary components the curves s, c 1 , . . . , c n . Let τ be a real number such that Rep τ (S n+1 ) and the next boundary conditions are non-empty. Let A : 0, 1 → PSL 2 (R) be a continuous path whose image is contained in the subset of hyperbolic isometries and fix ρ : π 1 S n+1 → PSL 2 (R) such that eu(ρ) = τ and A(0) = ρ(s). Then there exists a continuous path of representations

A : 0, 1 → Hom(π 1 S n+1 , PSL 2 (R)) such that A(0) = ρ and for all t ∈ 0, 1 , eu(ρ) = τ A t (s) = A(t), A t (c 1 ) ∈ C 1 , . . . , A t (c n ) ∈ C n .
Fix ρ with the hypothesis of the Lemma 3.8 is equivalent to fix C i ∈ C i for all i ∈ {1, . . . , n} with A(0)C 1 . . . C n = id. We hence fix these isometries. Up to restrict our reasoning to a Zariskiopen subset of the representation space, we can assume that the commutator [C , C i ] = id for all , i ∈ {1, . . . , n} distinct.

Proof. We are looking for continuous paths g i : 0, 1 → PSL 2 (R) such that for all t ∈ 0, 1 , we have

A(t) × g 1 (t)C 1 g 1 (t) -1 × • • • × g n (t)C n g n (t) -1 = id. Let K : PSL 2 (R) n → PSL 2 (R) be the map defined by K(g 1 , . . . , g n ) = g 1 C 1 g -1 1 × • • • × g n C n g -1 n .
We will prove the claim.

Claim 3.9. The map K is a submersion at the point (g 1 , . . . , g n ) such that each g does not send the fix point of C on the fix points of the other C i .

As in W. M. Goldman 1984, for j ∈ {1, . . . , g -1}, let Π j the product

j k=1 g k C k g -1 k
and by convention we let Π 0 = id. By Fox calculus, we compute that for all = 1, . . . , n:

∂ ∂g K(g 1 , . . . , g n ) = Π -1 (id -g C g -1 )
We refer to Fox 1953 and W. M. Goldman 1984, for basics on Fox calculus. Since:

d (g 1 ,...,gn) K = n =1 Ad ∂ ∂g K(g 1 , . . . , g n ) d g ,
we have that the orthogonal space d (g 1 ,...,gn) K sl 2 (R) n ⊥ is the intersection:

n =1 Ker id -Ad Π -1 g C g -1 Π -1 -1
.

Each kernel of the intersection is the centralizer in sl 2 (R) of the element Π -1 g C g -1 Π -1 -1 . Up to change the representative in the conjugacy class of C 1 , we can assume that g 1 = id. Hence we deduce that d (g 1 ,...,gn) K sl 2 (R) n ⊥ is the intersection

n =1 Z sl 2 (R) Π -1 g C g -1 Π -1 -1 . Finally, since {g 1 C 1 g -1 1 , . . . , Π n-1 g n C n g -1 n Π -1 n-1 } generates the same subgroup than {g 1 C 1 g -1 1 , . . . , g n C n g -1 n },
we conclude that if for all , g does not send the fix point of C on the fix point of a C i , the orthogonal space d (g 1 ,...,gn) K sl 2 (R) n ⊥ is trivial. Then the rank of K at the point (g 1 , . . . , g n ) for which each g does not send the fix point of C on the fix point of another C i , is maximal. This proves the claim.

In particular K is a submersion at the point (id, . . . , id).

Then for each t ∈ 0, 1 , the preimage K -1 (A(t) -1 ) is a codimension 1 submanifold of PSL 2 (R) n . There exists then a finite covering 0, 1 = ∪ j=0,...,κ-1 t j , t j+1 with t 0 = 0 and t κ = 1 such that on each t j , t j+1 , by Claim 3.9, there is paths g j : t j , t j+1 → PSL 2 (R), for = 1, . . . , n such that for all t ∈ t j , t j+1 , the relation

A(t) × g j 1 (t)C 1 g j 1 (t) -1 × • • • × g j n (t)C n g j n (t) -1 = id
is verified. In addition we have that for all ∈ {1, . . . , n}, g 0 (0) = id.

At each t j , since K t j is a submersion, there is a path which connect g j-1 (t j -ε) and g j (t j + ε). So up to a reparametrization, there exists continuous paths g : 0, 1 → PSL 2 (R) for = 1, . . . , n verifying the expected condition.

We hence define A t as the representation

π 1 S n+1 → PSL 2 (R) defined by A t (s) = A(t), A t (c 1 ) = g 1 (t)C 1 g 1 (t) -1 , . . . , A t (c n ) = g n (t)C n g n (t) -1 .
To conclude Proposition 3.5, we decompose Σ by pair of pants P i with i = 1, ..., p and c ⊂ ∂P 1 , g one-holded tori and a n + 1-sphere. By Lemma 3.7 we can find a path of representations of the g + 1-sphere by restricting ρ to each P i and gluing the extension we obtain. We extend this path extend to one-holded tori glued to g -1 boundary components by Lemma 3.6 and to the n + 1-sphere by Lemma 3.8. This gives a paths of representations of Σ which extend the path A and with initial condition ρ. Proof. Let (a, b) ∈ S 1 and let Σ 0 be a one-holded torus containing both a and b and let Σ be the surface of genus g -1 and with n + 1 boundary components c, c 1 , . . . , c n such that the surface Σ is the connected sum

Σ 0 Σ .
Let M 0 be the set of classes of representations π 1 Σ 0 → PSL 2 (R) sending a and b both on elliptic isometries which do not commute. We will denote by x ρ(σ) the fix point of ρ(σ) for σ ∈ {a, b}. The map M 0 → (0, 2π) × (0, 2π) × (0, +∞) defined by:

[ρ] → θ a (ρ), θ b (ρ), d H 2 (x ρ(a) , x ρ(b) )
is a homeomorphism. Let M 0 be the subset of M 0 of classes of representations which send a or b on an elliptic isometry of infinite order. Since in (0, 2π) × (0, 2π), the pairs (x, y) such that x / ∈ πQ or y / ∈ πQ form a connected space, the subspace M 0 is connected. We now prove the connectedness of EI e (a,b) ( Σ, C). Let [ρ], [ρ ] ∈ EI e (a,b) ( Σ, C) and let ρ 0 and ρ 0 be the restrictions to π 1 Σ 0 of ρ and ρ . The classes [ρ 0 ] and [ρ 0 ] lie in M 0 . Then, by connectedness of M 0 , there exists a continuous path A 0 : 0, 1 → M 0 such that A 0 (0) = [ρ 0 ] and A 0 (1) = [ρ 0 ]. this path induces a continuous path A : 0, 1 → PSL 2 (R), defined by A(t) = A 0 (t) ([a, b]), whose image is included in the set of hyperbolic isometries. Proposition 3.5 allows to conclude to the existence of a continuous path A in

EI e (a,b) ( Σ, C) such that A(0) = [ρ] and A(1)([a, b]) = ρ ([a, b]).
In other words, A(1) |π 1 Σ and ρ |π 1 Σ lie in the same relative character variety and have the same Euler class. By Mondello Theorem 7, there exists a continuous path joining A(1) |π 1 Σ and [ρ |π 1 Σ ]and thus there is a path joining A(1) and [ρ]. We then found a path joining [ρ] and [ρ ] in EI e (a,b) ( Σ, C), which conclude the connectedness of this space.

We introduce now the equivalence relation on S 1 generated by the relation

(a, b) ∼ (a , b ) ⇐⇒ EI e (a,b) ( Σ, C) ∩ EI e (a ,b ) ( Σ, C) = ∅. Lemma 3.13. Let (a, b) ∈ S 1 . For all n ∈ Z, (a, b) ∼ (a, a n b).
Proof. Let Σ 0 and Σ as in the previous proof. We will construct a class of representations in

EI e (a,b) ( Σ, C) ∩ EI e (a,a n b) ( Σ, C
) starting by construct it on Σ 0 and extend it by Proposition 3.5 to Σ (and thus to Σ). Let A and B be the matrices 0 -1 1 0 and 1 -1 1 -ε ε for ε ∈ 0, 1 . We then compute that A n B is elliptic. Up to change ε, we can assume that [A, B] is hyperbolic. We construct ρ 0 : a, b → PSL 2 (R) defined by ρ 0 (a) = A and ρ 0 (b) = B. Apply Proposition 3.5 to the constant path t → [A, B] allows to construct a class of representations [ρ] such that ρ |π 1 Σ 0 = ρ 0 and then

[ρ] ∈ EI e (a,b) ( Σ, C) ∩ EI e (a,a n b) ( Σ, C).
The two following lemmas come from Marché and Wolff 2016 (Lemmas 6.10 and 6.11).

Lemma 3.14. (Marché and Wolff 2016, Lemma 6.10) Let (a, b) ∈ S 1 and c be a simple, closed and non-separating which is disjoint of b such that (a, c) ∈ S 1 . Then (a, b) ∼ (a, c).

Proof. Let [ρ] ∈ EI e (a,b) ( Σ, C
) and assume that ρ(a) has infinite order. By lemma 3.10, there exists an integer n ∈ Z such that ρ(a n c) is elliptic. We necessarily have that tw n a • ρ (a) = ρ(a) is an elliptic isometry of infinite order and the isometries tw Proof. If a lies in the one-holded torus obtained by thickening a∪b, we apply Lemma 3.14 to concude.

If not, we process by induction on i(a, a ):

• If i(a, a ) = 0, we apply Lemma 3.14. b such that for i ∈ {0, . . . , n}, the curves b i and b i+1 are disjoint. In addition, we can find nonseparating curves a 0 = a, . . . , a n = a such that (a i , b i ) and (a i , b i+1 ) are in S 1 . Lemma 3.14 assures that (a i , b i ) ∼ (a i , b i+1 ) and hence Lemma 3.15 that (a i , b i ) ∼ (a i+1 , b i+1 ) for all i ∈ {0, . . . , n -1}. By Lemmas 3.11 and 3.12 we conclude that EI e ( Σ, C) is connected.

• If i(a, a ) = n > 0,
If we now assume that δ is separating, fix a base point on δ. then there is a curve σ such that i(γ, σ) = 1 and i(δ, σ) = 2. We can decompose σ as a product σ 1 σ 2 with i(γ, σ 1 ) = 1. We hence have that

f σ f δ = f σ 1 σ 2 f δ = f σ 1 f σ 2 δ -f σ 1 δ -1 σ -1 2 + f σ 2 f σ 1 δ -1 -f σ 1 σ -1 2 δ -1 .
Since the curves σ 2 and σ 2 δ are non-separating and do not intersect γ, by the non-separating case, we have that

df σ 2 , df σ 2 δ ∈ F [ρ] . Since the curves σ 1 , σ 1 δ -1 σ -1 2 , σ 1 δ -1 , σ 1 σ -1 2 δ -1
and σ 1 σ 2 are nonseparating and intersect γ once, Lemma 3.16 ensures that the differential of traces corresponding to these curves are in F [ρ] . We hence prove, by derivation, that df δ ∈ F [ρ] .

Lemma 3.18. There exists a curve δ such that i(γ, δ) = 1 and such that the Poisson bracket {f γ , f δ } := df γ X δ = 0. In particular, the map f γ is a submersion at each class of representation of Rep e ( Σ, C) which send γ on elliptic isometry of infinite order and the subset

f -1 γ 2 cos θ (C) e := Rep e ( Σ, C) ∩ f -1 γ 2 cos θ is a codimension 1-submanifold of Rep e ( Σ, C). Proof. Let ρ(γ) = A(θ) and ρ(δ) = B.
Then for all n ∈ Z, the Poisson bracket {f γ , f δn } = 0 if and only if tr A (0)A(nθ)B = 0. This is impossible since θ / ∈ πQ.

We will denote by A the conjugacy class corresponding to the matrices of trace 2 cos θ and by

C = (C 1 , . . . , C n , A, A -1 ).
We show now that F [ρ] coincides with the space T *

[ρ] Rep e ( Σ, C). Let X ∈ T [ρ]
Rep e ( Σ, C) be a vector orthogonal to F [ρ] . We have df γ X = 0 whose we deduce that X is tangent to f -1 γ 2 cos θ (C) e . Let r : f -1 γ 2 cos θ (C) e → Rep e ( Σ \ γ, C ) be the restriction map. It is well defined since the image by r of a Zariski-dense representation is Zariski-dense. Since the cotangent space of Rep e ( Σ \ γ, C ) is generated by all the linear form df σ with σ disjoint from γ and ∂ Σ, Lemma 3.17 implies that drX = 0 whose we deduce that X ∈ X γ . There exists u ∈ R such that X = uX γ . Let δ be a curve given by Lemma 3.18. Lemma 3.16 show that df δ ∈ F [ρ] and hence df δ X = 0. But

df δ X = u × df δ X γ = -u × {f γ , f δ }. Since {f γ , f δ } = 0, we have u = 0. This proves Proposition 3.3. 3.5 EI e ( Σ, C) has full measure in N H e ( Σ, C)
The proof of Marché and Wolff 2016 holds in our case and we write these to explain how to adapt the proof to boundary case. Let N e ( Σ, C) be the subset of X e ( Σ, C) of classes of representations which send simple and closed curves, not homotopic to a boundary component, either on elliptic isometries of infinite order or hyperbolic isometries. It is clearly a full measure subset because its complementary subset is the union:

γ t∈2 cos(πQ) f -1 γ (t)
which is a countable union of subset of measure zero.

We will then show that EI e ( Σ, C) ∩ N e ( Σ, C) has full measure in N H e ( Σ, C) ∩ N e ( Σ, C).

If [ρ] ∈
N H e ( Σ, C) ∩ N e ( Σ, C) then there exists a simple and closed curve γ / ∈ Σ \ ∂ Σ such that ρ(γ) is elliptic of infinite order. If γ is freely homotopic to a non-separating curve, then [ρ] ∈ EI e ( Σ, C) ∩ N e ( Σ, C). It is so sufficient to treat the case of γ is freely homotopic to a separating curve.

Lemma 3.19. Let Σ be a compact, connected and oriented surface of genus g ≥ 1 and with k ∈ {1, . . . , n + 1} boundary components denoted by c, c 1 , . . . , c k-1 . Let ρ : π 1 Σ → PSL 2 (R) such that:

• The condition ρ(c i ) ∈ C i holds.

• No simple and closed curve other than c, c 1 , . . . , c k-1 is send by ρ to an elliptic isometry of finite order, or on a parabolic isometry or on the identity.

• ρ(c) is an elliptic isometry of infinite order of fixed point x 0 ∈ H 2 .

• Every non-separating curve of Σ \ ∂Σ is sent on an hyperbolic isometry.

Then there exists a real number D > 0 and a sequence of non-separating and simple loops

(γ n ) n such that ρ(γ n ) is hyperbolic with displacement λ ρ(γ n ) = D and axis A n such that d H 2 x 0 , A n ---→ n→∞ +∞.
For f ∈ PSL 2 (R), we will denote by A f its invariant axis if f is hyperbolic and by x f its fixed point if f is elliptic.

Proof.

• If g ≥ 2, we refer to the proof of Lemma 6.16 inMarché and Wolff 2016.

• As in Marché and Wolff 2016, a triplet (x, y, z) ∈ R ≥0 3 satisfies the condition (Hex), if there exists a right-angled hexagon H ⊂ H 2 with three consecutive sides of respective lengths x, y and z.

Remark 3.20. For all sequences (x n ) n , (y n ) n , (z n ) n ∈ (R ≥0 N which do not accumulate on zero and such that either y n ---→ n→∞ +∞ or x n , z n ---→ n→∞ +∞, then for all n enough large, (x n , y n , z n ) satisfies (Hex). Moreover, (Hex) is an open condition and if (x, y, z) satisfies (Hex), then for all x ≥ x, y ≥ y and z ≥ z, (x , y , z ) satisfies (Hex).

Lemma 3.21. Let [ρ] ∈ X e ( Σ, C) such that ρ send a separating curve c on an elliptic isometry of infinite order. Assume ρ ∈ N e ( Σ, C). Then there exists a non-separating curve which is send by ρ on an elliptic element. is elliptic for a suitable N ∈ Z. If they disagree, we apply the case where the orientations agree to the isometries A and C N BC -N for a suitable N and we hence conclude. The Lemma 3.21 allows to conclude that EI e ( Σ, C) has full measure in N H e ( Σ, C). Indeed, N e ( Σ, C) has full measure in the character variety and we showed that EI e ( Σ, C) ∩ N e ( Σ, C) is equal to N H e ( Σ, C) ∩ N e ( Σ, C).

ρ : π 1 Σ -→ SU(2) with a dense image -Previte-Xia defined such a representation generic (see Previte and Xia 2000) -they firstly found a handle Σ 0 , namely a one-holed torus, such that the restriction of ρ to π 1 Σ 0 is dense. After obtaining a dense handle, they proceed to demonstrate the base density theorem for the (n + 2g -2)-holed torus. A similar process in the abelian case is not possible because dense handles do not always exist, see Example 4.24. In the light of Proposition 4.10, we shall bypass this issue by looking at the Sp 2g (Z)-action on the representation space as described above.

The present section is organised as follow. In subsection 4.2 we begin with a description of the T ncharacter variety and then subsequently introduce the homological representation space and show the identification with the character variety. We finally describe the action of the symplectic group Sp 2g (Z) on the homological representation space. As a consequence, we shall derive the Proposition 4.10 and the equivalence of Theorems D and F. In subsection 4.7 we shall give a complete characterisation of dense representations in the n-dimensional torus by proving Theorem 4.21. In subsection 4.8 we shall finally derive our main Theorem D. In the last section, we prove Proposition 4.9 and indeed Theorem E establishing the connection of our dynamical result with the Kronecker's Approximation Theorem. We finally conclude with a serie of appendix on which we shall discuss some further aspects related to our project. Appendix 4.10 we discuss about a direct approach to our problem which works for a fairly general class of representations. In Appendix 4.10.3 we digress a little by providing a brief description of the relative T n -character variety for surfaces with one puncture and then we claim that our main results extend to one-punctured surfaces.

T n -character variety

In this work we are interested in characterising the orbits of the Mod(Σ)-action on Hom(π 1 Σ, G)/G where G is a compact, connected and abelian Lie group. It is classical to see that any such a group is isomorphic to the n-dimensional torus T n . The specific interest for the abelian case comes from its connection with abstract harmonic analysis, the geometry of numbers and the theory of group actions on homogeneous spaces (connections with Ratner's Theorem, see section 4.8).

In the introduction we have given a very brief view of the character variety for a generic compact Lie group G. In this section we specialise the discussion for compact and connected abelian Lie groups. From the Lie theory, any such a group is known to be a n-dimensional torus, namely the product of n copies of the unit circle S 1 . In the present work S 1 is seen as e iθ | θ ∈ 0, 2π where 0, 2π carries the quotient topology obtained identifying the boundary points of the closed interval [0, 2π]. Consequently, the n-torus T n is defined as e iθ 1 , . . . , e iθn | θ i ∈ 0, 2π , for any i = 1, . . . , n endowed with the product topology.

Let Σ be a closed surface and let a 1 , b 1 , . . . , a g , b g be any standard generating system of the fundamental group. The choice of a representation ρ : π 1 Σ -→ T n amounts to choose for each gen-erator an element of T n such that these elements satisfy the condition imposed by the presentation of the fundamental group ofΣ. However, the abelian property of T n implies that the condition

[A 1 , B 1 ] • • • [A g , B g ] = 1
is automatically satisfied for any choice of 2g elements in (A 1 , B 1 , . . . , A g , B g ) ∈ T n . Thus, the representation space can be identified with the group T n 2g ∼ = T 2ng . Even more, thanks again to the abelian property, the action of T n on Hom π 1 Σ, T n by post-composition with inner automorphisms of T n is trivial. As a consequence, the T n -character variety coincides with the representation space.

Remark 4.2. Let Σ be any surface of genus g ≥ 2. The representation space Hom π 1 Σ, T n splits as the direct sum of g copies of Hom π 1 T, T n where T denote the 2-torus. The basis a 1 , b 1 , . . . , a g , b g of π 1 Σ we fixed satisfies moreover to the equalities i a i , a j = i b i , b j = 0, and i a i , b j = δ ij for all i, j with 1 ≤ i, j ≤ g. We may associate to any representation ρ : π 1 Σ -→ T n the g-tuple of representations ρ 1 , . . . , ρ g where ρ i is the restriction of ρ to the handle generated by a i , b i . Such a mapping defines then an isomorphism

Hom π 1 Σ, T n ∼ = g i=1 Hom a i , b i , T n .
which depends on the basis chosen. This decomposition is a consequence of the fact that a surface of genus g is the connected sum of a surface of genus g -1 and a torus T along with the property that each representation ρ sends all simple closed separating curves to the identity. A recursive argument leads to the desire conclusion.

Homological representations

Let H 1 (Σ, Z) be the first homology group. The close connection between the objects π 1 Σ and H 1 (Σ, Z) is well-known, indeed the latter is known to be isomorphic to the abelianization of π 1 Σ. As we have seen above, the representation space Hom π 1 Σ, T n naturally identifies with the 2gndimensional torus assigning to any representation ρ the 2g-tuple ρ(a 1 ), ρ(b 1 ), . . . , ρ(a g ), ρ(b g ) , where a 1 , b 1 , . . . , a g , b g is a basis for π 1 Σ. Every representation ρ fails to be injective and its kernel ker(ρ) always contains the subgroup generated by the commutators since the target is abelian. Therefore, ρ boils down to a representation

ρ : H 1 (Σ, Z) ∼ = π 1 Σ π 1 Σ, π 1 Σ -→ T n , ρ γ := ρ(γ).
In fact, let γ ∈ π 1 Σ and let γ be its image via the canonical projection p : π 1 Σ -→ H 1 (Σ, Z). Let γ σ 1 , σ 2 be a representative of γ . Since the following chain of equalities holds

ρ γ σ 1 , σ 2 = ρ γ ρ σ 1 , σ 2 = ρ γ , Definition 4.6. A collection of elements a i , b i , 1 ≤ i ≤ g of H 1 (Σ, Z) < H 1 (Σ, R) such that a i ∩ a j = b i ∩ b j = 0, a i ∩ b j = δ ij
for all i, j with 1 ≤ i, j ≤ g is called a symplectic basis of the group H 1 (Σ, Z) or a basis for the symplectic vector space H 1 (Σ, Z), ∩ . We define a collection of curves a i , b i such that [a i ], [b i ] is a symplectic basis as geometric symplectic basis for π 1 Σ.

The matrix associated to the antisymmetric bilinear form ∩ on the basis a i , b i is the 2g × 2g blockwise diagonal matrix

J =     J o . . . J o     with J o = 0 1 -1 0 .
The symplectic linear group Sp 2g (R) is defined as the group of invertible matrices A satisfying the relation AJA t = J and we denote by Sp 2g (Z) the subgroup of those matrices with integer coefficients. An orientation preserving homeomorphism induces an isomorphism in homology which preserves the intersection form ∩ defined above. Since isotopic homeomorphisms induce the same map in homology, there is a representation

µ : Mod(Σ) -→ Aut + H 1 (Σ, Z) ∼ = SL 2g (Z).
As each homeomorphism preserves the intersection form ∩, the image of µ lies also inside Sp 2g (R).

Therefore the image of µ lies inside SL 2g (Z) ∩ Sp 2g (R). The representation µ : Mod(Σ) -→ Sp 2g (Z)

usually called symplectic representation of Mod(Σ). The Torelli group is the kernel of µ.

Remark 4.8. In the genus one case the Torelli subgroup is trivial. Indeed, Mod(T ) ∼ = SL 2 (Z) = Sp 2 (Z).

4.4.2 Comparison of the Mod(Σ)-orbits with the Sp 2g (Z)-orbits.

We now consider the effect of changing the basis of H 1 (Σ, Z) pre-composing any homological representation with an automorphism φ ∈ Aut + H 1 (Σ, Z) such that any representation ρ is sent to ρ • φ -1 . We can, therefore, consider SL 2g (Z)-action on the space Hom H 1 (Σ, Z), T n . Of course, this action restricts to an action of the symplectic group Sp 2g (Z) on the same space. We are interested in studying the Sp 2g (Z)-orbits in the homological representation space. The main goal of this section is proving the following claim.

Proposition 4.9. Let ρ 1 , ρ 2 : π 1 Σ -→ T n be two representations and let ρ 1 , ρ 2 : H 1 (Σ, Z) -→ T n be the induced representations. Suppose there is φ ∈ Mod(Σ) such that ρ

2 = ρ 1 • φ. Then ρ 2 = ρ 1 • µ(φ),
where µ is the symplectic representation of Mod(Σ).

Proof. Let φ : π 1 Σ -→ π 1 Σ be any element of Out(π 1 Σ). As the image of any commutator is also a commutator, the mapping φ boils down to a isomorphism in homology µ(φ) : H 1 (Σ, Z) -→ H 1 (Σ, Z).

Two mappings φ 1 and φ 2 boil down to the same isomorphism in homology if and only if φ 2 • φ -1 1 descends to the identity map in homology, that is

φ 2 • φ -1
1 is an element of the Torelli subgroup. Therefore, the association φ -→ µ(φ) defines the symplectic representation µ seen above. Look at now the following commutative diagram

H 1 (Σ, Z) T n π 1 Σ π 1 Σ T n H 1 (Σ, Z) H 1 (Σ, Z) T n ρ 2 =ρ 1 •φ p p φ ρ 1 id id µ(φ) ρ 1 (3)
where p is the canonical projection. As ρ 2 = ρ 1 • φ by assumption, it turns out

ρ 2 = ρ 1 • φ = ρ 1 • µ(φ)
as desired.

Direct consequences.

Proposition 4.9 leads to some interesting consequences that we are going to show. The first one concerns the action of the Torelli subgroup Tor(Σ) on the representation space Hom π 1 Σ, T n .

Proposition 4.10. The action of the Torelli group Tor(Σ) on the representation space Hom π 1 Σ, T n is trivial.

Proof. Let ρ 1 ∈ Hom π 1 Σ, T n be any representation and let φ ∈ Tor(Σ). Set

ρ 2 = φ • ρ 1 = ρ 1 • φ -1 .
Proposition 4.9 implies that ρ 1 = ρ 2 because µ(φ) = 1. We now invoke Lemma 4.5 to conclude ρ 1 = ρ 2 , namely the action of φ is trivial.

The n-torus T n is a compact and connected Lie group and hence mapping class group Mod(Σ) acts ergodically on the representation space -see Theorem 1 in the introduction. As the action of the Torelli subgroup Tor(Σ) is trivial, the action of the quotient group is also well-defined and the following holds. As the homological representation spaces identifies with the representation space, it also carries a finite measure. Calling ı the identifying map, this finite measure can be seen as the pull-back measure ı * µ Σ , where µ Σ is the finite measure carried by the representation space.

Corollary 4.12. The action of the symplectic group Sp 2g (Z) on the space Hom H 1 (Σ, Z), T n is ergodic with respect to the finite measure ı * µ Σ .

As a final consequence we have the following characterisation. Proof. Proposition 4.9 implies that the mapping class group orbit of ρ coincides with the symplectic group orbit of ρ via the identification ρ → ρ. Therefore, one orbit is dense if and only if the other is dense.

Corollary 4.14. Let Σ be a surface of genus g ≥ 1. Then Theorem D holds if and only if Theorem F holds.

The matrix presentation

The n-torus T n can be seen also as the quotient of R n by the standard action of the lattice 2πZ n , indeed the exponential map provides an identification between R n /2πZ n and the n-torus described above. We shall define the map exp : R n -→ T n θ 1 , . . . , θ n -→ e iθ 1 , . . . , e iθn (4) as the canonical projection. We define 2πZ n as the the standard latticenotice that this lattice is 2π times the usual standard lattice.

Fix a set of generators a 1 , b 1 , . . . , a g , b g and let us consider T n as the quotient of R n by the action of the standard lattice. Let ρ : π 1 Σ -→ T n be any representations and set ρ(a i ) = e iθ 1,2i-1 , . . . , e iθ n,2i-1 ρ(b i ) = e iθ 1,2i , . . . , e iθ n,2i

for any i = 1, . . . , n. The elements ρ(a 1 ), ρ(b 1 ), . . . , ρ(a g ), ρ(b g ) generate the image of the representation ρ. Any element γ ∈ π 1 Σ may be seen as a word in the letters a i , b i for i = 1, . . . , 2g. Hence, by the Abelian property, ρ(γ

) = ρ w(a 1 , b 1 , . . . a g , b g ) is equal to ρ(a 1 ) k 1 • • • ρ(b g ) k 2g for some k 1 , . . . , k 2g ∈ Z.
In particular, the element ρ(γ) can be computed with the following matrix multiplication

    θ 1,1 • • • θ 1,i • • • θ 1,2g . . . . . . θ n,1 • • • θ n,i • • • θ n,2g              k 1 . . . k i . . . k 2g          .
Definition 4.15. Let Θ ρ be the matrix having as entries the values θ i,j ∈ 0, 2π with i = 1, . . . , n and j = 1, . . . , 2g. We define Θ ρ as the matrix associated to ρ with respect the basis a 1 , b 1 , . . . , a g , b g and the standard lattice 2πZ n .

In what follows, we shall often identify a representation ρ with its associated matrix. Let us briefly see the reason why we are legitimated to do that. Consider the topological vector space M n, 2g; R and introduce an equivalence relation where A ∼ B if and only if A-B = 2πH ∈ M n, 2g; 2πZ . The mapping ı associating to any ρ its associated matrix Θ ρ provides an homeomorphism between the representation space Hom π 1 Σ, T n and the quotient space M n, 2g, T . Moreover, the post-composition of the mapping ρ → ρ with ı -1 defines a homeomorphism between the spaces Hom H 1 (Σ, Z), T n and M n, 2g, T .

Given a representation ρ, the matrix Θ ρ depends on the choice of a set of generators for π 1 Σ and also on the choice of a lattice Λ < R n . Let us see how these choices affect definition 4.15. We begin describing the effect of changing the set of generators of π 1 Σ.

The effect of changing basis.

Given two basis B = a 1 , b 1 , . . . , a g , b g and B = a 1 , b 1 , . . . , a g , b g of π 1 Σ, we define Θ ρ and Θ ρ the matrices associated to ρ : π 1 Σ -→ T n with respect to B and B respectively. Every generator a l and b l is a finite word in the letters a 1 , b 1 , . . . , a g , b g , so there are integers a ij with i, j ∈ 1, . . . , 2g such that

ρ(a l ) = ρ(a 1 ) a 2l-1 1 • • • ρ(b g ) a 2l-1 2g and ρ(b l ) = ρ(a 1 ) a 2l 1 • • • ρ(b g ) a 2l 2g .
Setting A as the integral matrix a ij with i, j ∈ 1, . . . , 2g , a direct computation shows that Θ ρ equals Θ ρ • A. Likewise, a l , b l are also finite words in the letters a 1 , b 1 , . . . , a g , b g . Hence, there exist integers b ij such that

ρ(a l ) = ρ(a 1 ) b 2l-1 1 • • • ρ(b g ) b 2l-1 2g and ρ(b l ) = ρ(a 1 ) b 2l 1 • • • ρ(b g ) b 2l 2g .
Setting B as the integral matrix b ij with i, j ∈ 1, . . . , 2g , the same computation implies Θ ρ equals

Θ ρ • B.
It is worth noticing Θ ρ = Θ ρ • AB and the matrices A, B satisfy the equation AB = I 2g implying that A, B are unimodular. As the matrix Θ ρ can be singular we cannot directly deduce that AB = I 2g , hence let us give a glimpse of why this is true.

Instead of working in π 1 Σ we are going to look at the situation in the first homology group

H 1 (Σ, Z) ∼ = Z 2g . Let a 1 = w a 1 , b 1 , . . . , a g , b g , in particular a 1 = a 1 b 1 1 b 1 b 1 2 • • • a g b 1 2g-1 b g b 1 2g
where b 1j with j = 1, . . . , 2g are as above. On the other hand, any a l and b l is of the form

a l = a 1 a 2l-1 1 b 1 a 2l-1 2 • • • a g a 2l-1 2g-1 b g a 2l-1 2g , b l = a 1 a 2l 1 b 1 a 2l 2 • • • a g a 2l 2g-1 b g a 2l 2g
.

where a ij are as above. Replacing each a l and b l inside w = a 1 , for any l = 1, . . . , g, we obtain

a 1 = a 1 k 1 b 1 k 2 • • • a g k 2g-1 b g k 2g .
As Z 2g is torsion-free, we may deduce that k 1 = 1 and k 2 = • • • = k 2g = 0. On the other hand, it is straightforward to see that k m = 2g r=1 b 1r a rm . Applying the same reasoning to any other generator we get the desire conclusion.

Remark 4.16. The matrices A and B found above may not have any geometrical meaning. Indeed, for closed surfaces the action of Aut(π 1 S) is not transitive on the set of basis of π 1 Σ and then two different basis may not be related by any automorphisms of π 1 Σ. This means that not all matrices in SL 2g (Z) have a geometrical interpretation. As we shall see, a matrix has a geometrical meaning, that is induced by a homeomorphism of Σ, if and only if it is symplectic -see Proposition 4.11 above.

The effect of changing the basis of the lattice.

We begin noticing that the j-th column of the matrix Θ ρ corresponds to the vector of coordinates of a lift of the j-th generator of ρ π 1 Σ with respect to the standard lattice. Given any lattice Λ with basis v 1 , . . . , v n there is a matrix g ∈ GL n (R) such that Λ = g • 2πZ n . In particular g e i = v i . Change the basis means to change the coordinates of the vectors forming the columns of the matrix Θ ρ . Therefore, with respect to the lattice Λ, the matrix associated to ρ has the following form gΘ ρ . In the sequel we shall need to consider the matrix Θ ρ with respect to a lattice Λ different to the standard one. We therefore extend the notation in the following way: We denote by Θ ρ (Λ) the matrix associated to with respect to Λ. We shall use again the notation Θ ρ when the lattice is the standard one.

The modular action commutes with the change of lattice

Given a representation ρ : π 1 Σ → T n , the main goal of the present paper is to study its orbit under the action of the mapping class group. This reduces to study the orbit of the matrix Θ ρ naturally attached to ρ in the space M n, 2g; T . However the matrix Θ ρ depends on the lattice chosen and hence the orbit also. The aim of this paragraph is to point out that this is not the case; indeed the change of lattice commutes with the modular action. Each element Θ in the space M n, 2g; T can be thought as the datum of n vectors Θ i ∈ T 2g corresponding to the rows of Θ. By adopting this point of view, the space M n, 2g; T identifies with T 2g × • • • × T 2g . There is a left action of the group

G =            A . . . A     : A ∈ Sp 2g (Z)        ∼ = Sp 2g (Z) < SL 2gn (Z)
on the 2gn-dimensional torus induced by the natural right action of the symplectic group in the matrix space M n, 2g; T . Using this new perspective, one can easily verify that any change of lattice commutes with the Sp 2g (Z) action. Indeed, any change of lattice h ∈ SL n (Z) can be seen as an element of the group H defined as

H =            h 1 1 I 2g • • • h 1 n I 2g . . . . . . . . . h n 1 I 2g • • • h n n I 2g     where     h 1 1 • • • h 1 n . . . . . . h n 1 • • • h n n     ∈ SL n (Z)        ∼ = SL n (Z) < SL 2gn (Z)
Since H commutes with the group G defined above, the Sp 2g (Z) action commutes with the change of lattice.

Characterising dense representations

In this section we provide a complete characterisation of representations with dense image by providing necessary and sufficient conditions. We have seen in the previous section that, upon choosing a basis of the fundamental group and a lattice, each representation is represented by a well-define matrix Θ ρ . Along this section we fix an arbitrary basis for the fundamental group and we consider T n as the quotient of R n with the standard lattice.

Theorem 4.21. Let ρ : π 1 Σ -→ T n be a representation. Then ρ has dense image in T n if and only if rk Z Θ ρ = n and the rows of Θ ρ generate a πQ-free Z-module.

Notice that the necessary condition means that the Z-module generated by the rows of the matrix Θ ρ does not intersect πQ 2g and is equivalent to say that row rank over Z of the matrix

Θ ρ π • I 2g =             θ 1,1 • • • θ 1,2g . . . . . . θ n,1 . . . θ n,2g π • • • 0 . . . . . . . . . 0 • • • π             (5)
is maximal, namely 2g + n. Before proving the Theorem, we need a preliminar Lemma.

Lemma 4.22. Suppose that ρ :

π 1 Σ -→ T n is dense. Then each representation ρ k = π k • ρ, where
π k is the projection to k th factor, is dense.

Proof of Lemma 4.22. Suppose there is k for which the representation ρ k is not dense. Then there is an open subset A ⊂ S 1 such that A ∩ ρ k π 1 Σ = ∅. Suppose without loss of generality that k = 1.

Then A × T n-1 ∩ ρ π 1 Σ = ∅. In particular ρ is not dense, hence a contradiction.

Proof of Theorem 4.21. Assume ρ has a dense image and suppose the Z-module generated by the rows intersect πQ 2g that is rk Z Θ ρ π • I 2g < 2g + n. Thus, there is a row Θ i of Θ such that:

n j=1 j =i -λ j Θ j + λ n+1 π, . . . , λ n+2g π = λ i Θ i
with λ i different to zero. Such a summation can be rewritten as:

n j=1 λ j Θ j = λ n+1 π, . . . , λ n+2g π (6)
for some λ j ∈ Z and not all zero. Consider the matrix M ∈ M n, Z ∩ GL n (R) defined as:

M = I n + (λ i -1)E ii - n j=1 j =i λ j E ij =            1 0 • • • • • • • • • 0 0 1 • • • • • • • • • • • • • • • • • • . . . • • • • • • • • • -λ 1 -λ 2 • • • λ i • • • -λ n • • • • • • • • • • • • 1 0 0 0 • • • • • • 0 1           
where the E ij = e kl are the matrices with coefficients e kl = δ ki δ lj . The matrix M defines a linear homeomorphism, say f M of R n with respect to the canonical basis because det M = λ i which is different to zero. The mapping f M sends Z n to itself and descends to a finite-degree covering f M : T n -→ T n -in fact the degree coincides with the determinant of M . In particular the following equation holds: π • f M = f M • exp, where exp : R n -→ T n denotes as usual the canonical projection. Consider now the Z-module Θ j | j = 1, . . . , n Z generated by the rows of Θ ρ . By Equation 6, a straightforward computation shows that its image via the mapping f M is the Z-module generated by the vectors

        θ 1,1 . . . λ n+1 π . . . θ n,1         , . . . ,         θ 1,2g . . . λ n+2g π . . . θ n,2g         Z
Let us point out the following fact: As ρ is assumed to be dense in the torus, the image via the canonical projection in T n of Z-module generated by the rows of Θ ρ fills a dense subset of T n , namely the image of ρ. As M commutes with the action of 2πZ n and pass through to the quotient as a finite-degree covering map of the T n , the Z-module M • Θ j | j = 1, ..., n Z is mapped on a dense subset of the torus. On the other hand, the projection of the i-th factor is discrete. Lemma 4.22 implies the desire contradiction.

We now prove the opposite implication and again we argue by contradiction. Suppose ρ does not have a dense image in the n-torus, then its closure is a k-dimensional sub-manifold, say S 0 , of dimension k < n. We note that S 0 may not be connected in general. Indeed any closed subgroup of T n is homeomorphic to

T d × Z m 1 Z × • • • × Z m n-d Z .
Assume first S 0 be connected; we shall deduce the general case later on. The subspace S 0 lifts to a linear subspace S 0 of R n which of course contains the Z-module Θ j | j = 1, ..., 2g Z generated by the columns of Θ ρ . We now invoke the following lemma.

Lemma 4.23. There is g ∈ SL 2g (Z) such that :

g • Θ j | j = 1, . . . , 2g Z < e 1 , . . . , e k R
where the e i 's are the vectors of the canonical basis of R n .

Assume the lemma holds. The Z-module g • Θ j | j = 1, ..., 2g Z is contained in the first factor of T n = T k × T n-k and then Θ ρ cannot have maximal row rank over Z. As a consequence the matrix given in the equation 5 cannot have maximal row rank over Z. The general case follows by applying the same reasoning to the component S o 0 of S 0 containing the identity which contains a finite-index Z-module of Θ j | j = 1, ..., 2g Z . In the general case, g

• Θ j | j = 1, ..., 2g Z is contained in T n = T k × F , where F is isomorphic to the finite group Z m 1 Z × • • • × Z m n-d Z .
Let us proceed with the proof of Lemma 4.23.

Proof of Lemma 4.23. If S 0 o is contained in e σ(1) , . . . , e σ(k) R for some σ ∈ S n then it is sufficient to rename the coordinates. This corresponds to a matrix g obtained by product of elementary matrices.

Assume S 0 o is not contained in any such a space. Let x i be the intersection of S 0 o with the affine space e i + R n-k and let d i its Euclidean distance to R k . Then x i has the following form:

x i = 0, . . . , 1, . . . , 0, t 1 , . . . , t n-k where t i ∈ Q. In fact, if this had been not true then S o 0 would have been a dense subspace of dimension k + 1 in the torus. As a consequence d 2 i ∈ Q for any i = 1, . . . , k and S 0 o is described by n -k equations with integer coefficients. Look at the set S 0 o ∩ Z n . This is a lattice in S 0 o and there is a basis v 1 , . . . , v k made of integer vectors. We invoke Cassels 1997, Corollary 3, pag.14 to claim the existence of n -k vectors v k+1 , . . . , v n such that the vectors v 1 , . . . , v n gathered together form a basis for Z n . Since SL 2g (Z) acts transitively on the space of lattices, there is g such that g • Θ j | j = 1, . . . , 2g Z < g • S 0 = e 1 , . . . , e k R .

This concludes the proof of Lemma 4.23 and indeed the proof of Theorem 4.21.

From the proof we deduce that the row rank of the matrix Θ ρ has a very explicit geometric interpretation, in fact it coincides with the dimension of the subspace containing the image of ρ. Of course, the proof does not depend on the presentation of π 1 Σ either on the lattice chosen. Let us prove these facts.

Independence on the chosen basis. Let Θ j | j = 1, . . . , 2g Z be the Z-module generated by the columns of Θ ρ . In section 4.5.1 we have seen that the change of basis corresponds to multiply on the right the matrix Θ ρ with a matrix A ∈ SL 2g (Z). Since the row rank of Θ ρ is invariant under the action by right-multiplication of SL 2g (Z), the matrices Θ ρ and Θ ρ A have the same row rank. Furthermore, in the light of Proposition 4.20, πQ-freedom is also invariant under the right action of SL 2g (Z). On the other hand, let Ξ be the closure of the subspace of T n generated by the columns of Θ ρ . Its lift S is a linear subspace of R n described by n -k equations. As the columns of Θ ρ A satisfy the same equations, the image of unaffected by the change of basis. This proves the independence on the basis chosen.

Independence on the lattice chosen. Given two lattices Λ 1 and Λ 2 = A • Λ 1 for some element A ∈ SL n (Z). Such a map A descends to a homeomorphism of the n-torus and hence the Z-module Θ j | j = 1, . . . , 2g Z projects to a dense subset of the torus if and only if its image via A projects to dense subset as well. On the other hand, the row rank of the associated matrix Θ ρ (Λ 1 ) equals the one of Θ ρ (Λ 2 ) because the row rank is invariant under the action by left-multiplication of SL n (Z).

Again, Proposition 4.20 implies πQ-freedom is invariant under the left action of SL n (Z). Hence the conclusion.

We finally provide a couple of explicit examples.

Ratner's Theorem. Let G, P, Γ as above, let p ∈ M n, 2g; R = G/P and let γ ∈ G such that p = γ P . Then there is a closed and connected subgroup H γ such that the following holds.

• P γ = γ P γ -1 ≤ H γ ,

• Γ ∩ H γ is a lattice in H γ and

• Γ • p = Γ H γ p.
Notice that γ can be taken as (I, p). Since our goal here is to classify the closures of Γ-orbits of any point in the space M n, 2g; R , we just need to figure out which subgroups of G may be provided by Ratner's Theorem. To this purpose, let us consider the projection Φ : Sp 2g (R) M n, 2g; R -→ Sp 2g (R). Given a point p in M n, 2g; R , the group H γ is isomorphic to the semidirect product H γ = P γ K γ , where K γ is defined as ker Φ ∩ H γ , that is the kernel of the mapping Φ restricted to H γ . Notice that the image of H γ under the mapping Φ is the whole group Sp 2g (R) because

H γ ≥ P γ ∼ = Sp 2g (R).
In particular, H γ ∼ = Sp 2g (R) K γ .

Let us proceed on understanding K γ . The first thing we notice is that any change of lattice h ∈ SL 2g (Z) extends to a homeomorphism φ h of G defined as

φ h : G -→ G, φ h (A, a) = (A, ha).
This is an automorphism of G and its restriction to ker Φ, where Φ is the projection just defined above, is linear and corresponds to a change of lattice in M n, 2g; R . In particular, the relation

Γ • (h p) = φ h Γ • p (7)
holds for any p ∈ M n, 2g; R . As a consequence of Lemma 4.23, there is an element h ∈ SL 2g (Z) such that h • p is of the following form

            θ 1,1 • • • θ 1,2g . . . . . . θ k,1 . . . θ k,2g π q k+1,1 • • • π q k+1,2g
. . . . . .

π q n,1 • • • π q n,2g             = Θ o πQ (8)
where

• Θ o ∈ M k, 2g; R for some 0 ≤ k ≤ n,
• πQ ∈ M n -k, 2g; πQ ,

• (θ i,1 , . . . , θ i,2g ) : i = 1, . . . , k is πQ-free, and thus where A is the matrix having a (j) 's as columns, H ∈ M n, l; Z and C is a constant depending only on l, n. That is

A -B K mod 2π < ε. (12)
Let Σ be a closed surface of genus greater than zero, let ρ : π 1 Σ -→ T n be a representation and let Θ ρ be the associated matrix in the sense of Definition 4.15.

Proposition 4.9. The following are equivalent.

1. Mod(Σ) • ρ is dense in the representation space.

2. For any matrix A ∈ M n, 2g; R and any ε > 0 there is a matrix g ∈ Sp 2g (R) such that A -Θ ρ g mod 2π < ε.

Proof of Proposition 4.9. Each representation is identified with its associated matrix and the representation space with M n, 2g; T . Suppose Mod(Σ)•Θ ρ is not dense in the representation space. there is an element g ∈ Mod(S) such that g -1 • Θ ρ = Θ ρ g ∈ B ε (A). In particular, A -Θ ρ g mod 2π < ε. Once again, by Proposition 4.10, the matrix g can be taken in Sp 2g (R) and so Theorem D implies Theorem E as desired.

Dense Orbits and further discussion

In this subsection we are going to prove Theorem D for almost every representation without relying on Ratner's Theorem. We begin consider the genus one case and we shall use it to extend the discussion to surfaces of arbitrary genus.

Direct proof of Theorem F for almost every representations

The set of matrices M n, 2g; T contains, as a proper subset, the space D of all of those matrices of the following form

            θ 1 θ 2 • • • θ 2i-1 θ 2i • • • θ 2g-1 θ 2g λ 2 θ 1 λ 2 θ 2 • • • λ 2 θ 2i-1 λ 2 θ 2i • • • λ 2 θ 2g-
λ n θ 1 λ n θ 2 • • • λ n θ 2i-1 λ n θ 2i • • • λ n θ 2g-1 λ n θ 2g             (13)
where θ 1 , θ 2 , . . . , θ 2g-1 , θ 2g ∈ R 2g \ πQ 2g is the lift of e iθ 1 , e iθ 2 , . . . , e iθ 2g-1 , e iθ 2g ∈ T 2g contained in 0, 2π 2g and the reals {1, λ 2 , . . . , λ n } ⊂ R are linearly independent over Q.

Lemma 4.29. D is dense in M n, 2g; T .

Proof. Let λ 2 , . . . , λ n real numbers such that 1, λ 2 , . . . , λ n are linearly independent over Q. Consider the mapping ϕ : R 2g -→ T 2gn defined as θ 1 , θ 2 , . . . , θ 2g-1 , θ 2g → e iθ 1 , . . . , e iθ 2g , e iλ 2 θ 1 , . . . , e iλ 2 θ 2g , . . . , e iλnθ 1 , . . . , e iλnθ 2g .

This mapping factors through a mapping ϕ : R 2g -→ R 2gn such that ϕ = exp • ϕ and exp is the exponential mapping thought as in equation (4) introduced in section 4.5. The image of ϕ is a 2gdimensional linear subspace. As 1, λ 2 , . . . , λ n are linearly independent over Q, then the projection via the exponential mapping is dense in T 2gn . The space D is defined as the union of the images for each possible subset {λ 2 , . . . , λ n } ⊂ R such that 1, λ 2 , . . . , λ n are linearly independent over Q. Therefore D is dense.

The following lemma is easy to establish.

Lemma 4.30. D is Sp 2g (Z)-invariant.

Let us consider first surfaces of genus one. Let T be the torus, let ρ : π 1 T -→ T n be a dense representation and let Θ ρ be its associated matrix with respect to some basis a, b and the standard lattice of R n . Let Ω ρ be the SL 2 (Z)-orbit of Θ ρ in M n, 2; T . The associated matrix Θ ρ has the following form:

Θ ρ =          θ 1 θ 2 . . . . . . λ i θ 1 λ i θ 2 . . . . . . λ n θ 1 λ n θ 2          (14)
where θ 1 , θ 2 ∈ R 2 is the lift of e iθ 1 , e iθ 2 ∈ T 2 contained in 0, 2π

2 and λ i ∈ R, for any i = 2, . . . , n, are linearly independent over Q. Set

Θ ρ = Θ ρ π • I 2 .
Since the representation ρ is assumed to have a dense image, the matrix Θ ρ has maximal row rank, that is rk Z = n + 2. This implies the following properties of the matrix Θ ρ above.

14.i The real numbers θ 1 and θ 2 cannot be both elements of πQ. If this were the case, the row rank of the matrix Θ ρ would fail to be maximal, contradicting our assumptions. In the case one of them is an element of πQ, we can change the basis in such a way they are both elements of R \ πQ. Indeed, assume without loss of generality that θ 2 ∈ πQ. The Dehn twist tw a along a maps the curve b to ab and hence ρ(b) is mapped to ρ(ab). The second column of Θ ρ changes accordingly and the element of place (1, 2) of Θ twa•ρ is nothing else that θ 1 + θ 2 . As θ 1 / ∈ πQ the same necessarily holds for θ 1 + θ 2 . In what follows, we shall assume both θ 1 , θ 2 / ∈ πQ.

14.ii The real numbers π, θ 1 , . . . , λ i θ 1 , . . . , λ n θ 1 are linearly independent over Q. Indeed, if this were not the case then one can easily check that Θ ρ has not maximal rank. This implies that the subgroup of T n generated by the vector θ 1 , . . . , λ i θ 1 , . . . , λ n θ 1 is dense in T n , see Bekka and Mayer 2000, Exercise 1.13. The same holds also for the real numbers π, θ 2 , . . . , λ i θ 2 , . . . , λ n θ 2 .

14.iii The real numbers 1, λ 2 , . . . , λ n are linearly independent over Q. If this were not the case, then there would be a 1 , . . . , a n ∈ Q such that a 1 + a 2 λ 2 + • • • + a n λ n = 0. In particular, n i=1 a i λ i θ 1 , θ 2 = 0, with λ 1 = 1. That is the rows of Θ ρ are linearly dependent over Q. Therefore the row rank cannot be maximal, a contradiction. In particular, λ i / ∈ Q for every i = 2, . . . , n. In what follows, we shall sometimes refer to 1 as λ 1 for simplify the formulas.

We begin with considering the SL 2 (Z) action on the space M n, 2; R seen as the universal cover of M n, 2; T , see also Remark 4.15. Given the matrix Θ ρ as in 14, there is a unique lift, say Θ(ρ), in M n, 2; R which is still of the form of Equation 14. Notice that such a matrix is the unique one who has all the entries in the interval [0, 2π). Let us finally denote with Ω(ρ) the SL 2 (Z)-orbit of Θ(ρ) in M n, 2; R .

Since Θ(ρ) is of the form of Equation 14, an easy computation shows that the matrix A•Θ(ρ) ∈ Ω(ρ) is still of the form 14 for any A ∈ SL 2 (Z), that is the i-th row of A • Θ(ρ) is λ i -times θ 1 , θ 2 A -1 .

a 1

 1 , b 1 , . . . , a g , b g | g i=1 [a i , b i ] .

Figure 1

 1 Figure 1

  que cet invariant est borné : Théorème 4. (Milnor 1957/58) Pour tous les [ρ] ∈ X : |eu([ρ])| ≤ |χ(Σ)|. Une représentation dont la classe d'Euler est égale à ±|χ(Σ)| est appelée maximale. Dans W. M. Goldman 1988, Goldman a classifié les composantes connexes par la classe d'Euler et dans W. M. Goldman 1982 il a donné une caractérisation des représentations maximales : Théorème 5. (W. M. Goldman 1988 et W. M. Goldman 1982, Goldman) 1. Pour tout k ∈ Z∩[χ(Σ), -χ(Σ)], le sous-espace eu -1 (k) est non vide et les composantes connexes de X sont exactement les eu -1 (k) pour k ∈ Z ∩ [χ(Σ), -χ(Σ)]. Nous les noterons par X k et par Rep k leurs intersections avec Rep.

  dans la voie de cette conjecture, Marché et Wolff introduisent dansMarché and Wolff 2016, pour k ∈ Z ∩ χ(Σ), -χ(Σ) , le sous-espace N H k := [ρ] ∈ Rep k | ∃γ ∈ π 1 Σ simple , tr ρ(γ) ∈ [-2, 2] .Ils ont prouvé ce qui suit : Théorème 6. (Théorème 1.5 et Théorème 1.6, Marché and Wolff 2016) 1. Supposons que Σ est de genre 2. Alors l'action de Mod + (Σ) sur X ±1 est ergodique par rapport à λ.2. Supposons que g ≥ 3 et soit k un entier tel que |k| < |χ(Σ)|, alors l'action de Mod + (Σ) sur N H k est ergodique par rapport à λ.

  c 1 , . . . , c n sont des courbes sur Σ qui sont homotopes à un cercle autour de chaque composante de bord.

Figure 2 Définition 2 .

 22 Figure 2

  Pour chaque i ∈ {1, . . . , n}, notons r i le nombre de translation de ρ(c i ). La classe d'Euler de ρ est le nombre eu(ρ) La classe d'Euler d'une représentation ne dépend pas des choix des relevés et est constant sur sa classe de conjugaison. On définit ainsi la classe d'Euler d'une classe de représentations [ρ] par la classe d'Euler d'un représentant de cette classe. Cela donne une application eu : X ( Σ) → R qui induit par restriction eu C : X ( Σ, C) → R qui vérifie la propriété eu C ([ρ]) + {r} 1 ∈ Z où {r} est le vecteur des parties fractionnaires des r i . La préimage de eu -1 C (e), pour e ∈ R, sera notée X e ( Σ, C). Dans Mondello 2017, Gabriele Mondello explicite les composantes connexes de X ( Σ, C). Théorème 7. (Théorème 2.20,Mondello 2017) 1. L'image de eu : X ( Σ) → R est l'intervalle χ( Σ), -χ( Σ) . Si eu(ρ) = -χ( Σ), alors tous les ρ(c i ) sont hyperboliques et ρ est la monodromie d'une structure hyperbolique sur Σ avec des composantes de bord géodésiques. 2. Supposons e > 0. Alors l'espace X e ( Σ, C) est non vide si et seulement si e + {r} 1 + s 0 + s -∈ Z ∩ 0, -χ( Σ) , où s 0 est le cardinal de {i ∈ {1, . . . , n} | C i = id} et s -est le cardinal de {i ∈ {1, . . . , n}|C i unipotent negatif} où une matrice unipotente négative est conjuguée à la matrice 1 -1 0 1 . Dans ce cas, l'espace X e ( Σ, C) est connexe et lisse.

  C), des classes de représentations [ρ] qui admettent une courbe simple γ ⊂ Σ non homotope à une composante de ∂ Σ, telle que ρ(γ) est non-hyperbolique et telle que ρ(c i ) ∈ C i , est non-vide. De plus : Théorème C. Le groupe modulaire pur PMod( Σ) agit ergodiquement sur l'espace N H e ( Σ, C) par rapport à la mesure de Goldman.

  , ce fait est un résultat probabiliste et aucune description des points de X pour lesquels l'orbite est dense n'est donnée en général. Dans W. Goldman 2006, Goldman pose la question suivante : Question 1. Pourrait-on trouver une condition sur une représentation ρ : π 1 Σ → SU(2) pour que son orbite pour l'action de Mod + (Σ) soit dense ? Dans Previte and Xia 2000 et dans Previte and Xia 2002, Previte et Xia donnent une condition nécessaire et suffisante : Théorème 8. Une représentation ρ : π 1 Σ → SU(2) a une Mod + (Σ)-orbite dense si et seulement si ρ a une image dense dans SU(2). Avec Gianluca Faraco, nous traitons le cas abélien. Un groupe de Lie compact, abélien et connexe est isomorphe à un tore de dimension n. En utilisant le théorème de Ratner, voir Ratner 1991, nous prouvons dans Bouilly and Faraco 2021 : Théorème D. (Bouilly and Faraco 2021) Une représentation ρ : π 1 Σ → T n a une orbite dense sous l'action de Mod + (Σ) si et seulement si ρ a une image dense dans T n . Nous conjecturons qu'un tel résultat est vrai lorsque G = SU(3) et Σ est un tore troué.

Figure 3

 3 Figure 3

Theorem 4 .

 4 (Milnor 1957/58) For all [ρ] ∈ X :|eu([ρ])| ≤ |χ(Σ)|.A representation whose Euler class is equal to ±|χ(Σ)| is called maximal. In W. M. Goldman 1988, Goldman classified the connected components by Euler class and in W. M. Goldman 1982 he gave a characterisation of maximal representations: Theorem 5. (W. M.[START_REF] Goldman | Invariant functions on Lie groups and Hamiltonian flows of surface group representations[END_REF] W. M. Goldman 1982)

  depends only of the C i . We will denote by N H( Σ, C) the space we define to be the subspace of Rep( Σ, C whose elements are the classes of representations [ρ] for which there existes a simple and closed curve γ ⊂ Σ \ ∂ Σ such that ρ(γ) is a non-hyperbolic isometry of H 2 . For e ∈ R, we denote by N H e ( Σ, C) the intersection N H( Σ, C) Rep e ( Σ, C .

Figure 5

 5 Figure 5

  almost everywhere constant and hence that the restriction F |T 2 •[ρ] is almost everywhere constant. Since the Dehn twist tw x acts as a translation of the torus on U x × U c 2 • [ρ], we deduce that on a full measure subset of U x × U c 2 • [ρ] the function F and F • tw x are equal. This fact is true for almost every [ρ] ∈ M mx (π 1 Σ, SU(2)) which has full measure by Proposition 2.5. It follows that F |Mm x (π 1 Σ,SU(2)) is almost everywhere invariant by the Dehn twist tw x .

  Proposition 3.1. For e ∈ R >0 , the subset EI e ( Σ, C) is non-empty if and only if the integer k verifies the inequality {r} 1 < |k| ≤ -χ( Σ) -1. Proposition 3.1 use Mondello Theorem 7. Indeed, fixing a curve γ verifying the assumptions asking in the definition of EI e ( Σ, C) and a conjugacy class of a elliptic isometry of infinite order, we study the existence of representations of Σ \ γ with conditions on boundaries.

Figure 7 Following

 7 Figure 7

Figure 8 3. 3

 83 Figure 8

  n a • ρ (a -n b) = ρ(b) and tw n a • ρ (c) are elliptic isometry. We hence have that tw n a •[ρ] ∈ EI e (a,a -n b) ( Σ, C)∩EI e (a,c) ( Σ, C) and so (a, a -n b) ∼ (a, c). By Lemma 3.13 we conclude that (a, b) ∼ (a, c). Lemma 3.15. (Marché and Wolff 2016,Lemma 6.11) Let b be a simple, closed and non-separating curve and let a and a be two curves such that (a, b) and (a , b) are both in S 1 . Then (a, b) ∼ (a , b).

  starting from the lower intersection point of a and b, we follow a until the upper line of a and go directly to hit b. This constructs a new curve a which do not intersect a . Then (a , b) ∼ (a , b) and by induction hypothesis, (a, b) ∼ (a , b). The conclusion of the connectedness of EI e ( Σ, C) of Proposition 2.5 use the connectedness of 1skeleton of curves complex. Let (a, b), (a , b ) ∈ S 1 . There exists non-separating curves b 0 = b, . . . , b n =

  If g = 1, let a, b as on the next picture. By assumption, ρ(a) and ρ(b) are hyperbolic. If A ρ(a) and A ρ(b) are disjoint, then the sequence (γ n ) n := (a n ba -n ) n satisfies the conclusion of Lemma 3.19. If A ρ(a) and A ρ(b) intersect, then, by assumption on ρ, [ρ(a), ρ(b)] is either elliptic of infinite order or hyperbolic. In the first case, we can apply the case of disjoint axes to ρ(a) and ρ([a, b] k • b • [a, b] -k ) for a suitable k ∈ Z. In the second one, ρ(a) and ρ([a, b]) have disjoint axes. We so apply the corresponding case to ρ(a) and ρ([a, b]).

Figure 9

 9 Figure 9

Remark 4. 7 .

 7 Here, the symplectic group Sp 2g (R) is the subgroup of SL 2g (R) of matrices preserving the alternating2-form ω = e 1 ∧ e 2 + • • • + e 2g-1 ∧ e 2g . By Remark 4.2, Sp 2g (R) contains the g-times product SL 2 (R)ו • •×SL 2 (R)as a proper subgroup. In turns, the group Sp 2g (R) contains the g-times product SL 2 (Z) × • • • × SL 2 (Z) as a proper subgroup. This property will be useful in the sequel.

  Proposition 4.11. The action of Sp 2g (Z) ∼ = Mod(Σ) Tor(Σ) on Hom π 1 Σ, T n is ergodic with respect to the Haar measure.

Proposition 4. 13 .

 13 Let ρ : π 1 Σ -→ T n be a representation and let ρ : H 1 (Σ, Z) -→ T n the homological representation induced by ρ. Then the mapping class group orbit Mod(Σ) • ρ is dense if and only if the symplectic group orbit Sp 2g (Z) • ρ is dense.

  Then there is an open set U such that Mod(Σ) • Θ ρ ∩ U = φ. Let A be any matrix in U and ε a strictly positive real number such that the open ball B ε (A) ⊂ U . Then, for any g ∈ Mod(S) the following estimate A -Θ ρ g > ε mod 2π holds. As the action of the Torelli subgroup is trivial by Proposition 4.10, the action of the mapping class group coincides with the action of Sp 2g (Z). Therefore, Theorem E implies Theorem D. Suppose Mod(Σ) • Θ ρ dense in the representation space. Then, for any A ∈ M n, 2g; T and for any ε > 0 the mapping class group orbit intersects the open set B ε (A) ⊂ M n, 2g; T , i.e.

  est appelée Zariski-dense si son image est Zariski-dense dans PSL 2 (R). En particulier, une représentation Zariski-dense est régulière. Le sous-espace Rep( Σ) des classes de représentations Zariski-denses est un sous-ensemble de mesure pleine, ouvert et dense de X ( Σ). Cet espace est partitionné par le comportement sur les composantes de bord. Précisément, soient C 1 , . . . , C n des classes de conjugaison dans PSL 2

  j θ 1 λ j θ 2 • • • λ j θ 2i-1 λ j θ 2i • • • λ j θ 2g-1 λ j θ 2g . . .

						1 λ 2 θ 2g
	. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .

λ

Remerciements

The data of such a ρ and such a B gives a representation ρ : π 1 Σ → PSL 2 (R) which send the curve obtained by gluing d 1 and d -1 2 into the conjugacy class A. We thus have that for e > 0:

EI e ( Σ, C) = ∅ ⇐⇒ e = k -{r} 1 with {r} 1 < k ≤ -χ( Σ) -1 which is the sought for condition in Proposition 3.1.

Arc-connectedness of relative character varieties

Let us consider some arc-connectedness properties for relative character varieties.

Proposition 3.5. Let Σ be a compact, connected and oriented surface of genus g -1 and with n + 1 boundary components denoted by c 1 , . . . , c n , c. Let be a real number such that Rep (Σ ) and the next boundary conditions are non-empty. Let A : 0, 1 → PSL 2 (R) be a continuous path whose image is contained in the subset of hyperbolic isometries and let ρ : π 1 Σ → PSL 2 (R) be a representation of Euler class such that ρ(c i ) ∈ C i for all i = 1, . . . , n and ρ(c) = A(0). Then there exists a continuous path A : 0, 1 → Hom(Σ , PSL 2 (R)) such that A(0) = ρ and for all t ∈ 0, 1 , eu( A(t)) = and Ãt (c) = A(t), A t (c 1 ) ∈ C 1 , . . . , A t (c n ) ∈ C n .

We use a pairs of pants decomposition of Σ . The proposition 3.5 will hence follow from the three next lemmas.

The first one was proved by Goldman in W. M. Goldman 1988:

Lemma 3.6. Let Σ 0 be a one-holded torus. Let k = -1, 0 or 1, A : 0, 1 → PSL 2 (R) be a continuous path whose image is contained in the subset of hyperbolic isometries and fix ρ : π 1 Σ 0 → PSL 2 (R) of Euler class k and such that ρ(∂Σ 0 ) = A(0). Then there exists a continuous path of representations A : 0, 1 → Hom(π 1 Σ 0 , PSL 2 (R)) such that A(0) = ρ and for all t ∈ 0, 1 , eu( A(t)) = k and

Such an extension by representations is true for pair of pants and for n + 1-punctured sphere with elliptic conditions on some boundaries. It is the aims of next lemmas: Lemma 3.7. Let P be a pair of pants with as boundary components the curves d, d 1 , d 2 . Let A : 0, 1 → PSL 2 (R) be a continuous path whose image is contained in the subset of hyperbolic isometries. Then there exists a continuous path of representations A : 0, 1 → Hom(π 1 P, PSL 2 (R)) such that for all t ∈ 0, 1 , A t (d) = A(t) and f or all i ∈ {1, 2}, A t (d i ) hyperbolic.

3.4 EI e ( Σ, C) ⊂ U e ( Σ, C) Let [ρ] ∈ EI e ( Σ, C) and take a non-separating curve γ such that ρ(γ) is an elliptic isometry of infinite order and of angle θ. We will denote by F [ρ] the vector subspace of T *

[ρ] Rep( Σ, C) spanned by the linear forms df σ for σ all the non-separating curves which are send on a elliptic isometry by ρ. The following lemmas will use the well-known relation which holds for all A, B ∈ SL 2 (R): tr(AB) + tr(AB -1 ) = tr(A)tr(B).

These lemmas and their proofs are done in Marché and Wolff 2016 and we will reproduce the proofs for the reader's convenience.

Lemma 3.16. Let δ be a non-separating curve such that i(γ, δ) = 1.

Proof. We will denote by δ n the curve γ n δ = tw n γ (δ). The trace relation gives directly that for all n ∈ Z, the equality

We deduce from this equality the relation

Then, modulo F [ρ] , the sequence of linear forms df δn n satisfies the order two recursive equation u n+1 +u n-1 = 2 cos(θ)u n and there hence exists linear forms Λ, Θ in the cotangent space T * [ρ] Rep( Σ, C) such that for all n ≥ 0:

By Lemma 3.10, there exists infinitly many integers n ∈ Z such that ρ(δ n ) is elliptic. For these n, the linear form df δn is null up to F [ρ] . We hence deduce, since θ / ∈ πQ, that Λ and Θ are both the trivial linear form modulo F [ρ] . Then, for all integer n ∈ Z, df δn ∈ F [ρ] . In particular for n = 0. Lemma 3.17. Let δ be a curve such that i(γ, δ) = 0.

Then there is a curve σ, such that i(γ, σ) = i(δ, σ) = 1. We have the formula

Moreover, Lemma 3.16 ensures that df δσ , df δσ -1 , and df σ ∈ F [ρ] . We hence conclude that df δ ∈ F [ρ] .

4 Toplogical dynamic for modular action on representation spaces in Abelian compact Lie groups; with Gianluca Faraco

The proof of Theorem D strongly relies on the explicit knowledge of the objects involved. In fact, the n-torus has a well-known description and, thanks to the abelian property, the character variety coincides with the representation space since the action of T n by conjugation is trivial. Even better, the representation space can be identified with a torus of suitable dimension, hence the description of the representation spaceand then of the character varietyis very explicit. The main difficulties in the abelian case concern questions coming from number theory and ergodic theory.

Strategy of the proof and related results

Each given representation ρ : π 1 Σ -→ T n induces a homological representation ρ : H 1 (Σ, Z) -→ T n . The map associating to any representation ρ its homological representation defines a bijection between the representation space and the homological representation space. This essentially follows because the commutator group [π 1 Σ, π 1 Σ] is trivially a subgroup of kerρ in the abelian case, and such a property is no longer true for a generic non-abelian Lie group. There is also a well-defined action of the symplectic group Sp 2g (Z) on the space Hom H 1 (Σ, Z), T n by precomposition. Given a representation ρ : π 1 Σ -→ T n and its induced representation ρ : H 1 (Σ, Z) -→ T n , the Mod(Σ)-orbit of ρ coincides with the Sp 2g (Z)-orbit of ρ. As an immediate consequence we obtain an equivalent version of the main theorem D, namely we have the following.

Theorem F. Let Σ be a surface of genus g ≥ 1 and let ρ : H 1 (Σ, Z) -→ T n be a representation. Then the image of ρ is dense in T n if and only if the symplectic group orbit Sp 2g (Z) • ρ is dense in the homological representation space.

Along the way of our investigation we shall remark that the action of the Torelli group Tor(Σ) on the representation space Hom π 1 Σ, T n is trivial which contrasts with Theorem A. Given a representation, we are then reduced to consider its Sp 2g (Z)-orbit of instead of its modular orbits. This makes the study of orbits more understandable because the symplectic group is linear. We shall make the action even more explicit by identifying a representation with a matrix in the space M n, 2g; T . After these reductions, we shall see that we are in the position to apply Ratner's Theorem for studying orbit closures. In particular, we shall derive Theorem D.

Remark 4.1. When the genus of Σ is 1, the reader may notice that theorems D and F are not only equivalent but actually the same statement in the strict sense. Indeed, in this very particular case the following equalities π 1 Σ = H 1 (Σ, Z) and Mod(Σ) = SL 2 (Z) = Sp 2g (Z) holds.

The strategy we propose for Theorem D is different to the one developed by Previte-Xia to show their main theorem Previte and Xia 2000. Let us briefly give some more details. Given a representation the representation ρ is well-defined and the image does not depend on the choice of the representative. Furthermore, the image of ρ agrees with the image ρ by contruction. Definition 4.3. We define the homological representation space as the set Hom H 1 (Σ, Z), T n of representations of H 1 (Σ, Z) in T n endowed with the compact-open topology.

Lemma 4.4. The homological representation space Hom H 1 (Σ, Z), T n identifies with the 2gn-dimensional torus T 2gn . Proof. To any representation ρ we can assign the 2g-tuple ρ

, as T n is abelian, the universal property of free abelian groups implies the existence of a unique group homomorphism from

The implications of this lemma are quite simple, but of crucial importance. Upon choosing a standard generating system for π 1 Σ; the representation space Hom π 1 Σ, T n identifies with the homological representation space Hom H 1 (Σ, Z), T n and the identification is explicitely given by the association ρ → ρ. According to this property, we derive the following lemma.

Proof. This is just a matter of definitions given so far. The necessary condition follows trivially. The sufficient condition follows from ρ γ = ρ(γ) for any γ ∈ π 1 Σ.

Actions of the symplectic group Sp 2g (Z)

In this section we are going to describe the action of the symplectic group Sp 2g (Z) both on the representation space and on the homological representation space.

The symplectic group Sp 2g (Z).

We begin with recalling some standard notions. The algebraic intersection number

extends uniquely to a nondegenerate, alternating bilinear map

which realises H 1 (Σ, R) as a symplectic vector space.

4.5.3

The Z-row rank of the associated matrix.

We now introduce the following numerical invariant concerning the associated matrix Θ ρ . As we shall see, such an invariant give us a way to characterise dense representations in T n completely.

Definition 4.17. Let M ∈ M n, m; R . We define the Z-row rank of M as the dimension of the Z-module generated by the rows of M . We shall denote it as rk Z M .

We remark that the Z-row rank is not invariant by transposition.

Proof. We begin showing the first claim. Let k = rk Z M ≤ n. Define Z as the subset of Z n of those vectors v such that vM = 0. Notice that Z is a Z-module of rank n -k. Let A be any matrix in SL n (Z) and compute AM . It is easy to check that the j-th row is given by the linear combination n i=1 a ji m i1 , . . . , m im . Suppose there is a vector µ = µ 1 , . . . , µ n such that µAM = 0, then a straightforward computation shows that µA ∈ Z, that is µ = vA -1 for some v ∈ Z. The subset Given a representation ρ : π 1 Σ -→ T n , the following claims are direct consequences of the lemma above applied to the matrix Θ ρ .

Corollary 4.19. Let ρ : π 1 Σ -→ T n be a representation and let Θ ρ be the matrix associated to ρ with respect to some basis of π 1 Σ. The Z-row rank of Θ ρ is well-defined and it does not depend on any choice of a basis for π 1 Σ either on the choice of any lattice.

Let v 1 . . . , v k be vectors in R n . In the sequel, we shall say that a Z-module generated by v 1 . . . , v k is πQ-free if and only if v 1 , . . . , v k Z ∩ πQ n = (0, . . . , 0) . Keeping this definition in mind we finally state the following consequence.

Proposition 4.20. Let ρ : π 1 Σ -→ T n be a representation and let Θ ρ be the matrix associated to ρ with respect to some basis of π 1 Σ. The Z-module Θ j | j = 1, . . . , n Z generated by the rows of Θ ρ is πQ-free if and only if the Z-module AΘ ρ j | j = 1, . . . , n Z generated by the rows of AΘ ρ is πQ-free, where A ∈ SL n (Z). Similarly, for B ∈ SL 2g (Z), Θ j | j = 1, . . . , n Z is πQ-free if and only if the Z-module Θ ρ B j | j = 1, . . . , n Z is πQ-free since the tow Z-modules are equal.

Proof of Proposition 4.20. Look at the matrix AΘ ρ and suppose there are λ 1 , . . . , λ n ∈ Z, not all zero, such that

A simple manipulation of the formula above shows that

implying the existence of some µ 1 , . . . , µ n ∈ Z, not all zero, such that

The proof of the second claim works similarly: Suppose there are λ 1 , . . . , λ n ∈ Z, not all zero, such that

The same manipulation shows that

Remarks and comments on the modular action

In this section we collect a couple of final remarks about the Sp 2g (Z)-action.

Explicit description of the modular action

The action of the mapping class group on the representation space is defined by pre-composition of any representation with an automorphism φ ∈ Out(π 1 Σ) such that any representation ρ is sent to ρ • φ -1 . Since the Torelli subgroup acts trivially on the representation space by our Proposition 4.10, the action of mapping class group boils down to an action of the group Sp 2g (Z) which agrees with the Sp 2g (Z)-action on Hom H 1 (Σ, Z), T n ). In section 4.5, we have identified the representation space with M n, 2g; T by using the mapping ı associating to any representation ρ its matrix Θ ρ . We use such a mapping to transfer the action of Mod(Σ) on Hom π 1 Σ, T n to an action of Sp 2g (Z) on M n, 2g; T . Since any φ ∈ Tor(Σ) leaves ρ fixed, the matrix associated to ρ = φ • ρ = ρ • φ -1 agrees with Θ ρ , this is a consequence of Proposition 4.10. Any coset φ Tor(Σ) defines a unique matrix A in Sp 2g (Z). In the light of the discussion given at subsection 4.5.1, the matrix associated to

As the mapping ı is a homeomorphism, it is clear that Mod(Σ)-orbit of ρ is dense in the representation space if and only if the Sp 2g (Z)-orbit of Θ ρ is dense in M n, 2g; T .

Example 4.24. Let Σ be a surface of genus 2, and let ρ : π 1 Σ -→ T 2 ∼ = S 1 × S 1 be the representation such that ρ(a 1 ) = ρ(a 2 ) = e iϕ , e iϕ , where ϕ ∈ R \ πQ, and ρ(b 1 ) = ρ(b 2 ) = (1, 1).

The matrix Θ has the following form ϕ 0 ϕ 0 ϕ 0 ϕ 0

The image of ρ is densely contained the main diagonal. Both projections are dense in S 1 , but the image does not fill T 2 . Notice that the row rank of Θ over Z is 1 as the dimension of the smallest subspace containing ρ π 1 Σ .

Example 4.25. Let Σ be a surface of genus 2, and let ρ :

The matrix Θ has the following form

The image of ρ densely fills the torus. Notice that the rank of Θ is 2 in this case and both projections are dense.

Sp 2g (Z)-action and orbit closures

The symplectic group Sp 2g (Z) acts on the homological representation space Hom H 1 (Σ, Z), T n by precomposition. We have seen in section 4.5 that, up to a choice of a symplectic basis, this latter space identifies with the space M n, 2g; T . In this section we would like to study the orbit closures of an element of M n, 2g; T under the action of Sp 2g (Z). The first thing we notice is that a subset p ∈ M n, 2g; R may be regarded as the couple (I, p). The space M n, 2g; R naturally identifies with the G/P , where P is the stabiliser of any point. It is straighforward to check that the stabiliser of the zero matrix is nothing but Sp 2g (R). The subgroup Γ = Sp 2g (Z) M n, 2g; 2πZ is a lattice in G and acts in the obvious way on G/S. Under these conditions we are in the right position to apply Ratner's Theorem, see RM, which we state as follows according to our setting.

• the vectors (θ i,1 , . . . , θ i,2g ) i=1,...,k are linearly independent over Z;

and hence it is sufficient to study K γ for γ = (I, p) and p is a matrix given in Equation 8. Furthermore, it will be sufficient to study the closures of Γ-orbits for matrices in these form. The second thing we notice is that K γ is a linear subspace of M n, 2g; R invariant under the action of P γ by conjugation. Indeed, suppose γ = (I, p), let q = (I, q) ∈ K γ be any point and let (A, p -pA -1 ), we can write as the element (I, p) • (A, id) • (I, -p), be a generic element of P γ . Then

The following Lemma implies Theorem D for representations of closed surface groups into the unit circle S 1 .

Lemma 4.26.

Proof. Let Λ be the subgroup of R generated by the entries of θ and consider Λ 2g . The first claim is easy to establish. In this case Λ 2g is a lattice in R 2g containing 2πZ 2g and preserved by Sp 2g (Z). Now observe that the Sp 2g (Z)-orbit of θ is contained in Λ 2g . Suppose θ ∈ R 2g \ πQ 2g , hence there exists

There is an element in Sp 2g (R) such that all the entries are R \ πQ. We may assume θ i ∈ 0, 2π for all i = 1, . . . , 2g. Let θ * ∈ T 2g be any point. For each couple (θ 2i-1 , θ 2i ), where i = 1, . . . , g, there are two integers k i , h i such that the couple

). Therefore the Sp 2g (R)-orbit of θ is dense in T 2g and hence Sp 2g (Z) 2πZ 2g • θ is dense in R 2g as desired.

Before proving the general case we need the following proposition on which we describe the group

Proposition 4.27. Let p ∈ X be any point in the form given in the equation (8) and let k the number of lines not in πQ 2g . Let H γ be the group provided by Ratner's Theorem, where γ = (I, p). Then K γ is trivial or

Proof of Proposition 4.27. Let p be any point in M n, 2g; R . Assume p be different from the zero matrix for which the claim trivially holds. Let us begin with the case p = πQ ∈ M n, 2g; πQ , that means k = 0. We claim K γ to be trivial. Let γ = (I, p) and let H γ be the group provided by Ratner's Theorem. The orbit Γ•p lies in the subgroup of M n, 2g; R generated by the matrices πq ij E ij , where πq ij are the entries of p, which is discrete and closed. This means that Γ • p = Γ • p and implies H γ is the stabiliser of p. Therefore H γ = P γ and hence K γ is trivial. Notice that this argument generalises the first case of the previous Lemma 4.26. Let us now assume k > 0. The linear space K γ is completely determined by Θ o , indeed the block πQ does not give any contribution. In this case, the orbit Γ • p is no longer closed and the Sp 2g (Z)-orbit of p is contained in some linear subspace of M k, 2g; R of dimension 2g l, where l is the dimension of the linear space generated by the rows of Θ o . Hence K γ contains V as a proper subspace. We can notice that V is Sp 2g (Z)-invariant but V ∩ M k, 2g; 2πZ is not a lattice because the Z-module generated by the rows of Θ o is πQ-free. For the same reason, the minimal linear space containing V and a lattice is M k, 2g; R , hence K γ = M k, 2g; R .

From the proof of Proposition 4.27 we can deduce the following corollary.

Corollary 4.28. Let p ∈ M n, 2g; R be any point in the form given in the equation (8) and let k the number of lines not in πQ 2g . There exists a closed connected subgroup H ≤ T n of dimension k such that Γ • p projects to a finite union of inhomogeneous torii of dimension k corresponding to cosets of H. In particular, the modular orbit of a dense representation ρ : H 1 (Σ, Z) -→ T n is dense in the representation space.

This corollary implies Theorem F and indeed Theorem D. In the appendix, we shall study the modular orbits by applying a direct approch without rely on Ratner's Theory.

An application: Approximation result

The aim of this final section consists in showing Proposition 4.9 and indeed Theorem E. Let us begin by recalling the statement of Kronecker's Theorem as formulated in Hewitt and Ross 1963, Section 26.19(e). The reader may also consult Bekka and Mayer 2000, Section 1.12(iii) for another one-dimensional version of Kronecker's theorem.

m , with i = 1, . . . , n, be vectors of R m such that b (1) , . . . , b (n) , πe 1 , . . . , πe m are linearly independent over Q in the vector space R m (where the e j 's form the canonical basis of R m ). Let a 1 , . . . , a n be any real numbers and let ε be a positive number. Then there is an element k 1 , . . . , k m ∈ Z m such that

for every i = 1, . . . , n.

For a real a, the expression |a| < ε mod 2π means that |a -2kπ| < ε for some integer k. From the equation ( 9) above, one can easily infer the equivalent estimate

where h 1 , . . . , h n ∈ Z n , B is the matrix having b (i) 's as rows, C is a real constant depending only on n and • is any norm on R n . Kronecker's theorem generalises to simultaneous approximation of l given real vectors a (j) = a 1j , . . . , a nj t where j = 1, . . . , l. Indeed, for any ε > 0 there is a matrix

Therefore, we can deduce that Ω(ρ) is contained in some proper linear subspace S of R 2n . In fact, the coefficients of any matrix A • Θ(ρ) = (θ i,j ) i,j ∈ Ω(ρ) satisfy the following homogeneous linear system S :

in 2n-2 equations and 2n variables. Hence, S is defined as the full space of solutions of the linear system S. Let us consider then the subspace S. Since each of the λ i is taken as an element of R \ Q, the subspace S meets the lattice M n, 2; 2πZ only at the origin. Therefore, the projection of the subspace S into the space M n, 2; T densely fills a closed subspace K of M n, 2; T . We finally claim that K cannot be a proper subspace. To this end, we begin with noting that, due to the nature of the linear system S, the subspace S splits as the direct product

Therefore, the image of S into the space M n, 2; T lies inside a closed subgroup of the form

Therefore the proof of the final claim boils down to show that H i cannot be a proper subgroup for both i = 1, 2. As the group H 1 contains the vector exp θ 1 , λ 2 θ 1 , . . . , λ n θ 1 , then it contains also the subgroup exp t θ 1 , λ 2 θ 1 , . . . , λ n θ 1 | t ∈ Z and thus its closure which we know to be equal to the full space T n . In the same fashion, we can prove H 2 = T n . Therefore K = M n, 2; T and the SL 2 (Z)-orbit of Θ ρ is dense in M n, 2; T as desired.

The general case for surfaces of genus greater than one works similarly. Given any matrix of the form as in equation ( 13), up to change the matrix with any element of Sp 2g (Z) we may assume, without loss of generality, that at least one of θ 2i-1 , θ 2i / ∈ πQ. Under this condition all the observations of subsection 5.10.1 hold for each pair of columns

Therefore the action of the g-times product SL 2 (Z) × • • • × SL 2 (Z) < Sp 2g (Z) provides a dense orbit inside the space M n, 2g; T as desired.

Finding curve generating dense subgroups

All the representations ρ considered in subsection 4.10.1 above are characterized by the following property: each column of the associated matrix Θ ρ generates a dense subgroup of T n . Actually, for any such a representation one can find infinitely many curves whose image generates a dense subgroup in T n . This lead us to ask: given a dense representation ρ : π 1 Σ -→ T n , can we find a simple closed curve γ such that ρ(γ) is dense in T n ? Remind that a vector e iθ 1 , . . . , e iθn ∈ T n generates a dense subgroup if and only if π, θ 1 , . . . , θ n are linearly independent over Q. As a corollary of Lemma we deduce the following Lemma.

Lemma 4.31. Let ρ : π 1 Σ -→ S 1 be a dense representation. Then there always exists a simple closed curve γ such that ρ(γ) = S 1 .

However, for n ≥ 2, the scenario changes completely. Indeed, for any n we can find examples of dense representations which do not have any curve generating a dense subgroup in T n .

Example 4.32. Let Σ be a surface of genus g and ρ : π 1 Σ -→ T 2 be the representation associated to the matrix Θ ρ ∈ M 2, 2g; T defines as follow.

One can show that Θ ρ has maximal rank and hence ρ is a dense representation. However, no curve is applied by ρ to a vector generating a dense subgroup.

Example 4.33. Let Σ be a surface of genus g and ρ : π 1 Σ -→ T n be the representation associated to the matrix Θ ρ ∈ M n, 2g; T defines as follow.

where 1, θ 3 , . . . , θ n are linearly independent over Q. One can show that Θ ρ has maximal rank and hence ρ is a dense representation. However, no curve is applied by ρ to a vector generating a dense subgroup.

Surfaces with one puncture

Let us now discuss the case of the one-holed torus Σ. We shall denote π 1 Σ ∼ = a, b the fundamental group of Σ. Also in this case the choice of a representation consists in choosing for each generator an element of T n . The representation space Hom π 1 Σ, T n trivially identifies with the space T n × T n .

For each choice of an element c, the relative representation variety Hom c π 1 Σ, T n is defined as the preimage of c via the commutator map k : T n × T n → T n . Thus, as a consequence of the abelian property, the relative representation space is empty for any c = (1, . . . , 1) and coincides with the full representation variety when c = (1, . . . , 1). Once again, the action of T n by inner automorphisms is trivial and hence the character variety trivially coincides with the representation space. As a consequence, the space Hom π 1 Σ, T n naturally identifies with the space Hom π 1 T, T n . The equalities Mod(T ) = Mod(Σ) = SL(2, Z) are well-known and the actions of Mod(T ) and Mod(Σ) on the representation spaces associated to T and Σ respectively coincide. Therefore, we have the following proposition.

Proposition 4.34. Theorem D and Theorem F hold for the torus T if and only if they hold for the one-holed torus Σ.

More generally, the main results of the present work extend to surfaces of higher genus and with one boundary component. Indeed, let Σ g,1 be a surface a surface of genus g and one boundary component. We have already seen above that this is true for the one-holed torus Σ, Proposition 4.34.

The general claim follows because, as a consequence of the abelian property of T n , one can establish an identification between the representations spaces Hom π 1 Σ, T n and Hom π 1 Σ g,1 , T n . Since the mapping class group coincides with the pure mapping class group for one-puncture surfaces the following proposition also holds.

Proposition 4.35. Theorem D and Theorem F hold for a closed surface of genus g if and only if they hold for the one-holed surface of genus g.

L'objectif de cette thèse est de répondre à des problématiques dynamiques quant à l'action du groupe modulaire sur les variétés de caractères de surfaces. Nous étudions d'abord le cas des variétés de caractères à valeurs dans des groupes de Lie semi-simples et compacts. L'action du groupe modulaire est alors connue pour être ergodique. Nous renforçons cette propriété dynamique en démontrant que le sous-groupe de Torelli agit ergodiquement sur ces variétés de caractères. Ce fait généralise un corollaire d'un résultat de Funar-Marché pour les variétés de caractères à valeurs dans SU(2). Ensuite nous adaptons les arguments de la preuve d'un théorème de Marché-Wolff au cas des surfaces à bord. Le groupe modulaire pur agit sur les variétés de caractères relatives. En imposant des conditions aux bords et en considérant les classes d'Euler qui le permettent, nous introduisons un sous-espace analogue à celui de Marché-Wolff et montrons que le groupe modulaire pur agit ergodiquement sur celui-ci. Enfin, l'ergodicité des actions considérées implique que presque toutes les orbites sont denses. Prévite-Xia donne une condition nécessaire et suffisante pour que la classe de conjugaison de ρ : π 1 Σ → SU(2) ait une orbite dense dans la variété des caractères de Σ à valeurs dans SU(2). Avec Gianluca Faraco, nous démontrons un résultat analogue pour l'espace des représentations d'une surface dans un groupe de Lie connexe, compact et Abélien. Nous montrons ainsi qu'une représentation a une orbite dense si et seulement si son image est dense et relions cet énoncé à des problèmes d'approximations diophantiennes.

The aim of this thesis is to answer to some dynamical questions about the modular group action on character varieties of surfaces. We first study the case of character varieties with values in a semi-simple and compact Lie groups. The modular group action is then known to be ergodic. We improve this dynamical property in proving that the action of the Torelli subgroup is ergodic on such varieties. This fact generalizes a result of Funar-Marché for character varieties in SU(2). We then adapt the arguments of the proof of Marché-Wolff in the case of surfaces with boundaries. The pure modular group acts on the relative character varieties. Fixing conditions on the boundary and taking the Euler classes which allows it, we introduce a subspace which is analogue to the one of Marché-Wolff and prove that the pure modular group acts ergodically on it. Finally, the ergodicity of the actions we considered implies that almost every orbit is dense. Prévite-Xia gives a necessary and sufficient sufficient on a representation ρ : π 1 Σ → SU(2) such that its orbits is dense in the character variety in SU(2). With Gianluca Faraco, we prove an analogue result for the representations space in a compact, connected and Abelian Lie group. We hence show that a representation has a dense orbit if and only if its image is dense and relate this statement to problems of Diophantine approximations.