In this thesis we study various groups of piecewise continuous transformations, notably of the interval, consisting of piecewise translations. One of these is the group IET of interval exchange transformations. A larger one is the group IET of interval exchange transformations with flips. For every subgroup Γ of R, we consider a subgroup of IET, denoted by IET(Γ), consisting of elements of IET which are continuous outside Γ. We similarly define the group IET (Γ). This is a countable group when Γ is countable.

The preliminary Chapter 1 introduces these groups in detail.

In Chapter 2, we exhibit a natural isomorphism between the abelianization of IET(Γ) and the second skew-symmetric power of Γ over Z. The case Γ = R was treated by Arnoux-Fathi-Sah. The case when Γ is equal to 2Γ can be treated similarly, but otherwise there is a new difficulty, and indeed 2-torsion then appears in the abelianization (the above skewsymmetric power not just being the second exterior power). The idea is to construct a group homomorphism which "measures", for every f ∈ IET(Γ), the set of pairs flipped by f . This group homomorphism is inspired of the classical signature on finite permutation groups. In Chapter 3 we exhibit a natural isomorphism between the abelianization of IET (Γ) and an explicit 2-elementary abelian group. In the case Γ = R, Arnoux proved that IET is a perfect group.

In Chapter 4, we prove the vanishing of the Kapoudjian class of the group of piecewise continuous self-transformations of [0, 1[. This result is equivalent to the existence of a group homomorphism onto Z/2Z which extends the classical signature on finitely supported permutations.

In Chapter 5 we study a generalization of IET in higher dimension. For d ≥ 1, we consider the group Rec d of rectangle exchange transformations of [0, 1[ d . We introduce a family of transformations, called restricted shuffles, and we prove that they form a generating subset of Rec d . In dimension 1, this result is already known and easy, but the proof for d ≥ 2 is much more complicated, the underlying combinatorics being richer. We use this generation result to identify the abelianization of Rec d by generalizing the Arnoux-Fathi-Sah homomorphism to this higher-dimensional context.

iii 0 1 1 Le graphe d'un élément de PC (sans précisions sur les valeurs en les points de discontinuité).

Table des matières

Introduction en français

Cette thèse porte sur l'étude de certains groupes d'origine dynamique en s'intéressant à leur abélianisé, ce qui revient à classifier leurs invariants additifs (c'est-à-dire, leurs morphismes vers des groupes abéliens). Elle a été réalisée sous la supervision d'Yves Cornulier entre octobre 2018 et juin 2021. Les groupes d'origine dynamique en question sont des groupes agissant par isométries par morceaux sur un intervalle. Un exemple plutôt connu est le groupe des échanges d'intervalles, noté IET, constitué des translations par morceaux de l'intervalle [0, 1[.

1) Bref historique

Les éléments du groupe IET sont introduits par M. Keane [START_REF] Keane | Interval exchange transformations[END_REF] en 1975 et ce sont principalement leur dynamique et leurs propriétés ergodiques qui sont étudiés. L'étude de IET en tant que groupe est initiée par Arnoux-Fathi [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF] et Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] par l'identification de l'abélianisé de IET. Durant ces quinze dernières années, cette étude s'est poursuivie et renforce notre compréhension des sous-groupes de IET. On peut citer en particulier les travaux de C. Novak [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF], Dahmani-Fujiwara-Guirardel [DFG13, DFG17], Boshernitzan [START_REF] Boshernitzan | Subgroup of interval exchanges generated by torsion elements and rotations[END_REF] et Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF]. Ces travaux incluent des progrès sur la question de l'existence ou non d'un sous-groupe libre de rang 2 dans IET de Katok. Une autre question, posée par Y. Cornulier, porte sur le caractère moyennable ou non moyennable du groupe IET. Pour cette question aussi, des avancées ont été faites en particulier avec le travail de Juschenko-Monod [START_REF] Juschenko | Cantor systems, piecewise translations and simple amenable groups[END_REF] et un peu plus tard rejoints par Matte Bon et de la Salle [START_REF] Juschenko | Extensive amenability and an application to interval exchanges[END_REF]. Gauche : Le graphe d'un élément de IET (sans précisions sur les valeurs en les points de discontinuité). Droite : Son "action" sur [0, 1[.

Si on autorise toutes les transformations qui sont isométriques par morceaux, alors on obtient le groupe des échanges d'intervalles avec renversements, noté IET . Ce groupe possède de nombreuses connexions avec les feuilletages mesurés non orientés sur des surfaces et billards. L'étude de la dynamique et des propriétés ergodiques d'un élément a intéressé Gutierrez [START_REF] Gutiérrez | Smooth nonorientable nontrivial recurrence on two-manifolds[END_REF], Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF], Nogueira [START_REF] Nogueira | Almost all interval exchange transformations with flips are nonergodic[END_REF] et Danthony-Nogueira [START_REF] Danthony | Measured foliations on nonorientable surfaces[END_REF] et plus récemment O. Paris-Romaskevich et P. Hubert [START_REF] Paris | Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and rauzy gasket[END_REF] ou Skripchenko-Troubetzkoy [START_REF] Skripchenko | On the Hausdorff dimension of minimal interval exchange transformations with flips[END_REF]. Peu de choses sont connus sur le groupe lui-même, Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] a prouvé que c'est un groupe simple et N. Guelman et I. Liousse [START_REF] Guelman | Bounded simplicity of affine interval exchange transformations and interval exchange transformations[END_REF] ont prouvé que ce groupe est uniformément parfait (tout élément est produit d'au plus 6 commutateurs). Gauche : Le graphe d'un élément de IET qui contients des renversements (sans précisions sur les valeurs en les points de discontinuité). Droite : Son "action" sur [0, 1[.

Le groupe des transformations continues par morceaux et affines par morceaux apparaît dans différents travaux en particulier à cause de l'inclusion naturelle des groupes de Thompson. Il a été initialement définit par M. Stein [START_REF] Stein | Groups of piecewise linear homeomorphisms[END_REF] et récemment N. Guelman et I. Liousse ont prouvé que son sous-groupe des éléments qui préservent l'ordre par morceaux est simple.

Le groupe de tous les éléments continus par morceaux, dénoté par PC , a aussi été étudié. Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] a prouvé que son sous-groupe PC + des éléments qui préservent l'orientation des morceaux est simple.

2) Abélianisés des groupes IET(Γ) et IET (Γ) Le chapitre 1 introduit tous les groupes considérés dans cette thèse. On décrit des sous-familles d'éléments qui jouent un rôle important dans l'étude de ces groupes, par exemple l'ensemble des rotations restreintes (voir définition 1.2.7) est un système générateur du groupe IET.

On introduit aussi des familles de sous-groupes des groupes IET et IET qui sont à l'origine de cette thèse. Pour tout sous-groupe Γ de R/Z, on définit IET(Γ), respectivement IET (Γ), comme le sous-groupe de IET, respectivement de IET , des éléments qui sont continus en dehors de la pré-image Γ de Γ dans R. L'identification de leur abélianisé est le premier problème étudié dans cet thèse. Le cas où Γ est fini peut être réduit à l'étude d'un groupe de permutation fini, ainsi on comprend directement son sous-groupe dérivé et son abélianisé. Pour Γ = R, Arnoux-Fathi et Sah ont démontré le théorème suivant :

Théorème 1 (Arnoux-Fathi-Sah [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF]). Il existe un morphisme de groupes surjectif (explicite) ϕ : IET →2 Q R dont le noyau est égal au sous-groupe dérivé D(IET). Il est appelé le SAF-invariant.

On s'intéresse ici au cas de la restriction de ce morphisme aux sous-groupes IET(Γ) pour Γ infini. Que peut-on tirer de la preuve originale ? Celle-ci utilise le fait qu'on peut couper un intervalle en son milieu. Ainsi avec la condition Γ = 2 Γ on peut toujours couper un intervalle en son milieu et ainsi il est possible d'adapter la preuve originale pour obtenir que ϕ induit un isomorphisme :

IET(Γ) ab 2 Z Γ.
Quand cette condition n'est pas satisfaite on ne peut plus forcément couper un intervalle en son milieu. Pour Γ quelconque on a encore un morphisme surjectif IET(Γ) ab 2 Z Γ mais il n'est pas forcément injectif ; son noyau est en fait un 2-groupe. Pour le décrire plus précisément, on introduit la seconde puissance anti-symétrique de Γ au dessus de Z noté 2 Z Γ qui est généralement différente de la seconde puissance extérieure 2 Z Γ. Dans le chapitre 2 on prouve le résultat suivant : Théorème 2 (Voir théorème 2.3.24). Il existe un morphisme de groupes surjectif (explicite) ε Γ : IET(Γ) → 2 Z Γ dont le noyau est égal au sous-groupe dérivé D(IET(Γ)).

L'idée est de construire un morphisme de groupes qui mesure, pour tout f ∈ IET(Γ), son ensemble d'inversions : {(x, y) | x ≤ y, f (x) ≥ f (y)}. Il est inspiré du morphisme signature des groupes de permutations finis. Nous obtenons alors un morphisme de groupes qui est la "moitié" du SAF-invariant. Pour cela on considère une algèbre de Boole noté A Γ telle que l'ensemble des inversions appartient à A Γ ⊗A Γ . On construit aussi une mesure sur l'algèbre de Boole A Γ ⊗ A Γ , à valeurs dans 2 Z Γ, pour mesurer l'ensemble des inversions. Dans le cas où Γ est un groupe abélien libre de rang d, on obtient que l'abélianisé de IET(Γ) est naturellement isomorphe à Z d(d-1 ) 2

× (Z/2Z) d . Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] a prouvé que le groupe IET est simple et sa preuve utilise aussi le fait qu'on peut couper un intervalle en son milieu. On montre que si on a la condition Γ = 2 Γ alors IET (Γ) est toujours un groupe simple. Dans le cas général on montre le résultat suivant : Théorème 3 (Voir théorème 3.4.5). L'abélianisé IET (Γ) ab est naturellement isomorphe au groupe {a ⊗ a [mod 2] | a ∈ Γ} × { ∧ [mod 2] | ∈ Γ} , où le terme de gauche du produit appartient à 2 Z Γ/(2 2 Z Γ) et le terme de droite à L'ensemble des inversions d'un élément de IET (Γ) n'est plus à chaque fois dans A Γ ⊗ A Γ et ne peut donc plus être mesuré comme dans le cas de IET(Γ). On remarque alors qu'en considérant l'union de cet ensemble avec son symétrique (par rapport à l'axe y = x) on obtient un ensemble qui appartient toujours à A Γ ⊗ A Γ . Cela donne un premier morphisme de groupes mais dont le noyau est plus gros que le sous-groupe dérivé D(IET (Γ)). La deuxième idée est d'"approximer", en un certain sense, les éléments de IET (Γ) par des éléments de IET(Γ). Cela nous permet d'utiliser le morphisme construit pour IET(Γ) pour obtenir un autre morphisme sur IET (Γ). On prouve alors que l'intersection des noyaux des deux morphismes construits est égal au sous-groupe dérivé D(IET (Γ)).

3) Classe de Kapoudjian Soit X un ensemble (pour la suite on aura X = [0, 1[) et soit S(X) le groupe de toutes les permutations de X. On note S fin = S fin (X) son sous-groupe des permutations à support fini. On rappelle ici qu'avec la même preuve utilisée pour les groupes de permutations finis, on peut construire un morphisme de groupes surjectif de S fin sur Z/2Z que l'on appelle la signature classique. Son noyau, noté A fin , est l'unique sous-groupe d'indice 2 de S fin ; il est appelé le sous-groupe alterné.

On définit PC le sous-groupe de S(X) constitué des transformations qui sont continues en dehors d'un nombre fini de points et on note PC sa projection dans S(X)/S fin . Ainsi pour tout sous-groupe G de S(X)/S fin on note G sa pré-image dans S(X).

Dans cette partie, il est important de faire la distinction entre les groupes continus par morceaux (sous-groupes de PC ) et leur projection dans PC . En effet, dans [START_REF]Realizations of groups of piecewise continuous transformations of the circle[END_REF], Y. Cornulier a prouvé que le groupe PC ne se relève pas dans le groupe des permutations de l'intervalle [0, 1[. Cela motive alors sa question sur l'annulation d'un élément spécifique du second groupe de cohomologie appelé la classe de Kapoudjian ; en effet la non-annulation de cette classe impliquerait le résultat de non-relèvement ci-dessus. Cette classe apparaît dans le travail de Kapoudjian et Sergiescu [START_REF] Kapoudjian | Virasoro-type extensions for the higman-thompson and neretin groups[END_REF][START_REF] Kapoudjian | An extension of the Burau representation to a mapping class group associated to Thompson's group T , Geometry and dynamics[END_REF] où ils considèrent la suite exacte suivante : 0 → Z/2Z = S fin /A fin → S/A fin → S(X)/S fin → 1 De plus, pour tout sous-groupe G de S(X)/S fin on obtient la suite exacte suivante :

0 → Z/2Z = S fin /A fin → G/A fin → G → 1
Cette suite exacte correspond en fait à une extension centrale et donc à un élément du second groupe de cohomologie H 2 (G, Z/2Z) appelé la classe de Kapoudjian de G.

Dans le chapitre 4, on répond à la question posée par Cornulier [START_REF]Realizations of groups of piecewise continuous transformations of the circle[END_REF] en prouvant l'annulation de la classe de Kapoudjian pour PC . La première chose à noter est que cette question est équivalente à la construction d'un morphisme de groupes sur PC qui étend la signature classique. C'est le théorème suivant : Théorème 4 (Voir théorème 4.1.2). Il existe un morphisme de groupes ε : PC → Z/2Z qui étend la signature classique de S fin .

Ainsi la classe de Kapoudjian de PC est égale à la classe triviale et cela implique le même résultat pour tous les sous-groupes de PC . L'idée derrière ce théorème est d'écrire tous les éléments f ∈ PC comme un produit d'un élément de IET avec un produit de renversements et une permutation à support fini. En imposant un certain paramètre on arrive à avoir une telle décomposition qui est unique et on prouve que la somme modulo 2 du nombre de renversements apparaissant dans la décomposition avec la signature classique de la permutation à support fini est un nombre invariant de ce paramètre. Il reste alors à prouver que l'on a bien un morphisme de groupes.

Une application de ce résultat est la description complète des sous-groupes normaux de certains sous-groupes de PC : Théorème 5 (Voir théorème 4. 1.4). Soit G un sous-groupe de PC qui est simple et non abélien. Soit G sa pré-image dans PC . Alors G a exactement 5 sous-groupes normaux, qui forment le diagramme suivant :

G S fin
Ker(ε| G )

A fin {1}
4) Echanges de rectangles Une manière de généraliser le groupe IET en dimension d pour d ≥ 1 est de remplacer l'intervalle [0, 1[ par le rectangle [0, 1[ d . On peut alors considérer le groupe Rec d des bijections f de [0, 1[ d telles qu'il existe une partition finie en sous-rectangles de [0, 1[ d telle que f est une translation sur tous les rectangles de cette partition. On remarque alors que pour d = 1 on a l'égalité entre Rec 1 et IET. Dans le chapitre 5, on commence l'étude du groupe Rec d par donner un analogue des rotations restreintes appelé battage restreint (voir définition 5.1.3).

On prouve le théorème suivant :

Théorème 6 (Voir théorème 5.1.5). L'ensemble des battages restreints est un système générateur de Rec d .

En dimension 1, on peut interpreter un échange d'intervalle comme une permutation de segments et le résultat se ramène au fait que le groupe symétrique S n est engendré par les transpositions (i, i + 1). En dimension supérieure ou égale à 2, on a pas une interprétation combinatoire si simple : par exemple il n'existe pas forcément de quadrillage envoyé sur un autre quadrillage par translation des pièces. L'idée est alors de démontrer qu'avec certaines conditions sur une partition il est possible de la transformer en quadrillage grâce à des battages restreints. Ces conditions portent sur de la Q-liberté de sous-ensembles de R qui sont liés aux longueurs des côtés des pièces de la partition.

Après cela, on poursuit avec l'étude du sous-groupe dérivé D(Rec d ). On dit qu'un élément f ∈ Rec d est une transposition de rectangles s'il existe un sous-rectangle P de [0, 1[ d tel que P ∩ f (P ) = ∅ et f permute P avec f (P ) en fixant le reste de [0, 1[ d (voir définition 5.1.3) ; on note par T d l'ensemble de toutes les transpositions de rectangles et on prouve le résultat suivant : Théorème 7 (Voir théorème 5.1.6). Le sous-groupe dérivé D(Rec d ) est simple et est engendré par son sous-ensemble T d . De plus il est contenu dans tous les sous-groupes normaux non triviaux de Rec d .

Pour le cas d = 1, la simplicité est due à Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] (non-publié) et le résultat d'engendrement s'en déduit facilement. Une preuve détaillée est due à Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF], une preuve plus simple est presentée par Vorobets dans [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF]. Ce théorème repose en partie sur le théorème qui le précède. En effet on déduit de ce dernier que le groupe D(Rec d ) est engendré par les conjugés des commutateurs de deux battages restreints.

Le dernier théorème du chapitre 5 est une généralisation du théorème d'Arnoux-Fathi-Sah [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF] à propos de l'identification de l'abélianisé de IET. On note R ⊗k la k-ième puissance tensorielle de R au dessus de Q.

Théorème 8 (Voir théorème 5.7.21). Le groupe (Rec d ) ab est naturellement isomorphe à

(R ⊗d-1 ⊗ ( 2 Q R)) d .
L'idée est que chaque direction a une contribution qui est indépendante des autres directions et que cette contribution est calculée grâce à un analogue du morphisme construit par Arnoux-Fathi-Sah. Un exemple d'application est que le groupe IET d ∪T d = Rec d si d ≥ 2. En effet, on vérifie que la restriction de l'isomorphisme (R ⊗d-1 ⊗ ( 2 Q R)) d n'est pas surjective.

Introduction in English

This thesis focuses on the study of some groups with dynamical origins by looking at the abelianization, it is the same as classifying their additive invariants (in other words, group homomorphisms from these groups to abelian groups). It has been realized under the supervision of Yves Cornulier between October 2018 and June 2021. More precisely, we deal with groups which act piecewise isometrically on an interval. An important example of such group is the group of interval exchange transformations denoted by IET, consisting of piecewise translations.

1) Brief historical account.

The elements of IET were introduced by M. Keane [START_REF] Keane | Interval exchange transformations[END_REF] in 1975 and it is mostly their dynamics and ergodic properties which have been studied. The study of the group itself was initiated by Arnoux-Fathi [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF] and Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] by determinating its abelianization. In the past fifteen years, this study has been pursued and rises our understanding of IET's subgroups, notably with the work of C. Novak [START_REF] Christopher | Discontinuity-growth of interval-exchange maps[END_REF], Dahmani-Fujiwara-Guirardel [DFG13, DFG17], Boshernitzan [START_REF] Boshernitzan | Subgroup of interval exchanges generated by torsion elements and rotations[END_REF] and Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF]. These works contain some progress about the question, due to Katok, of the existence of a free subgroup of rank 2 in IET. The related question of the amenability of IET has been raised by Y. Cornulier and here also, some progress has been made by Juschenko-Monod [START_REF] Juschenko | Cantor systems, piecewise translations and simple amenable groups[END_REF] and further the two of them with Matte Bon and de la Salle [START_REF] Juschenko | Extensive amenability and an application to interval exchanges[END_REF]. If one allows all piecewise isometric transformations, we obtain the interval exchange transformations group with flips, denoted IET . This group has strong connections with non oriented measured foliations on surfaces and billiards and the study of the dynamical and ergodic properties of a single element has interested Gutierrez [START_REF] Gutiérrez | Smooth nonorientable nontrivial recurrence on two-manifolds[END_REF], Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF], Nogueira [START_REF] Nogueira | Almost all interval exchange transformations with flips are nonergodic[END_REF] and Danthony-Nogueira [START_REF] Danthony | Measured foliations on nonorientable surfaces[END_REF] and more recently by O. Paris-Romaskevich and P. Hubert [START_REF] Paris | Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and rauzy gasket[END_REF] or Skripchenko-Troubetzkoy [START_REF] Skripchenko | On the Hausdorff dimension of minimal interval exchange transformations with flips[END_REF]. Few things are known about the group itself, Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] proved that it is a simple group and N. Guelman and I. Liousse [START_REF] Guelman | Bounded simplicity of affine interval exchange transformations and interval exchange transformations[END_REF] proved that this group is uniformly perfect (every element is a product of 6 commutators). The group of piecewise continuous and affine transformations appears in different works because of the natural inclusion of Thompson's groups. It has been initially defined by M. Stein [START_REF] Stein | Groups of piecewise linear homeomorphisms[END_REF] and recently N. Guelman and I. Liousse [START_REF] Guelman | Bounded simplicity of affine interval exchange transformations and interval exchange transformations[END_REF] proved that its subgroup consisting of piecewise orientation-preserving elements is simple.

The whole group of piecewise continuous elements, denoted by PC , has been barely studied. Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] proved that its subgroup PC + , consisting of piecewise orderpreserving transformations, is a simple group. A graph of an element of PC (without specifying values at discontinuity points).

2) Abelianization of IET(Γ) and IET (Γ)

Chapter 1 introduces all the groups considered in the thesis. We describe subfamilies which play a main role in the study of subgroups of PC , for instance the set of all restricted rotations (see Definition 1.2.7) is a generating subset of IET.

We also introduce a family of subgroups of IET and IET which were the starting point of the thesis. For every subgroup Γ of R/Z, we define IET(Γ) (resp. IET (Γ)) as the subgroup of IET (resp. IET ) consisting of all elements that are continuous outside the preimage Γ of Γ in R. The identification of their abelianization was the first problem of my thesis. The case where Γ is finite can be reduced to the study of a finite permutation group and thus we already well-understand its derived subgroup and abelianization. For Γ = R, Arnoux-Fathi and Sah prove the following theorem:

Theorem 1 (Arnoux-Fathi-Sah [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF]). There exists an explicit surjective group homomorphism ϕ : IET → 2 Q R whose kernel is equal to the derived subgroup D(IET). It is called the SAF-invariant.

Here we try to understand the restriction of this morphism to subgroups IET(Γ) for Γ infinite. What can we deduce from the original proof? This proof uses the fact that we can cut an interval at its middle point, then with the condition Γ = 2 Γ it is still possible to cut an interval at its middle point thus it is possible to adapt the original proof to obtain that ϕ induces an isomorphism

IET(Γ) ab 2 Z Γ.
This is no longer true when this condition is not satisfied. For a general Γ, we still have a sujective group homomorphism

IET(Γ) ab 2 Z
Γ but it is not always injective; its kernel is in fact a 2-group. In order to describe it more precisely, we introduce the second skew-symmetric power of Γ over Z denoted by The idea is to construct a group homomorphism which for every f ∈ IET(Γ) measures the set of inversions: {(x, y) | x ≤ y, f (x) ≥ f (y)}. It is inspired by the signature group homomorphism for finite permutation groups. This brings us to obtain a group homomorphism that is the "half" of the SAF-invariant. For this we consider a Boolean algebra denoted by A Γ such that the set of inversions belongs to A Γ ⊗ A Γ . We construct a measure on the Boolean algebra A Γ ⊗ A Γ , with value in 2 Z Γ, to measure the set of inversions.

In the case where Γ is an abelian free group of rank d, we obtain that the abelianization of IET(Γ) is naturally isomorphic to Z [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] proved that the group IET is simple and its proof uses the fact that we can cut an interval at its middle point. The proof then extends to if Γ = 2 Γ = {0} then IET (Γ) is simple. In the general case we prove the following theorem: Theorem 3 (See Theorem 3.4.5). The abelianization group IET (Γ) ab is naturally isomorphic to the group {a ⊗ a

d(d-1) 2 × (Z/2Z) d . Arnoux
[mod 2] | a ∈ Γ} × { ∧ [mod 2] | ∈ Γ} , where the left term of the product is in 2 Z Γ/(2 2 Z Γ) and the right one is in 2 Z Γ/(2 2 Z Γ).
The set of inversions of an element of IET (Γ) is not always in A Γ ⊗ A Γ . But, if we considered the union of this set with its mirror (its symmetric image according to the axis y = x) we obtain a new set that belongs to A Γ ⊗A Γ . This gives a first group homomorphism but its kernel is larger than the derived subgroup D(IET (Γ)). The second idea here is to "approximate" in some sense an element of IET (Γ) by elements of IET(Γ). From this we manage to construct a second group homomorphism and we prove that the intersection of their kernels is equal to the derived subgroup D(IET (Γ)).

3) Kapoudjian class

Let X be a set (in the following we will have X = [0, 1[) and let S(X) be the group of all permutations of X. Let S fin = S fin (X) be its subgroup consisting of all finitely supported permutations. We recall that with the same proof used for finite permutation groups, we can construct a surjective group homomorphism called the classical signature from S fin onto Z/2Z. Its kernel, denoted by A fin , is the alternating subgroup and it is the unique subgroup of index 2 of S fin .

We denote by PC the subgroup of S(X) consisting of all piecewise continuous transformations and we denote by PC its projection in S(X)/S fin . Then for every subgroup G of mf S(X)/S fin we define G its preimage in S(X).

In this part, it is important here to distinguish between piecewise continuous groups (subgroups of PC ) and their image in PC . Indeed, in [START_REF]Realizations of groups of piecewise continuous transformations of the circle[END_REF], Y. Cornulier proved that the group PC cannot be lifted to a group of permutations of the interval [0, 1[. This motivates his question on the vanishing of a specific element of the second cohomology group called the Kapoudjian class; precisely the vanishing of this class implies the above nonlifting result. This class appears in the work of Kapoudjian and Sergiescu [Kap02,[START_REF] Kapoudjian | An extension of the Burau representation to a mapping class group associated to Thompson's group T , Geometry and dynamics[END_REF] where they considered the following exact sequence:

0 → Z/2Z = S fin /A fin → S/A fin → S(X)/S fin → 1
For every subgroup G of S(X)/S fin , we obtain similarly the following exact sequence:

0 → Z/2Z = S fin /A fin → G/A fin → G → 1
This exact sequence is in fact a central extension and thus can be seen as an element of the second cohomology group H 2 (G, Z/2Z) called the Kapoudjian class of G.

In Chapter 4, we answer the question of Cornulier [START_REF]Realizations of groups of piecewise continuous transformations of the circle[END_REF] by proving that the Kapoudjian class of PC vanishes. The first thing to notice is that this question is equivalent to the construction of a group homomorphism from PC which extends the classical signature on S fin , it is the following theorem: Theorem 4 (See Theorem 4.1.2). There exists a group homomorphism ε : PC → Z/2Z that extends the classical signature on S fin .

Hence the Kapoudjian class of PC vanishes and this implies the result for every subgroup of PC . The idea behind this theorem is to write every f ∈ PC into a product of an element of IET with a product of flips and a finitely supported permutation. Up to a parameter we manage to get that this decomposition is unique and we prove that the sum modulo 2 of the number of flips in the decomposition with the signature of the finitely supported permutation is independent of this parameter. It remains to prove that it is a group homomorphism.

An application of this result is the complete description of normal subgroups of some subgroups of PC : In dimension 1, we can interpret an interval exchange transformation as a permutation of segments and the result reduces to the fact that the symmetric group S n is generated by transpositions (i, i + 1). In dimension superior or equal to 2 , there is not such an easy combinatorial description: for instance there does not always exist a grid-pattern sent onto another grid-pattern by translation of the pieces. The idea is to prove that, with some conditions on a partition, it is possible de move pieces of this partition, thanks to restricted shuffles, in order to obtain a grid-pattern. These conditions are related to the Q-freeness of some subsets of R which are linked with the length of the side of pieces of the partition.

After that we pursue the study of the derived subgroup D(Rec d ) by giving a generating subset of the derived subgroup. We called an element f ∈ Rec d a rectangle transposition if there exists a subrectangle P of [0, 1[ d such that P ∩ f (P ) = ∅ and f permutes P with f (P ) while fixing the rest of [0, 1[ d For the case d = 1, the simplicity is due to Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] (unpublished) and the result of generation is deduced easily. A detailed proof is due to Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF], and a simpler one is presented by Vorobets in [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF]. This theorem relies on the previous one because we obtain that D(Rec d ) is generated by conjugates of commutators of two restricted shuffles.

The last theorem of Chapter 5 is a generalization of the theorem of Arnoux-Fathi-Sah [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF] about the identification of the abelianization of IET. We denote by R ⊗k the k-th tensor power of R over Q.

Theorem 7 (See Theorem 5.7.21). The group

(Rec d ) ab is naturally isomorphic to (R ⊗d-1 ⊗ ( 2 Q R)) d .
The idea is that every direction has a contribution which is independent from other directions. This contribution is calculated thanks to an analogue of the group homomorphism constructed by Arnoux-Fathi-Sah [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF]. An application is that the group

IET d ∪T d = Rec d if d ≥ 2. Indeed, we check that the restriction of the isomorphism (R ⊗d-1 ⊗ ( 2 Q R)) d is not surjective.
CHAPTER 1

Piecewise continuous groups

Résumé en français. Dans ce chapitre nous construisons les groupes continus par morceaux. On donne aussi des systèmes générateurs de ces groupes ainsi que de leur sousgroupe dérivé. On présente des résultats connus qui sont des préliminaires pour les autres chapitres.

English abstract. This chapter contains a formal definition for piecewise continuous groups. We define subfamilies which allow us to have generating subset for these groups or their derived subgroups. Also, we give known results about these groups that are preliminaries for next chapters. The group PC contains the subgroup S fin thus the quotient PC = PC /S fin is well-defined. Definition 1.1.2. We denote by IET the subgroup of S([0, 1[) consisting of all elements that are continuous outside a finite set and piecewise isometric. We also define IET + as the intersection IET ∩ PC + .

Then we notice that the group IET + is isomorphic to IET thus we freely use either both notations. The group IET is the group of interval exchanges with flips up to a finitely supported permutation.

Remark 1.1.3. Here the "+" means that we preserve locally the order of [0, 1[. We can also consider PC -the subset of PC consisting of piecewise-reversing elements. It is not a group however if we consider PC ± = PC + PC -we obtain two groups. Then PC + is a subgroup of index 2 in PC ± . The same remark can be done by defining similarly IET - and IET ± and we obtain the following property: Proposition 1.1.4. The normalizer of IET + in IET is IET ± . The same is true when we replace IET by PC.

Proof. Let g be an element of IET which is not the identity. There are two cases:

1. If g ∈ IET + IET then g = σg with σ ∈ S fin {Id} and g ∈ IET. Then for every f ∈ IET we have gf g -1 = σg fg -1 σ -1 . Thus it is enough to treat the case of S fin . Let us assume g ∈ S fin then let x in the support of g. There exist two consecutive right-open and left-closed intervals I and J of the same length such that x is the right endpoint of I (and the left endpoint of J). Up to reduce I and J we can assume that I does not intersect the support of g. Then let f ∈ IET which exchanges I and J while fixing the rest of [0, 1[. Then gf g -1 exchanges the interior of I with the interior of J but gf g -1 (x) is not equal to f (x) because f (x) is the left endpoint of I and I does not intersect the support of g. Then we deduce that gf g -1 is not right-continuous on J.

If g ∈ IET

IET + . Then we can find two consecutive subinterval I and J where g is continuous and order-reversing on I ∪ J. Let a be the right endpoint of J. Let f be the element in IET which exchanges I and J. Then gf g -1 exchanges the interior of g(J) with the interior of g(I). However the left endpoint of g(J) is send by g -1 on a which is fixed by f . Then gf g -1 fixes the left endpoint of g(J), thus gf g -1 is not right-continuous on g(J).

A natural family of subgroups of IET + and IET can be defined thanks to subgroups of R/Z. Let Γ be a subgroup of R/Z and let Γ be its preimage in R.

Definition 1.1.5. We denote by IET (Γ) (resp. IET + (Γ)) the subgroup of IET (resp. IET + ) consisting of all elements that are continuous outside Γ.

We describe below the case where Γ is finite. After that we always assume that the group Γ is dense in R/Z.

Remark when Γ is finite

We assume that Γ is a finite subgroup of R/Z. Then there exists n ∈ N ≥1 such that Γ is equal to 1 n Z. We deduce that for every 1 ≤ i ≤ n, every element of IET (Γ) is continuous on the interval [ i-1 n , i n [ up to a finitely supported permutation. Then the group IET(Γ) is naturally isomorphic to the finite permutation group S n . It is a Coxeter group of type A n-1 so its abelianization is

{1} if n = 1 and it is Z/2Z if n > 1.
The group IET (Γ) is isomorphic to the signed symmetric group Z/2Z S n . It is a Coxeter group of type B n so its abelianization is Z/2Z for n = 1 and it is (Z/2Z) 2 if n > 1.

Topological-full groups

A way to understand groups is to understand their actions on some spaces. Here we explicit how piecewise continuous groups and their image in PC act on a Stone space. Definition 1.1.6. A Stone space is a totally disconnected compact space.

For example every Cantor space is a Stone space. In this context we define topologicalfull groups.

Definition 1.1.7. Let X be a Stone space and G be a subgroup of Homeo(X). The group G is a topological-full group if for every n in N, every {X 1 , X 2 , . . . , X n } and {Y 1 , Y 2 , . . . , Y n } two partitions into clopen subsets of X and for every g i in G such that g i (X i ) = Y i ; the element g ∈ Homeo(X) which satisfies g = g i on X i is an element of G. The group G is said to be minimal if for every x ∈ X the orbit of x is dense in X.

Topological-full groups have connection with groupoids and they are intensively studied. We do not explain more these notions in this thesis and for more background about it we redirect to the work of Matui [START_REF] Matui | Some remarks on topological full groups of cantor minimal systems[END_REF][START_REF]Topological full groups of one-sided shifts of finite type[END_REF], of Nekrashevych [START_REF] Nekrashevych | Simple groups of dynamical origin, Ergodic Theory Dynam[END_REF] and the work of Matte Bon [START_REF] Matte | Rigidity properties of full groups of pseudogroups over the cantor set[END_REF]. We explicit a result of Nekrashevych about the smallest normal subgroups appearing in minimal topological-full groups, adapted to our cases. It gives the simplicity of a lot of derived subgroups.

Theorem (Nekrashevych [Nek19]). Let X be an infinite Stone space and let G be a subgroup of Homeo(X) such that G is a topological-full group acting minimally on X. Then the subgroup of G generated by the subset of all elements of order 3 is simple and contained in every normal subgroup of G.

Next, we present two Stone spaces and we explain how PC and PC act on one of them.

Examples.

1. We denote by S (-,+) the space obtained from R/Z by "doubling" all points. We replace every point x ∈ R/Z by two copies x -, x + where x -< x + and where we endow S (-,+) with the topology induced by the natural circular order on it. For more clarity we think about [0, 1[ as S (-,+) where every interval [a, b[ has to be understand as the interval [a + , b -] in S (-,+) . As we have the order topology then the set of every such intervals is a basis of the topology, hence the space is Hausdorff and totally disconnected. The compactness is a consequence of the compactness of R/Z. Indeed let I be a directed set and ψ : I → S (-,+) be a net. Let p : S (-,+) → R/Z be the natural projection. Then ξ := p(ψ) is a net for R/Z which is compact so there exists a cluster point α ∈ R/Z for ξ. By the absurd, assume that α - and α + are not cluster points for ψ. Hence there exist a ∈ I and U -, U + open neighbourhoods of respectively α -and α + such that for every b ∈ I with b ≥ a we have ψ(b) / ∈ U -∪ U + . We deduce that there exists w,

x ∈ R/Z different from α such that α -, α + ∈ [w + , x -] and ψ(b) / ∈ [w + , x -] for every b ≥ a. Hence for every b ≥ a we have ξ(b) = p(ψ(b)) / ∈ ]w, x[ ⊂ p([w + , x -]). Or ]w, x
[ is an open neighbourhood of α which is in contradiction with the fact that α is a cluster point for ξ. We deduce that either α -or α + is a cluster point for ψ.

2.

The second example is when we triple every point. We denote by S (-,•,+) the space obtained from R/Z by "tripling" all points. We replace every point x ∈ R/Z by three copies x -, x • , x + where x -< x • < x + and where we endow S (-,•,+) with the topology induced by the natural circular order on it. Here an [a, b[ has to be understand as (-,•,+) . Similar proof can be done to prove that it is a Stone space. We just point that {a • } is also a clopen subset; thus S (-,•,+) is a Stone space with isolated points.

{a • } [a + , b -] in S
Then we can see that PC and its subgroups can also be seen as subgroups of Homeo(S (-,•,+) ). For every f ∈ PC and for every a ∈ [0, 1[ if f is continuous on a neighbourhood of a then if f is order-preserving on this neighbourhood we define f (a ε ) = (f (a)) ε with ε ∈ {+, -} and if f is order-reversing we define f (a ε ) = (f (a)) -ε . Also for every a ∈ [0, 1[, we put

f (a • ) = (f (a)) • .
Similarly, PC and its subgroups can be seen as subgroups of Homeo(S (-,+) ).

With this point of view it is immediate that every group define with the symbol PC or IET are topological-full groups. However those who are subgroups of Homeo(S (-,+) ) are minimal topological-full groups. In the other cases, the orbit of any point a • with a ∈ [0, 1[ is never dense.

Subfamilies and tools

We introduce the notion of partition associated to an element, this tool is really useful to describe how act an element. We pursue by describing different subfamilies and how they manage to describe some groups. After that we focus on the group IET by expliciting the decomposition into minimal and periodic elements done by Arnoux-Fathi [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF]. We finish by some properties between elements of finite order and those of order 2. We denote by λ the Lebesgue measure on R.

Partition associated and combinatorial description

From now on every partition into intervals of [0, 1[ is supposed to be finite and into rightopen and left-closed intervals. For every subinterval I of R we denote by I • its interior. Definition 1.2.1. Let f be an element of PC and P be a partition into intervals of [0, 1[. The partition P is said to be a partition associated with f if for every interval I ∈ P, the element f is continuous on I • . We denote by Π f the set of all partitions associated with f .

It is an essential partition associated with f if there exists a finitely supported permutation σ such that P is a partition associated with σ f . For every essential partition associated with f we denote by f (P ) the arrival partition of f associated with P the partition into right-open and left-closed intervals of [0, 1[ such that for every J ∈ f (P) there exists

I ∈ P such that J • = σ f (I • ).
It is necessary to define essential partitions in order to define partition associated with an element of PC . Because two representatives have the same set of essential partitions associated but not the same set of partitions associated. Definition 1.2.2. Let f ∈ PC and let f be a representative of f in PC . Let P be a partition into intervals of [0, 1[. The partition P is said to be a partition associated with f if it is an essential partition associated with f . We denote by Π f the set of all partitions into intervals associated with f . We denote by f (P) the arrival partition of f associated with P the arrival partition of f associated with P.

Let f ∈ PC and f be its image in PC . We precise that the set Π f is not equal to Π f , it is more larger.

Remark 1.2.3. For every f in PC there exists a unique partition P min f associated with f which has a minimal number of intervals. It is actually minimal in the sense of refinement: Π f consists precisely of the set of partitions refining P min f . This partition is the partition define by the set of discontinuities of the unique representative of f in PC that has the minimal number of discontinuities. Sometimes we will want to know where the length of the intervals live: Definition 1.2.4. Let S be a subset of R. Let f ∈ PC and P be a partition associated with f . The partition P is called a S-partition if for every I ∈ P the length of I is in S. Remark 1.2.5. Let Γ be a subgroup of R/Z and let Γ be its preimage in R. For the sake of notations, a Γ-partition is also called a Γ-partition.

Another description strongly used in IET is a description that gives the dynamic of the element: Definition 1.2.6. Let f ∈ IET and let P = {I 1 , . . . , I k } be a partition associated with f where we order the intervals to be consecutive. Let f (P) = {J 1 , . . . , J k } be the arrival of f associated with P where we order the intervals to be consecutive. Let μ := {λ(I 1 ), . . . , λ(I k )} and σ be the finitely supported permutation such that f (I i ) = J σ(i) up to a finite number of points. Then (μ, σ) is called a combinatorial description of f . If every component of μ is in Γ then (μ, σ) is a Γ-combinatorial description of f . The partition P is also called the partition associated with (μ, σ).

Subfamilies

One of the first thing we want to get, when we study a group, is a nice generating subset. We start in IET to finish in PC .

1.2.2.1 In IET Definition 1.2.7. For every a, b ∈ R, a restricted rotation of type (a, b) is an element r of IET such that there exists two consecutive intervals I and J of length a and b respectively with sup(I) = inf(J), where r is the translation by +b on I and -a on J. The intervals I and J are called the intervals associated with r.

For every subgroup Γ of R/Z, a Γ-restricted rotation is a restricted rotation in IET(Γ). We cite Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] and Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] for the following theorem. Proof. Let f, P and S be as in the statement. We numbered the intervals of P, then f act like a permutation on this numbering. Such a permutation can be written as a product of transposition and such a transposition on the numbering coincide with a restricted rotation with type inside S × S.

Corollary 1.2.9. The group IET is generated by its subset of all restricted rotations and for every subgroup Γ of R/Z, the group IET(Γ) is generated by its subset of all Γ-restricted rotations.

This gives one decomposition with a property on the length of the intervals. Another useful property is the decomposition into elements of small support. To obtain this result we introduce some elements of order 2 and we need to understand how we can reduce a restricted rotation. This is done by Vorobets in [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] in IET. Definition 1.2.10. Let a ∈ [0, 1 2 ] and f ∈ IET. The element f is an IET-transposition of type a if it permutes two non-overlapping subintervals of [0, 1[ of length a while fixing the rest of [0, 1[. For every subgroup Γ of R/Z, if in addition f ∈ IET(Γ) then f is a Γ-transposition of type a. Proof. This is immediate with Figure 1 Lemma 1.2.12. For every a, b > 0 and ε > 0 there exists sequences (a i ) i∈N and (b i ) i∈N such that:

.2.3. If a ≤ b then it appears a Γ-restricted rotation of type (a, b -a) and if a ≥ b it is a Γ-restricted rotation of type (a -b, b). a -b b b b a a b a a b -a
1. (a 0 , b 0 ) = (a, b); 2. either (a i+1 , b i+1 ) = (a i , b i -a i ) or (a i -b i , b i ); 3. both (a i ) i and (b i ) i converge to 0. Proof. By induction we define (a i+1 , b i+1 ) = (a i -b i , b i ) if a i > b i and (a i+1 , b i+1 ) = (a i , b i -a i ) if a i ≤ b i .
Then we notice that both (a i ) i and (b i ) i are decreasing sequences of positive real numbers. By construction we have

a i+1 + b i+1 = a i + b i -min(a i , b i ).
By iterating the equality we deduce that the sum of the minimum converges thus this minimum tends to 0. As we reduce at each step the maximum by the minimum we deduce that the minimum encounters both an infinite number of times (a i ) i and (b i ) i thus these two sequences tend to 0. Proposition 1.2.13. Let Γ be a subgroup of R/Z. For every ε > 0 and every f ∈ IET, the element f is a product of Γ-restricted rotations and Γ-transpositions with a support's length smaller than ε.

Proof. Let Γ be a subgroup of R/Z and let Γ be its preimage in R. Let ε and f as in the statement. If f is a Γ-transposition, let I be the interval such that f permutes I and f (I) while fixing the rest of [0, 1[. The interval I has endpoints in Γ hence we can partition I into intervals with endpoints in Γ and with support's length smaller than ε 2 . Then we obtain that f is a product of Γ-transpositions with support's length smaller than ε. By Theorem 1.2.8 it is enough to do the case where f is a restricted rotation. Thanks to Proposition 1.2.11 and Lemma 1.2.12 and the previous case we deduce the result.

In IET and IET

The only difference between the definition of IET + and IET is that we have piecewise isometric elements and not only piecewise translations. This means that the only thing we add is the possibility to reverse an interval. Definition 1.2.14. Let I be a subinterval of [0, 1[. We define the I-reflection map as the element r I of IET that reverses the interior I • while fixing the rest of [0, 1[. The type of an I-reflection map is the length of I. We define the I-reflection as the image of the Ireflection map in IET . The type of an I-reflection is the length of I. For every subgroup Γ of R/Z, a Γ-reflection is anI-reflection for some subinterval of [0, 1[ with endpoints in Γ.

Remark 1.2.15. A reflection is also called a flip and that is why IET is called the group of Interval Exchange Transformations with flips.

It is immediate that IET is generated by IET + and its subset of reflection maps. The similar is true for IET (Γ). Hence the group IET is generated by IET + and its subset of reflections. Also we can reduce to the subset of reflection: Proposition 1.2.16. For every subgroup Γ of R/Z, the group IET (Γ) is generated by its subset of Γ-reflections.

Proof. It is enough to show that IET + (Γ) is generated by flips, thus to show that every Γrestricted rotation is a product Γ-reflections. Let r be a Γ-restricted rotation and let I and J be the two intervals associated to r. We assume sup(I) = inf(J). For every right-open and left-closed subinterval I of [0, 1[, we define s I the I-reflection. Then r I,J = s I s J s I∪J .

We deduce that we have generating subsets consisting of elements with small support: Corollary 1.2.17. For every ε > 0 and every subgroup Γ of R/Z, the group IET (Γ) is generating by its subset of Γ-reflections and Γ-transpositions whose support's measure is less than ε.

In PC

The main thing to see here is that we can modify the length of every interval by an element of the homeomorphism group of [0, 1[ which preserves the orientation (this group is denoted by Homeo + ([0, 1[)). This allows us to link the group PC with IET and to give a generating subset consisting of elements which have small support. Proof. Let f ∈ PC and let P := {I 1 , . . . , I n } be a partition associated with f where we sorted the intervals such that for every 1 ≤ i ≤ n -1 we have sup(I i ) = inf(I i+1 ). We denote by a i the left endpoint of I i . Let f (P) = {J 1 , . . . , J n } and let σ be the permutation of {1, . . . , n} such that f (I

• i ) = J • σ(i) .
We denote by b i the left endpoint of J i . There exists a finitely supported permutation whose support is included in

{a i | 1 ≤ i ≤ n} such that f (a τ (i) ) = b i . Let {K 1 , . . . , K n } be the partition of [0, 1[ into intervals such that for every 1 ≤ i ≤ n the length of K i is equal to I σ -1 (i) .
Let g be the element of IET which sends

K • i on I • σ -1 (i)
and sends the left endpoint of K i on the left one of I σ -1 (i) . We also ask g to be order-reversing on K i if f is order-reversing on I σ -1 (i) , and order-preserving otherwise. Then the composition f • τ • g is an element of Homeo + ([0, 1[) by sending K i on J i . Hence there exists ψ ∈ Homeo + ([0, 1[) such that ψ • f ∈ IET . The other case is similar.

Corollary 1.2.19. We have the following equalities:

1. PC = Homeo + ([0, 1[) IET ; 2. PC + = Homeo + ([0, 1[) IET + .
Hence, to obtain that there exists a generating subset of PC consisting of elements of small support it is enough to show it for Homeo + ([0, 1[), a proof is given by Arnoux in [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF].

Proposition 1.2.20. Let ε > 0, then every f ∈ Homeo + ([0, 1[) is a product of elements of Homeo + ([0, 1[) with support's length less than ε. Proof. Let f ∈ Homeo + ([0, 1[). Let I = [a, a + b[ be a subinterval of [0, 1[ such that Supp(f ) ⊂ I. Let us assume that f (a + b 2 ) ∈ [a, a + b 2 [. Then there g ∈ Homeo + ([0, 1[) such that g(f (a + b 2 )) = a + b 2 and Supp(g) ∈ [a, a + 3 4 b[. Then a + b 2 is a fixed point of g • f hence there exist f 1 , f 2 ∈ Homeo + ([0, 1[) such that Supp(f 1 ) ⊂ [a, a + b 2 [ and Supp(f 2 ) ⊂ [a + b 2 , a+ b[. Then we deduce that f = g -1 • f 1 • f 2
and every homeomorphism of this product have a support's length less than λ(I)

3 . The same can be done if we assume that f (a + b

2 ) ∈ [a + b 2 , a + b[, thus by iterating this process a finite number of time we obtain the result.

Corollary 1.2.21. For every ε > 0, the group PC and PC + have a generating subset consisting of elements which have support's length less than ε.

Minimal and finite order elements in IET

The study of finite order elements appears to be useful for the study of the derived subgroup, see Subsection 2.3.3. First we give a nice decomposition into two kinds of elements done by Arnoux in [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] and then we describe how finite order elements and elements order 2 are related.

1.2.3.1 Decomposition into elements with disjoint support Definition 1.2.22. Let U be a subset of [0, 1[. An element f of IET is said to be minimal on U if U is invariant by f and for every x ∈ U , the orbit of x by f is dense in U .
The following decomposition into minimal and periodic elements done by Arnoux in [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF] is a decomposition into disjoint elements. Arnoux proved it in the case of IET and the result in the case of IET(Γ) is just a consequence. Proposition 1.2.23. Let Γ be a subgroup of R/Z ad let Γ be its preimage in R. Let f ∈

IET(Γ). There exist unions of right-open and left-closed disjoint intervals

U 1 , . . . , U n , V 1 , . . . , V k with endpoints in Γ, such that: 1. they form a partition of [0, 1[ into f -invariant subsets; 2. For every 1 ≤ i ≤ n the element f is minimal on U i ; 3. For every 1 ≤ i ≤ k there exists p i ∈ N such that every element x ∈ V i has minimal period p i .

Finite order elements

A consequence of Proposition 1.2.23 is the following Proposition proved by Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] in the case of IET:

Proposition 1.2.24. Let Γ be a subgroup of R/Z. Every finite order element f of IET(Γ) is a finite product of Γ-transpositions. In particular there exist n ∈ N and P := {I 1 , . . . , I n } a partition associated with f and a permutation σ of {1, . . . , n} such that f

(I i ) = I σ(i) .
Proof. Let f ∈ IET(Γ) be a finite order element. By Proposition 1.2.23 there exists k ∈ N and for every 1 ≤ i ≤ k there exist p i ∈ N and an element f i ∈ IET(Γ) of minimal period p i such that:

1. for every 1 ≤ i = j ≤ k the support of f i and the support of f j do not overlap;

2. f = f 1 . . . f k .
Then it is enough to assume f to have minimal period p ∈ N. In this case, there exist the union of the set of discontinuities of powers of f is finite, we denote by P the partition of [0, 1[ into intervals defined by these points. We notice that P is a Γ-partition associated with f . By construction there exists a partition of P into subsets p intervals such that f is a permutation of the intervals in every of these subsets. We deduce that f is a product of Γ-transpositions.

Now we want to understand how the product of two IET-transpositions is a finite order element. This is not immediate because for every n ∈ N we can construct examples of such a product with order n: Proposition 1.2.25. For every n in N ≥1 there exist two IET-transpositions f and g such that the product gf has order n.

Proof. We distinguish the case where n is even or odd. In both cases we illustrate the proof with Figure 1.2.4. The case n = 1 is given by the equality f 2 = Id for any IETtransposition f . Let n ∈ N ≥1 .

Let I and J be two consecutive intervals of the same length ∈ [0, 1 2 ] and let g be the IET-transposition that swaps I and J. Let A 1 , A 2 , . . . , A n-1 and C be consecutive intervals of length n such that the left endpoint of A 1 is the left endpoint of I (hence the right endpoint of C is the right endpoint of I). Let D and B 1 , B 2 , . . . , B n-1 be consecutive intervals of length n such that the right endpoint of B n-1 is the right endpoint of J (hence the left endpoint of D is the left endpoint of J). Let f be the IET-transposition that swaps A i and B i for every

1 ≤ i ≤ n -1. Hence by definition we get g(A 1 ) = D, g(A i ) = B i-1 for every 2 ≤ i ≤ n -1 and g(C) = B n-1 . So the composition gf is equal to the permutation (A 1 A 2 . . . A n-1 C B n-1 B n-2 . . . B 1 D). Thus gf has order 2n.
It remains the case of order 2n -1. Let I, J and K be three consecutive intervals with I and J of length ∈ [0, 1 3 [ and K of length ∈ ] n , [. Let g be the IETtransposition that swaps I and J. We define A 1 , A 2 , . . . , A n-1 consecutive intervals of length n such that the right endpoint of A n-1 is the right endpoint of I. We define also D and B 1 , B 2 , . . . , B n-1 consecutive intervals of length n such that the left endpoint of B n-1 is the left endpoint of K. Let f be the IET-transposition that swaps A i and B i for every 1 ≤ i ≤ n -1. One can check that the product gf in this case is the permutation

(A 1 A 2 . . . A n-1 B n-1 B n-2 . . . B 1 D) so gf has order 2n -1. I J A 1 A 2 A 3 C D B 1 B 2 B 3 A 4 B 4 I J A 1 A 2 A 3 D B 1 B 2 B 3 A 4 B 4 K Figure 1.2.4 -Illustration of Proposition 1.
2.25 with n = 5. Left: For the case "fg has order 2n".

Right: For the case "fg has order 2n -1".

To prove that every product of IET-transpositions is a fintie order element, we use the notion of reversible maps and some of their properties proved by N. Guelman and I. Liousse in [START_REF]Reversible maps and products of involutions in groups of iets[END_REF].

Definition 1.2.26. Let h, k be two elements of IET. We said that h reverses k if hkh -1 = k -1 .
A direct consequence of the definition is:

Proposition 1.2.27. Let h, k ∈ IET such that h reverses k. Then for every p ∈ Z both h and h -1 reverse k p . Thus k p h = hk -p .
For every f ∈ IET we denote by Per(f ) the set of periodic points of f and by Fix(f ) the set of fixed points of f . Lemma 1.2.28. Let h, k ∈ IET such that h reverses k. Then:

1. The set Per(k) is invariant by h; 2. The set Fix(h) is a subset of Per(k).
Proof.

1. Let x ∈ Per(k) and let p ∈ Z such that k p (x) = x. By Proposition 1.2.27 we have k p (h(x)) = hk p (x) = h(x), thus h(x) ∈ Per(k).

By contradiction let x ∈ Fix(h) Per(k).

As h is in IET we deduce that there exists c ∈ [0, 1[ such that the whole interval [x, c[ ⊂ Fix(h). Also x is not a periodic point of k thus the point x is a right cluster point of the k-orbit of x. This means that there exists a sequence (p n ) n of positive integers such that k pn (x) > x and lim

n→∞ k pn (x) = x.
Hence there exists an integer p > 0 such that

k p (x) ∈ [x, c[ thus it is a fixed point of h. Then k p (x) = h(k p (x)) = k -p (h(x)) = k -p (x)
and we deduce that x ∈ Per(k) which is a contradiction.

Corollary 1.2.29. Let f and g be two IET-transpositions. Then f and g reverse gf and for every

x / ∈ Per(gf ) the f, g -orbit of x is contained in [0, 1[ Per(gf ) ⊂ Supp(f )∩Supp(g).
Lemma 1.2.30. Let f and g be two IET-transpositions. Then gf has finite order.

Proof. Let I, J, A, B be the intervals such that sup(I) ≤ inf(J) and f swaps I with J while fixing the rest of [0, 1[ and sup(A) ≤ inf(B) and g swaps A with B while fixing the rest of

[0, 1[. Let α, β ∈ [0, 1[ such that J = I + α and B = A + β. A crucial property is that either J ∩ A or I ∩ B is the empty set.
By contradiction let x be a point that is not in Per(gf ). By Corollary 1.2.29 we know that the f, g -orbit of x is included in :

Supp(f ) ∩ Supp(g) = (I ∩ A) (I ∩ B) (J ∩ A) (J ∩ B)
We distinguish three cases:

1. If the f, g -orbit of x does not intersect (I ∩ A) (J ∩ B) then it is included in (I ∩ B) ∩ (J ∩ A)
and as one of them is empty it is included either in I or J which is impossible.

2. Now assume that the f, g -orbit of x intersects I ∩A. Up to change x by a point of its gf -orbit we can assume that x ∈ I ∩A. Then we deduce that f (x) ∈ (J ∩A) (J ∩B).

a If J ∩ A = ∅ then f (x) ∈ J ∩ B thus gf (x) ∈ I ∩ A hence for every n ∈ N we have (gf ) n (x) ∈ I ∩ A. Then (gf ) n (x) = x + n(α -β) hence x has to be inside Per(gf ) which is a contradiction. b If J ∩ A = ∅ then I ∩ B = ∅ thus g(x)
∈ J ∩ B and we deduce that fg(x) ∈ I ∩ A. Then for ever n ∈ N we obtain that (fg) n (x) = x + n(βα) hence x ∈ Per(fg). Also Per(fg) = Per(g -1 f -1 ) = Per(gf ) thus x ∈ Per(gf ) which is a contradiction.

3. Let now assume that the f, g -orbit of x intersects J ∩ B and does not intersect I ∩ A. Up to change x by a point of its gf -orbit we can assume that x ∈ J ∩ B. Then Proof. The proof is only a consequence of the following equalities between commutators:

f (x) ∈ I ∩ B, in particular I ∩ B = ∅ thus J ∩ A = ∅
1. [f 1 f 2 , g] = f 1 [f 2 , g]f -1 1 [f 1 , g]; 2. [f, g 1 g 2 ] = [f, g 1 ]g 1 [f, g 2 ]g -1 1 .
Thanks to Theorem 1.1.2 It is enough to prove that the derived subgroup is the smallest normal subgroup to obtain that the derived subgroup is simple. This come from the Epstein's simplicity criterion [START_REF] Epstein | The simplicity of certain groups of homeomorphisms[END_REF] and it is also described by Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF].

Theorem 1.3.2. Let Γ be a dense subgroup of R/Z. Let G be a subgroup of PC which contains IET + (Γ) and such that for every ε > 0 there exists a generating subset of G consisting of elements with support's length less than ε. Then D(G) is the smallest normal subgroup of G.

Proof. Let G be a subgroup of PC which contains IET + (Γ) and let N be a normal subgroup of G. It is enough to show that there exists ε > 0 such that every commutator of two element in G with a support's length less than ε is in N . Let f ∈ N {Id}. There exists an interval I such that f is continuous on I and

I ∩ f (I) = ∅ and I ∪ f (I) = [0, 1[. Let ε = λ(I)
3 and let g 1 , g 2 ∈ PC + (resp. IET + ) with support's length less than ε. By density of Γ, there exists t ∈ IET + (Γ) that translates Supp(g 1 ) ∪ Supp(g 2 ) inside

I. Let h = t -1 • f • t.
As N is normal we deduce that h ∈ N . Also we notice that h(Supp(g 1 ) ∪ Supp(g 2 )) ∩ (Supp(g 1 ) ∪ Supp(g 2 )) = ∅ thus Supp(g 1 ) ∩ Supp(hg 2 h -1 ) = ∅ thus these two elements commute. We deduce that the commutator

[g 1 , [g 2 , h]] = [g 1 , g 2 ] also [g 2 , h] ∈ N and so [g 1 , [g 2 , h]] ∈ N . Then [g 1 , g 2 ] ∈ N and we deduce that D(G) is a subgroup of N . Corollary 1.3.3. Let G be a subgroup of PC . Assume that there exists a dense subgroup Γ of R/Z such that IET + (Γ) is a subgroup of G. Then the derived subgroup D(G) is simple.
In particular this is true for PC , PC + , IET (Γ) and IET + (Γ) for every dense subgroup Γ of R/Z.

Abelianization

Simple groups

We explicit here some groups that are perfect and thanks to Corollary 1.3.3 these groups will be simple. Theorem 1.3.4 (Arnoux [START_REF]Un invariant pour les echanges d'intervalles et les flots sur les surfaces[END_REF]). Both PC + and IET are simple.

The case of IET results of the fact that a reflection is a commutator thanks to Figure 1.3.1. For PC + it results from the simpleness of the group Homeo + (R/Z) (see Schreier and Ulam [START_REF] Schreier | Eine bemerkung über die gruppe der topologisehen abbildungen der kreislinieauf sich selbst[END_REF]); this implies that a restricted rotation is a commutator as well as every homeomorphism of [0, 1[ with small support. Theorem 1.3.4 has for direct consequence: Z A, which is injective if A = 2A. In general, its kernel is a 2-elementary abelian group, which is not trivial in general. For instance, if

A Z d , then 2 Z A is isomorphic to Z d(d-1)/2
, while this kernel is isomorphic to (Z/2Z) d . In the case where A = R we recall that every map is Z-bilinear if and only if it is Q-bilinear,

thus 2 Q R = 2 Z R. 1.3.2.3 Abelianization of IET Arnoux-Fathi [Arn81a]
and Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] proved the following theorem that identifies the abelianization of IET. ]). There exists a surjective group homomorphism from IET onto the second skew-symmetric algebra of R over Q whose kernel is the derived subgroup of IET.

Theorem 1.3.6 (Arnoux-Fathi-Sah [Arn81a, Sah81
Definition 1.3.7. This group homomorphism is called the SAF-invariant, or the SAFhomomorphism, and it is define by the following:

ϕ := IET -→ 2 Q R f -→ a∈R a ∧ λ((f -Id) -1 ({a}))
In particular, for every restricted rotation f of type (a, b) we have ϕ(f ) = 2b ∧ a.

Remark 1.3.8. Let Γ be a subgroup of R/Z and let Γ be its preimage in R. If Γ = 2 Γ then the restriction of the SAF-invariant to IET + (Γ) is surjective on 2 Z Γ and its kernel is equal to the derived subgroup D(IET + (Γ)).

We explicit how this homomorphism can be expressed thanks to a combinatorial description:

Proposition 1.3.9. Let f ∈ IET and (μ, σ) be a combinatorial description of f . Let k be the number of coordinates of μ. Then we have:

ϕ(f ) = k i=1 ( i: σ(i)<σ(j) μ i - i<j μ i ) ∧ μ j
Proof. Let f ∈ IET and (α, τ ) be a combinatorial description of f . Let n be the length of α and let {I 1 , I 2 , . . . , I n } be the partition associated with (α, τ ). For each j we denote by v(j) the value of f -Id on I j . Thus we deduce that ϕ(f The injectivity of the SAF-homomorphism relies on the fact that we can always cut an interval into two intervals with the same size. This is the main obstruction to obtain the abelianization of IET(Γ). This is the same obstruction to understand the abelianization of IET (Γ) because to show that a reflection is a commutator we cut an interval into four intervals of the same size.

) = n j=1 v(j) ∧ λ(I j ) = n j=1 v(j) ∧ α j . Also we know that v(j) = i τ (i)<τ (j) α i - i<j α i (see

CHAPTER 2

Abelianization of IET(Γ)

Résumé en français. Soit Γ un sous-groupe dense de R/Z et Γ sa pré-image dans R. Dans ce chapitre on établit un isomorphisme naturel entre l'abélianisé de IET(Γ) et la seconde puissance anti-symétrique de Γ sur Z notée par 2 Z Γ. Ce groupe possède souvent de la 2-torsion qui n'est pas détectée par le morphisme de groupes SAF. L'idée est de construire un morphisme de groupes qiu est inspiré par le morphisme signature sur les groupes de permutations finis. On introduit la notion d'inversion pour un élément du groupe IET et on essaye de mesurer l'ensemble de ses inversions. Ce chapitre correspond à la première moitié de l'article [START_REF] Lacourte | Abelianization of some groups of interval exchanges[END_REF].

English abstract. Let Γ be a dense subgroup of R/Z and Γ be its preimage in R. In this chapter, we establish a natural isomorphism between the abelianization of IET(Γ) and the second skew-symmetric power of Γ over Z denoted by 2 Z Γ. This group often has non-trivial 2-torsion, which is not detected by the SAF-homomorphism. The idea is to construct a group homomorphism that is inspired by the classical signature on finite permutation group. We introduce the notion of inversions for an element of IET and we try to measure the set of inversions. This chapter corresponds with the first half of the article [START_REF] Lacourte | Abelianization of some groups of interval exchanges[END_REF]

Γ

One difficulty encountered to understand the abelianization of IET(Γ) is that R is a Qvector space and Γ is only a Z-module. Hence understanding the positive cone of Γ will give informations about how cutting an interval into smaller intervals. Let us introduce some classical terminology from the theory of ordered abelian groups. An ordered abelian group is an abelian group endowed with an invariant partial ordering.

For any subgroup H of R we denote by H + := {x ∈ H | x ≥ 0} the positive cone of H. A difficulty is that H + is not, in general, finitely generated as a subsemigroup. For every subset B of R we denote by Vect N (B) the subsemigroup generated by B.

A subsemigroup of an abelian group is simplicial if it is generated, as a subsemigroup, by a finite Z-independent subset, and it is ultrasimplicial if it is the direct limit of an upward directed set of simplicial subsemigroups. An ordered abelian group is simplicially ordered if its positive cone is simplicial and is ultrasimplicially ordered if its positive cone is ultrasimplicial. The next theorem is proved by G.A. Elliott in [Ell79]: Theorem 2.1.1. Every totally ordered abelian group is ultrasimplicially ordered.

We deduce the following corollary for the totally ordered abelian group Γ: Corollary 2.1.2. There exist an upward directed set (M Γ , ≤) and a finite Z-linearly independent subset S a of Γ for every a ∈ M Γ such that for every a, b ∈ M Γ with a ≤ b we have Vect N (S a ) ⊂ Vect N (S b ) and Γ+ is equal to the direct limit lim

-→ Vect N (S a ).
We will use the Theorem 2.1.1 in the form of the following corollary which specifies the finite rank case: Corollary 2.1.3. Let H be an abelian subgroup of R of finite rank d. Then there exists a sequence (B n ) n∈N of Z-basis of H such that for each n we have Vect N (B n ) ⊂ Vect N (B n+1 ) and H + is equal to the increasing union n Vect N (B n ). Furthermore for every k ∈ N and L 1 , L 2 , . . . , L k ∈ H + there exists a basis { 1 , 2 , . . . , d } ⊂ H + of H such that for every 1 ≤ i ≤ k the element L i is a linear combination of 1 , 2 , . . . , d with coefficients in N.

Kernel of the SAF-homomorphism's restriction to IET(Γ)

See Definition 1.3.7 for the definition of the SAF-homomorphism. We denote by ϕ Γ the restriction of the SAF-homomorphism to IET(Γ). We follow the idea of Y. Vorobets in [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] and introduce the notion of balanced product of restricted rotations. The aim is to show that the kernel of ϕ Γ is generated by its subset of all balanced products of Γ-restricted rotations.

Balanced product of Γ-restricted rotations

Definition 2.2.1. Let n ∈ N and let r i be a restricted rotation for every 1 ≤ i ≤ n. For every a, b ∈ Γ+ let n a,b be the number of restricted rotation r i of type (a, b). The tuple (r 1 , r 2 , . . . , r n ) is said to be a balanced tuple of restricted rotations if n a,b = n b,a for every a, b ∈ Γ+ . We say that a product g of restricted rotations is a balanced product of restricted rotations if there exists a balanced tuple of restricted rotations (r 1 , . . . , r n ) such that g = r 1 . . . r n .

Example 2.2.2. Let a be an element in Γ+ with a ≤ 1 2 . Every Γ-restricted rotation of type (a, a) is a balanced product of restricted rotations. It is also a Γ-transposition which swapped two consecutive intervals of same length a.

Example 2.2.3. Let a, b ∈ Γ+ with a + b ≤ 1. If h is a Γ-restricted rotation of type (a, b) then h -1 is a Γ-restricted rotation of type (b, a). Thus every element of D(IET(Γ)) is a balanced product of Γ-restricted rotations.
In order to get the decomposition of the elements of Ker(ϕ Γ ), we need to know the freeness of some families of

2 Z V . Lemma 2.2.4. Suppose V is a subgroup of R (then it is a Z-module). Let k in N ≥1 and v 1 , v 2 , . . . v k elements of V which are Z-linearly independent. Then the wedge products v i ∧ v j for 1 ≤ i < j ≤ k are Z-linearly independent in 2 Z V . Proof. Let v 1 , v 2 , . . . v k in V which are Z-linearly independent. It is sufficient to prove the lemma for V = R because being Z-linearly independent in 2 Z R implies being Z-linearly independent in 2 Z V . We know that 2 Z R is isomorphic to 2 Z R. Let us assume that v 1 , v 2 , . . . , v k are in R.
Then being Z-linearly independent is the same that being Q-linearly independent. Indeed if there exist p 1 , p 2 , . . . , p k in Z and q 1 , q 2 , . . . , q k in N ≥1 such that

k i=1 p i q i v i = 0 then k i=1 ( j=1 j =i q j p i )v i = 0 is an equality in Z.
Thus for each i ∈ {1, 2, . . . , k} we have j=1 j =i q j p i = 0. Or q j = 0 for every j then p i = 0 for every i.

The Q-vector space generated by all the v i for Proof. Let f ∈ Ker(ϕ Γ ). This is trivial if f = id; assume otherwise. Let (μ, σ) be a Γ-combinatorial description of f (see Definition 1.2.6), let k ∈ N and {I 1 , I 2 , . . . , I k } be the partition into intervals associated to (μ, σ) ( we have k ≥ 2 as f is not the identity). We recall that μ i is the length of I i for every 1 ≤ i ≤ k.

1 ≤ i ≤ k is isomorphic to Q k . We can complete the Q-linearly independent set {v 1 , v 2 , . . . , v k } in a basis S of R seen as a Q-vector space. Thus in 2 Z R the elements v i ∧ v j for 1 ≤ i < j ≤ k are Q-linearly independent so they are Z-linearly independent in
We treat the case where Γ is finitely generated. Then Γ is also finitely generated and we denote by d its rank. By Corollary 2.1.3 there exist 1 , 2 , . . . , d in Γ+ such that L := { 1 , 2 , . . . , d } is a basis of Γ and such that μ i is a linear combination of 1 , 2 , . . . , d with non-negative integer coefficients for every 1 ≤ i ≤ k. Then I i can be partitioned into smaller intervals with length in L for every 1 ≤ i ≤ k. We obtain a partition associated with f whose intervals have length in L. By Theorem 1.2.8 there exist n ∈ N and a restricted rotation

f i of type (a i , b i ) with a i , b i ∈ L for 1 ≤ i ≤ n such that f = f 1 f 2 . . . f n .
For any i, j ∈ {1, 2, . . . , d} let s ij be the number of Γ-restricted rotation of type

( i , j ) in the tuple (f 1 , f 2 , . . . , f n ). As ϕ Γ (f i ) = j ∧ i -i ∧ j = 2 j ∧ i (see Definition 1.3.7), we obtain that: ϕ Γ (f ) = d i=1 d j=1 2s ij ( j ∧ i ) = 1≤i<j≤d 2(s ij -s ji )( j ∧ i )
We know that { 1 , 2 , . . . , d } is a basis of Γ thus by Lemma 2.2.4 we obtain that { j ∧ i } 1≤i<j≤d is a free family of 2 Z Γ. Then the assumption ϕ Γ (f ) = 0 implies s ij = s ji for every 1 ≤ i < j ≤ d. This means that the product of Γ-restricted rotations f 1 f 2 . . . f n is balanced.

We do not assume Γ finitely generated any more. Hence we only know that ϕ Γ (f ) =

k j=1 i: σ(i)<σ(j) μ i - i<j μ i ∧ μ j = 0 in 2 Z Γ (see Proposition 1.3.9). We denote by ϕ Γ (f ) := k j=1 i: σ(i)<σ(j) μ i - i<j μ i ⊗ μ j . It is a representative of ϕ Γ (f ) in 2 Z Γ.
Then there exist a finite set J and x j , y j ∈ Γ for every j ∈ J, such that ϕ Γ (f ) = j∈J x j ⊗ y j + y j ⊗ x j . We denote by à the subgroup of Γ generated by {μ i } 1≤i≤k ∪ {x j , y j } j∈J . Then à is a finitely generated subgroup of R which contains Z. Its image A in R/Z is a finitely generated subgroup of R/Z. Also we know that f is in IET(A) and (μ, σ) is also a A-combinatorial description of f and ϕ Γ (f ) is an element of 2 Z Ã. Thus in 2 Z à we have:

ϕ A (f ) = k j=1 i: σ(i)<σ(j) μ i - i<j μ i ∧ μ j = [ϕ Γ (f )] 2 Z Ã = [ j∈J x j ⊗ y j + y j ⊗ x j ] 2 Z Ã = 0
Then we can applied the previous case and conclude that f is a balanced product of A-restricted rotations, thus a balanced product of Γ-restricted rotations.

Ker(ϕ Γ ) is generated by Γ-transpositions

The work of Y.Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] done for IET can be adapted to show the next two lemmas. We reproduce here their proof. Lemma 2.2.6. Let f and g be two Γ-restricted rotations. If they have the same type then f -1 g is finite a product of Γ-transpositions.

Proof. The case f = g is immediate and we assume that f = g. Thanks to Proposition 1.2.11, we can reduce to the case where f and g have a support whose length is less than 1 5 thanks to a finite number of Γ-transpositions. Then there exists an interval I disjoint from Supp(f ) ∪ Supp(g) and such that λ(I) = λ(Supp(f )). Let h be the restricted rotation which has the type of f and such that its support is equal to I. Then f -1 g = f -1 hh -1 g. Then f and h has non-overlapping support, same is true for g and h, and Figure 2.2.1 proves that f -1 h and h -1 g are products of three Γ-transpositions. Lemma 2.2.7. Let f be a Γ-restricted rotation and g be any transformation in IET(Γ). Then the commutator [f, g] is a product of Γ-transpositions.

Proof. Let I be the support of f . If g is continuous on I then gf -1 g -1 is a Γ-restricted rotation of support g(I) with the same type of f -1 and by Lemma 2.2.6 we obtain the result. Thanks to Proposition 1.2.11 there exists a product h of Γ-transposition such that hf is a Γ-restricted rotation such that g is continuous on Supp(hf ). Then it is enough to see that we have the following equalities:

[f, g] = fgf -1 g -1 = h -1 (hf gf -1 h -1 g -1 )(ghg -1 ) = h -1 [hf, g]ghg -1
Hence by the previous case [hf, g] is a product of Γ-transpositions. Also we can always write a Γ-transposition into a product of Γ-transpositions with smaller supports. Hence the conjugate ghg -1 is a product of Γ-transpositions.

These lemmas with

k ∈ {2, . . . , n} such that f k is a Γ-restricted rotation of type (b, a). Let g 1 = f 2 . . . f k-1 or g 1 = Id if k = 2. Let g 2 = f k+1 . . . f n or g 2 = Id if k = n. Then we can write f 1 f 2 . . . f n = f 1 g 1 f k g 2 = (f 1 f k )(f -1 k g 1 f k g -1 1 )(g 1 g 2 )
Hence, the induction assumption and Lemmas 2.2.6 and 2.2.7 give the result.

Corollary 2.2.9. The kernel Ker(ϕ Γ ) is generated by the set of all Γ-transpositions.

Description of the abelianization of IET(Γ)

In this section we construct a group homomorphism ε Γ : IET(Γ) → 2 Z Γ whose kernel is the derived subgroup D(IET(Γ)).

Boolean measures

In finite permutation groups there is a natural signature. One way to describe the signature is as follows: the signature of a finite permutation f , viewed in Z/2Z is the number modulo 2 of pairs (x, y) such that x < y and f (x) > f(y). In our context where f ∈ IET(Γ), while this set is infinite, the idea is to measure it in a meaningful sense. We denote by Itv(Γ) the set of subintervals of R whose endpoints are in Γ.

Definition 2.3.1. Let A be a Boolean algebra and G be an abelian group. Let μ : A → G be a finitely additive map: It might be useful to notice that for k in N ≥1 and every I 1 , I 2 , . . . , I k intervals in Itv(Γ), the intersection i I i is still an element of Itv(Γ). Moreover for every I in Itv(Γ), the complement of I is the disjoint union of two elements of Itv(Γ). Thus any Boolean combination of elements of Itv(Γ) is a finite disjoint union of such elements.

∀ U, V ∈ A disjoint, μ(U V ) = μ(U ) + μ(V ).
a i × b i . So ω 1 (c) = k i=1 ω 1 (a i × b i ) = k i=1 ω 2 (a i × b i ) = ω 2 (c).
Thus ω 1 = ω 2 and the unicity is proved.

It is sufficient to prove the existence for every finite Boolean subalgebra of C. Indeed if we assume that for every D finite Boolean subalgebra of C there exists a Boolean algebra measure Then let D be the finite Boolean algebra generated by all the a i × b j with 1 ≤ i ≤ k and 1 ≤ j ≤ . Let U be the finite Boolean subalgebra of A generated by all a i and let V be the finite Boolean subalgebra of B generated by all b j . Then U and V are atomic. Let u 1 , . . . u n be the atoms of U and v 1 , . . . v m be the atoms of V . Hence D is atomic with atoms given by u i × v j for every 1 ≤ i ≤ k and 1 ≤ j ≤ . Then for each element d in D there exists a unique We need to check some IET(Γ)-invariance for the measure ω Γ . For this we define an action of IET(Γ) on [0, 1[ × [0, 1[ by f.(x, y) = (f (x), f(y)). Hence for every P in A Γ we have f.P in A Γ , this gives us a new Boolean algebra measure f * ω Γ .

m D for D in G ⊗ H such that m D (a × b) = μ(a) ⊗ ν(b) for every a in A and b in B with a × b in D. Let c be an element of C. Then {0 C , c, ¬c, 1 C } is a finite Boolean subalgebra of C non-trivial. Moreover if c is in D 1 ∩ D 2
J d ⊂ {1, 2, . . . , n} × {1, 2, . . . , m} such that d = (i,j)∈J d u i ⊗ v j . Hence the map m D defined by m D (d) = m D ( (i,j)∈J d u i × v j ) = (i,j)∈J d μ(u i ) ⊗ ν(v j ) is well-defined,
Proposition 2.3.5. For every f in IET(Γ) and every P in A Γ ⊗ A Γ we have f.P :

= {(f (x), f(y)) | (x, y) ∈ P } in A Γ ⊗ A Γ . Furthermore we have f * ω Γ = ω Γ .
Proof. Let f ∈ IET(Γ) and (μ, σ) be a Γ-combinatorial description of f and let {I 1 , . . . , I n } be the partition into intervals associated. Let P be an element of A Γ ⊗ A Γ . There exist m in N and pairwise disjoint elements p 1 , p 2 , . . . , p m of Itv(Γ) × Itv(Γ)

such that P = k i=1 p i . As f is a permutation of [0, 1[ we get f. m i=1 p i = m i=1 f.p i , so it is enough to show that f.p belongs to A Γ ⊗ A Γ . For i ∈ {1, . . . , k}, let a i , b i , c i , d i ∈ Γ such that p i = [a i , b i [ × [c i , d i [. Then f.p i = (k,l) f ([a i , b i [ ∩ I k ) × f ([c i , d i [ ∩ I l ) which is a finite disjoint union of elements of Itv(Γ) × Itv(Γ) because a i , b i , c i , d i and extremities of I l are in Γ for each 1 ≤ l ≤ n. In conclusion f.p i is in A Γ ⊗ A Γ thus f.P is in A Γ ⊗ A Γ .
Also f is piecewise a translation and λ is the Lebesgue measure, so for any J in Itv(Γ) we have λ(J) = λ(f (J)). Thus:

ω Γ (f.p i ) = (k,l) ω Γ (f ([a i , b i [ ∩ I k ) × f ([c i , d i [ ∩ I l )) = (k,l) λ(f ([a i , b i [ ∩ I k )) ⊗ λ(f ([c i , d i [ ∩ I l )) = ( k λ([a i , b i [ ∩ I k )) ⊗ ( l λ([c i , d i [ ∩ I l )) = λ([a i , b i [) ⊗ λ([c i , d i [) = ω Γ (p i ) This gives us ω Γ (f.P ) = m i=1 ω Γ (f.p i ) = m i=1 ω Γ (p i ) = ω Γ (P ). Hence ω Γ = f -1 * ω Γ . As
f -1 is also in IET(Γ) we can do the same to deduce ω Γ = f * ω Γ .

Creation of a signature

Definition 2.3.6. Let f be a transformation in IET(Γ). Every pair (x, y) in [0, 1[ × [0, 1[ such that x < y and f (x) > f(y) is called an inversion of f . We denote by E f the set of all inversions of f . Proposition 2.3.7. Let f be a transformation in IET(Γ) and (μ, τ ) be a combinatorial description of f . We have

E f = n j=1 i<j τ (i)>τ (j) I i × I j .
Proof. Let (μ, τ ) be a Γ-combinatorial description of f and let {I 1 , I 2 , . . . , I k } be the partition into intervals associated. Let (x 0 , y 0 ) be an element of E f . Then there exist i, j ∈ {1, 2, . . . , k} such that x 0 ∈ I i and y 0 ∈ I j . As x 0 < y 0 we have i ≤ j. Furthermore if i = j then as f is an isometry which preserves the order on I i we get f (x 0 ) < f(y 0 ) which is a contradiction, we deduce that i < j. By definition of f we have f (I i ), f(I j ) ∈ Itv(Γ) and they are disjoint. Thus as f (x 0 ) > f(y 0 ) we deduce that for every x ∈ I i and y ∈ I j we have x < y and f (x) > f(y), so I i × I j ⊂ E f . Also, this implies σ(i) > σ(j) and we deduce that n j=1 i<j τ (i)>τ (j)

I i × I j = E f . Corollary 2.3.8. For every f ∈ IET(Γ) we have E f ∈ A Γ ⊗ A Γ .
We denote by p the projection from 2 Z Γ into 2 Z Γ.

Definition 2.3.9. The signature for IET(Γ) is the following map:

ε Γ : IET(Γ) -→ 2 Z Γ f -→ p • ω Γ (E f )
Proposition 2.3.10. For every A and B in A Γ we have:

p • ω Γ (A × B) = -p • ω Γ (B × A) Proof. Let A, B ∈ A Γ then: p • ω Γ (A × B) = λ(A) ∧ λ(B) = -λ(B) ∧ λ(A) = -p • ω Γ (B × A) Theorem 2.3.11. The map ε Γ is a group homomorphism.
Proof. Let f, g ∈ IET(Γ). We denote by s the symmetry of axis y = x. We remark that every element

I of A Γ ⊗ A Γ satisfies s(I) ∈ A Γ ⊗ A Γ . Then {(x, y) | x < y, g(x) > g(y), fg(x) < fg(y)} = E g ∩ sg -1 (E f ) is an element of A Γ ⊗ A Γ . We also notice that {(x, y) | x < y, g(x) < g(y), fg(x) > fg(y)} = E f •g ∩ E c g and {(x, y) | x < y, g(x) > g(y), fg(x) > fg(y)} = E f •g ∩ E g are two elements of A Γ ⊗ A Γ .
For more clarity we do some calculus first. By Proposition 2.3.5 and Proposition 2.3.10 we get:

-p • ω Γ ({(x, y) | x < y, g(x) > g(y), fg(x) < fg(y)}) = -p • ω Γ ({(g(x), g(y)) | x < y, g(x) > g(y), fg(x) < fg(y)}) = p • ω Γ ({(g(y), g(x)) | x < y, g(x) > g(y), fg(x) < fg(y)}) = p • ω Γ ({(u, v) | g -1 (u) > g -1 (v), u < v, f(u) > f(v)}) and p • ω Γ ({(x, y) | x < y, g(x) < g(y), fg(x) > fg(y)}) = p • ω Γ ({(g(x), g(y)) | x < y, g(x) < g(y), fg(x) > fg(y)}) = p • ω Γ ({(u, v) | g -1 (u) < g -1 (v), u < v, f(u) > f(v)})
In addition:

p • ω Γ ({(u, v) | g -1 (u) > g -1 (v), u < v, f(u) > f(v)}) + p • ω Γ ({(u, v) | g -1 (u) < g -1 (v), u < v, f(u) > f(v)}) = p • ω Γ (E f ) = ε Γ (f )
Hence by adding and remove the same quantity at the fourth equality we obtain:

ε Γ (f • g) = p • ω Γ ({(x, y) | x < y, fg(x) > fg(y)} = p • ω Γ ({(x, y) | x <

y, g(x) > g(y), fg(x) > fg(y)} {(x, y) | x < y, g(x) < g(y), fg(x) > fg(y)})

= p • ω Γ ({(x, y) | x < y, g(x) > g(y), fg(x) > fg(y)})

+ p • ω Γ ({(x, y) | x < y, g(x) < g(y), fg(x) > fg(y)}) = p • ω Γ ({(x, y) | x < y, g(x) > g(y), fg(x) > fg(y)}) + p • ω Γ ({(x, y) | x < y, g(x) > g(y), fg(x) < fg(y)}) -p • ω Γ ({(x, y) | x < y, g(x) > g(y), fg(x) < fg(y)}) + p • ω Γ ({(x, y) | x < y, g(x) < g(y), fg(x) > fg(y)}) = p • ω Γ (E g ) + p • ω Γ (E f ) = ε Γ (g) + ε Γ (f )
In conclusion, ε Γ is additive thus it is a group homomorphism.

Proposition 2.3.12. Let a ∈ Γ+ with a ≤ 1 2 and f be a Γ-transposition of type a. Then

ε Γ (f ) = a ∧ a. Proof. Let u, v, b ∈ Γ such that ((u, a, b, a, v), ( 2
4)) is a Γ-combinatorial description of f (see Figure 2.3.1). Let {I 1 , . . . , I 5 } be the partition into intervals associated. We already proved in 2.3.7 that it is sufficient to check if a pair (x, y)

∈ I i × I j is in E f to know that I i × I j is in E f . We also have I i × I j / ∈ E f if j ≤ i.
Thus one can look at the graph of f to find that E f is equal to the tiling space on Figure 2 In the general case, let w ∈ Γ with 0 ≤ w ≤ 1 2 . Then there exist k, ∈ Z and a , b ∈ Γ with 0 ≤ a , b < w such that a = kw + a and b = w + b . Then a ∧ b = k w∧w+kw∧b + a ∧w+a ∧b . By the previous case and as ε Γ is a group homomorphism, we deduce that a ∧ b is in Im(ε Γ ).

.3.1. We deduce that ε Γ (f ) = a ∧ b + a ∧ a + b ∧ a = a ∧ a.

Description of Ker(ε Γ )

The aim of this part is to conclude that Ker(ε Γ ) = D(IET(Γ)) and the induced morphism IET(Γ) ab → 2 Z Γ is an isomorphism. We recall that ϕ Γ is the restriction of the SAFhomomorphism ϕ to IET(Γ).

Lemma 2.3.15. We have 2ε Γ = -ϕ Γ .

Proof. Let f ∈ IET(Γ) and (α, τ ) be a Γ-combinatorial description of f . Let n be the length of α. Thanks to Propositions 1.3.9 and 2.3.13 we have:

ϕ Γ (f ) = n j=1 i τ (i)<τ (j) α i - i<j α i ∧ α j = n j=1 i>j τ (i)<τ (j) α i ∧ α j + n j=1 i<j τ (i)<τ (j) α i - i<j α i ∧ α j = n i=1 j<i τ (j)>τ (i) α i ∧ α j - n j=1 i<j τ (i)>τ (j) α i ∧ α j = -2 n i=1 j<i τ (j)>τ (i) α j ∧ α i = -2ε Γ (f ) Corollary 2.3.16. We have the inclusion Ker(ε Γ ) ⊂ Ker(ϕ Γ ).
By Corollary 2.2.9 we know that Ker(ϕ Γ ) is generated by the set of all Γ-transpositions. Thus it is natural to look at these elements who are also in Ker(ε Γ ). If σ is a Γ-transposition of type a in Ker(ε Γ ) then we have the equality a ∧ a = 0. We want to prove that σ is in D(IET(Γ)) if and only if a ∈ 2 Γ.

We denote by S 2 Γ the second symmetric power of Γ and we denote by a a image of a ⊗ a in S 2 Γ. For every group G and every w ∈ G we use the notation w [mod 2] for the image of w in G/2G. Lemma 2.3.17. For every group G, the group

2 Z G/2( 2 Z G) is naturally isomorphic to S 2 G/2(S 2 G).
Proof. We have a surjective group homomorphism from G ⊗ G onto S 2 G/2(S 2 G). We notice that for every x, y ∈ G, the element x ⊗ y + y ⊗ x is sent to 2(x y) [mod 2] = 0.

Hence we deduce a surjective group homomorphism from

2 Z G onto S 2 G/2(S 2 G). It is immediate that 2( 2 Z G
) is sent to zero. Thus we have a surjective group homomorphism from

2 Z G/2( 2 Z G) onto S 2 G/2(S 2 G).
In order to show that it is injective we take

w ∈ G ⊗ G such that [w] S 2 G [mod 2] = 0 = 0 and we show that [w] 2 Z G [mod 2] = 0. There exist n ∈ N and a 1 , . . . a n , b 1 , . . . , b n ∈ G such that w + n i=1 a i ⊗b i -b i ⊗a i = 0. Hence [w] 2 Z G + 2 a i ∧ b i = 0. Finally [w] 2 Z G [mod 2] = 0 thus we have an isomorphism between 2 Z G/2( 2 Z G) and S 2 G/2S 2 G. Proposition 2.3.18. Let a ∈ Γ, if a ∧ a [mod 2] = 0 then a belongs to 2 Γ.
Proof. For every group G, the group

2 Z G/2( 2 Z G
) is naturally isomorphic to the second symmetric power S 2 G/2(S 2 G). This comes from the fact that these groups satisfy the following universal property: for every group G and every abelian elementary 2-group We denote i the natural inclusion of G × G into S 2 G/2(S 2 G). For every bilinear symmetric group homomorphism b : G × G → A there exists a unique group homomorphism f :

S 2 G/2(S 2 G) → A such that for every g, h ∈ G we have b(g, h) = f (i(g, h)).
Let a ∈ Γ with a = 0 (because we already have 0 = 2×0). We denote by a a the image of a ⊗ a in S 2 G/2(S 2 G), and we assume that a a [mod 2] = 0. The projection Γ → Γ/2 Γ gives rise to a morphism ζ :

S 2 Γ → S 2 ( Γ/2 Γ). As 2(S 2 Γ) ⊂ Ker(ζ) we obtain a morphism ζ : S 2 Γ/2S 2 Γ → S 2 ( Γ/2 Γ). Hence if a / ∈ 2 Γ then ζ(a a) = 0 thus ζ (a a [mod 2
]) = 0 which is a contradiction with the assumption. In conclusion a ∈ 2 Γ.

Corollary 2.3.19. Every Γ-transposition f in Ker(ε Γ ) is in D(IET(Γ)).
Proof. Let a ∈ Γ+ , with a ≤ 1 2 , be the type of f and let u, v ∈ Γ+ such that In order to show that Ker(ε Γ ) = D(IET(Γ)) we prove that any element f ∈ Ker(ε Γ ) can be written as f = σh where h ∈ D(IET(Γ)) and σ is a Γ-transposition. This concludes because we just show that a Γ-transposition which is also in Ker(ε Γ ) is in D(IET(Γ)). We begin by a particular case of a product of Γ-transpositions with pairwise disjoint support. The aim will be to reduce the general case to this one. We recall that the identity is considered as a Γ-transposition.

I 1 = [u, u + a[ and I 2 = [v, v + a[ are the two intervals swapped by f . From f ∈ Ker(ε) we deduce that ε Γ (f ) = a∧a = 0. Hence a∧a [mod 2] = 0 in 2 Z Γ/2( 2 Z Γ).

. . τ k = σh where σ is a Γ-transposition and h is an element of D(IET(Γ)).

Moreover the support of h and σ do not overlap and are included in the union of the supports of the τ i .

Proof. By induction it is enough to show the result in the case k = 2 Let respectively a 1 and a 2 be the type of τ 1 and τ 2 . As their support do not overlap we know that τ 1 and τ 2 commute. Hence we can assume a 1 ≥ a 2 without loss of generality. If a 1 = a 2 then there exists f ∈ IET(Γ) such that We can now treat the case of finite order elements: Lemma 2.3.21. If f ∈ IET(Γ) has finite order then there exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that the support of σ and h are inside the support of f and f = σh.

τ 1 = fτ 2 f -1 . Thus τ 1 τ 2 is in D(IET(Γ)). If a 1 > a 2 let u, v ∈ Γ such that [u, u + a 1 [ and [v, v + a 1 [
Proof. Let f ∈ IET(Γ) as in the statement. By Proposition 1.2.24 there exist n ∈ N and P = {I 1 , I 2 , . . . , I n } ∈ Π f and σ ∈ S n such that f (I i ) = I σ(i) . Let k ∈ N and σ = c 1 c 2 . . . c k be the disjoint cycle decomposition for σ. Let f i be the element of IET(Γ) that is equal to f on I j for every j ∈ Supp(c i ) while fixing the rest of [0, 1[. Then f = f 1 . . . f k and f i commutes with f j for every 1 ≤ i = j ≤ k. Then if the statement is true for every f i we can write f i = τ i h i with τ i a Γ-transposition and h ∈ D(IET(Γ)), both of them with support inside the support of

f i . Then f = f 1 . . . f k = τ 1 h 1 . . . τ k h k = τ 1 . . . τ k h 1 .
. . h k because the support of τ i does not overlap with the support of τ j of h j for every 1 ≤ j ≤ k and j = i. We conclude with Lemma 2.3.20 applied to τ 1 . . . τ k . Let c by a cycle of length n ≥ 2 and let I 1 , I 2 , . . . , I n be non-overlapping intervals of Itv(Γ) of same length. Let f ∈ IET(Γ) be the element that permutes the set {I 1 , I 2 , . . . ,

I n } by c. Then if c ∈ D(S n ) = A n we deduce that f ∈ D(IET(Γ)). If c / ∈ D(S n
) then let g be the unique Γ-transposition that swaps I 1 with I c(1) (we notice that the support of g is included in the support of f ). By the previous case, gf ∈ D(IET(Γ)) and we conclude that f = g(gf ) is a wanted decomposition. Proof. The proof is by induction on k. The initialisation k = 1 is immediate. The case k = 2 is Corollary 2.3.22. Now if we assume the result for k ≥ 2 let τ 1 , τ 2 , . . . , τ k , τ k+1 be Γtranspositions. Then by assumption, applied to τ 2 τ 3 . . . τ k+1 , there exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that τ 2 τ 3 . . . τ k+1 = σh. Hence τ 1 τ 2 . . . τ k+1 = τ 1 σh. By using the case k = 2 we deduce that there exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that τ 1 σ = σ h . Thus τ 1 τ 2 . . . τ k+1 = σ g, with g = h h ∈ D(IET(Γ)), which is a wanted decomposition.

Finally we can prove the main theorem of the section: Theorem 2.3.24. We have the equality Ker(ε Γ ) = D(IET(Γ)), and the induced morphism

IET(Γ) ab → 2 Z Γ is an isomorphism.
Proof. The inclusion from right to left is immediate. For the other inclusion let f ∈ Ker(ε Γ ). By Corollary 2.3.16 we know that f ∈ Ker(ϕ Γ ), then by Corollary 2.2.9 there exists k ∈ N such that f is equal to the product τ 1 τ 2 . . . τ k where τ i is a Γ-transposition. By Lemma 2.3.23 there exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that f = σh.

Then ε Γ (f ) = ε Γ (σ)ε Γ (h) = ε Γ (σ) = 0. By Corollary 2.

we deduce σ ∈ D(IET(Γ)). Hence f = σh ∈ D(IET(Γ)).

We deduce that the induced group homomorphism ε Γ : IET(Γ) ab → 2 Z Γ is injective. Furthermore ε Γ is surjective by Proposition 2.3.14 thus ε Γ is surjective and we conclude that ε Γ is an isomorphism.

Z Γ/(2 2 Z Γ) × 2 Z Γ/(2 2 Z Γ).
Pour obtenir ce résultat, on construit deux morphismes de groupes. L'un des deux est un analaogue du morphisme signature ε Γ construit dans la section 2.3. L'ensemble des inversions ne peut plus tous le temps être mesuré avec la mesure construite dans la section 2.3.1, toutefois cet ensemble avec son symétrique peut être mesuré. Le noyau de ce morphisme est légèrement plus gros que le sous-groupe dérivé D(IET (Γ)) ; en fait, il manque quelques renversements d'un certain type. On remarque que ces renversements qui manquent sont conjugués à une rotation restreinte. On veut alors utiliser le morphisme de groupes ε Γ directement. Pour cela, on a besoin de voir un élémént de IET (Γ) comme une "limite" d'éléments de IET(Γ) en un certain sens.

Ce chapitre correspond à la seconde moitié de l'article [START_REF] Lacourte | Abelianization of some groups of interval exchanges[END_REF].

English abstract. For every subgroup IET (Γ) we establish a natural isomorphism between its abelianization and {a ⊗ a

[mod 2] | a ∈ Γ} × { ∧ [mod 2] | ∈ Γ} which is a 2-elementary abelian subgroup of 2 Z Γ/(2 2 Z Γ) × 2 Z Γ/(2 2 Z Γ).
In order to obtain this we construct two group homomorphisms. One is an analogue of the signature homomorphism ε Γ constructed in Section 2.3. The set of inversions cannot be measured anymore with the measure created in Section 2.3.1, however this set with its symmetric still can be measured. Its kernel will be slightly larger than D(IET (Γ)); in fact, it will miss some reflections of a certain type. We notice that these missing reflections are conjugate to a restricted rotation, thus we will try to use directly the group homomorphism ε Γ . To do this we need to see that an element of IET (Γ) is a "limit" of elements in IET(Γ) in some sense.

This chapter corresponds with the second half of the article [START_REF] Lacourte | Abelianization of some groups of interval exchanges[END_REF].

Contents

Positive substitute

We refer to Section 1.2.1 for the different kinds of partition associated. Here we introduce the notion of positive substitute. The idea is to get a unique decomposition for every element of IET . This can be obtain by having a parameter on which the decomposition will depend and this parameter is the partition associated chosen. Also it will be important to understand in which cases this decomposition satisfies the group homomorphism relation. For this we introduce the notion of a partition associated with a tuple.

Definition

We recall here that for every subinterval I of R we denote by r I the I-reflection map and by r I the I-reflection. Proof. Let f and P as in the statement.We know that f is continuous on every interior I • with I ∈ P. We define:

A(f, P) = {J | ∃I ∈ P such that J • = f (I • ) and f | I • is order-reversing } It is the unique subset of f (P) such that ( I∈A( f,P) r I ) • f

Compatibility with the composition

We need to define what is a partition associated with a tuple. We also want to talk about order-preserving and order-reversing for elements in IET .

Definition 3.1.6. Let f ∈ IET and P ∈ Π f . Let I ∈ P, we say that f is orderpreserving on I (resp. order-reversing on I) if there exists a representative of f in IET that is order-preserving on I (resp. order-reversing on I).

Thanks to this when we have a partition P associated with an element f ∈ IET we can always say that f is either order-preserving or order-reversing on every interval of P.

With some conditions we deduce how positive substitute behaves with composition.

E f ∈ A Γ ⊗ A Γ .
Proof. Let f ∈ IET and let P be a partition associated with f . Let f + P be the positive P-substitute of f . Let J ⊂ P be the subset of intervals where f is order-reversing. By seeing A Γ as its image in A * R then we deduce that

E f = E f + P ∪ I∈J I × I is an element of A Γ ⊗ A Γ .
Then this set can be measured with the same measure ω Γ used in Section 2.3. Here we expect a 2-group for the image of our group homomorphism. We denote by π the projection of 2 Z Γ onto 2 Z Γ/2( 2 Z Γ) and for every a ∈ Γ we denote π(a ⊗ a) by a ⊗ a [mod 2].

Definition 3.2.8. We define the signature for IET (Γ) as the map:

ε Γ : IET (Γ) -→ 2 Z Γ/2( 2 Z Γ) f -→ w Γ (E f ) [mod 2]
For more clarity we explicit some equalities used to show that ε Γ is a group homomorphism. Lemma 3.2.9. Let f, g ∈ IET (Γ). We have the following equalities:

1. E f •g ∪ (E g ∩ g -1 (E f )) = E g ∪ g -1 (E f ), 2. E f •g ∩ E g ∩ g -1 (E f ) = ∅. Remark 3.2.10. We notice that E g ∩ g -1 (E f ) is an element of A Γ ⊗ A Γ .
Proof.

1. We proceed by double inclusions. From left to right, we know that

E g ∩ g -1 (E f ) ⊂ E g ∪ g -1 (E f ) hence it is sufficient to show the inclusion E f •g ⊂ E g ∪ g -1 (E f ). Let (x, y) be in E f •g
. We can assume that x < y, the case x > y is similar. Then we deduce that f (g(x)) > f(g(y)). We have two cases, if g(x) > g(y) then (x, y) ∈ E g else we have g(x) < g(y) and f (g(x)) > f(g(y)) thus (x, y) = g -1 (g(x), g(y)) ∈ g -1 E f . From right to left, let (x, y) ∈ E g ∪g -1 (E f ). We can assume that x < y the case x > y is similar. If (x, y) ∈ E g ∩ g -1 (E f ) then it is done. We have two cases, if (x, y) ∈ E g and (x, y) / ∈ g -1 (E f ) then as x < y we have g(x) > g(y) and f (g(x)) > f(g(y)) thus (x, y) ∈ E f •g . If (x, y) ∈ g -1 (E f ) and (x, y) / ∈ E g then as x < y we have g(x) < g(y) and f (g(x)) > f(g(y)) thus (x, y) ∈ E f •g .

2.

By contradiction let us assume that there exists (x, y) ∈ E f •g ∩ E g ∩ g -1 (E f ). We can assume that x < y, the case x > y is similar. As x < y and (x, y) ∈ E f •g ∩ E g we know that g(x) > g(y) and f (g(x)) > f(g(y)). However g.(x, y) = (g(x), g(y)) ∈ E f thus as g(x) > g(y) we have f (g(x)) < f(g(y)) which is a contradiction.

Theorem 3.2.11. The signature ε Γ is a group homomorphism.

Proof. Let f, g ∈ IET (Γ). Thanks to Lemma 3.2.9 we have:

ε Γ (f • g) =ω Γ (E f •g ) [mod 2] =ω Γ (E f •g ) + ω Γ (E g ∩ g -1 (E f )) -ω Γ (E g ∩ g -1 (E f )) [mod 2] =ω Γ (E f •g (E g ∩ g -1 (E f ))) -ω Γ (E g ∩ g -1 (E f )) [mod 2] =ω Γ (E g ∪ g -1 (E f )) -ω Γ (E g ∩ g -1 (E f )) [mod 2] =ω Γ (E g ) + ω Γ (g -1 (E f )) -2ω Γ (E g ∩ g -1 (E f )) [mod 2] =ω Γ (E g ) + ω Γ (g -1 (E f )) [mod 2] = ε Γ (g) + ε Γ (f )
We notice that every finitely supported permutation of support in Γ is in Ker( ε Γ ).

Corollary 3.2.12. There exists a group homomorphism

ε Γ : IET (Γ) → 2 Z Γ/2( 2 Z Γ) such that for every f ∈ IET (Γ) we have ε Γ (f ) = ε Γ ( f ) for every representative f of f in IET (Γ).
Example 3.2.13. We give the value of ε Γ for two kinds of elements:

1. Let r be a Γ-reflection of type a then ε Γ (r) = a ⊗ a [mod 2], 2. Let s be a Γ-restricted rotation of type (p, q) then:

ε Γ (s) = p ⊗ q + q ⊗ p [mod 2]
As IET is generated by reflections, we deduce the image of ε Γ : Corollary 3.2.14. We have the following isomorphism:

IET (Γ)/ Ker(ε Γ ) Im(ε Γ ) = {a ⊗ a [mod 2] | a ∈ Γ} , where {a ⊗ a [mod 2] | a ∈ Γ} is a subgroup of 2 Z Γ/2( 2 Z Γ).

Description of Ker(ε Γ )

As

2 Z Γ/2( 2 Z Γ
) is an abelian group we know that D(IET (Γ)) is included in Ker(ε Γ ). We will see later that the other inclusion is false in general.

With Figure 1.3.1 we already know that for every a ∈ Γ such that a 4 ∈ Γ then every Γ-reflection of type a is in the derived subgroup D(IET (Γ). Definition 3.2.15. We denote by Ω Γ the conjugate closure of the group generated by the set of all Γ-reflections of type 2 with ∈ Γ 2 Γ (the closure inside IET (Γ)). The aim here is to show the equality Ker(ε

The inclusion

Ω Γ ⊂ Ker(ε Γ ) is immediate. With
Γ ) = D(IET (Γ))Ω Γ .
We begin by proving the result in the specific case where Γ has finite rank. We will reduce the general case to this one. Lemma 3.2.17. Let Γ be a finitely generated subgroup of R/Z. Then for every f ∈ Ker(ε Γ ) there exist δ ∈ D(IET (Γ)) and h ∈ Ω Γ such that f = δh.

Proof. As Γ is finitely generated, we know that Γ is finitely generated. Let d be the rank of Γ.

Let f ∈ Ker(ε Γ ). Let n ∈ N and P := {I 1 , I 2 , . . . , I n } be a partition into Γ-intervals associated with f . We denote by L i the length of I i for every 1 ≤ i ≤ n. By Corollary 2.1.3 there exists B := { 1 , 2 , . . . , d } a basis of Γ with elements in Γ+ such that L i ∈ Vect N (B) for every 1 ≤ i ≤ n. Hence we can cut each I i into smaller intervals with length in B. This operation gives us a new partition Q := {J 1 , J 2 , . . . , J k }, with k ∈ N, into Γ-intervals associated with f .

For every 1 ≤ i ≤ k we define r i as the J i -reflection if f is order-reversing on J i else we put r i = Id. Let g be the product fr 1 r 2 . . . r k ; it is an element of IET(Γ) and Q is a partition into Γ-intervals associated with g. By Theorem 1.2.8 the element g can be written as a finite product of Γ-restricted rotations with type inside {(l p , l q ) | p, q ∈ {1, 2, . . . , d}}. Thanks to an element of D(IET (Γ)) we can organize this product to put all Γ-restricted rotations of type (l p , l p ) together (with 1 ≤ p ≤ d; they are elements in Ω Γ ): there exist w 1 ∈ D(IET (Γ)) and h ∈ Ω Γ and m ∈ N and s 1 , s 2 , . . . , s m some Γ-restricted rotations with type inside {(l p , l q ) | p, q ∈ {1, 2, . . . , d}, p = q} such that g = w 1 hs 1 s 2 . . . s m . Then

f = w 1 hs 1 s 2 . . . s m r k r k-1 . . . r 1 .
We define u p := Card{i ∈ {1, 2, . . . , k} | r i = Id, type(r i ) = l p }. Let v p,p = 0 for every 1 ≤ p ≤ d and let v p,q := Card{j ∈ {1, 2, . . . , m} | type(s j ) = (l p , l q )} for every 1 ≤ p = q ≤ d. Then we have:

ε Γ (f ) = k i=1 ε Γ (r i ) + m j=1 ε Γ (s j ) = d p=1 u p l p ⊗ l p + d p=1 d q=1 v p,q (l p ⊗ l q + l q ⊗ l p ) [mod 2] = 0 We notice that d p=1 d q=1 v p,q (l p ⊗ l q + l q ⊗ l p ) = d p=1 d q=1 (v p,q + v q,p )l p ⊗ l q . Furthermore B is a basis of Γ so {l p ⊗ l q } 1≤p,q≤d is a basis of 2
Z Γ, thus we deduce that 2 divides u p for every 1 ≤ p ≤ d and 2 divides v p,q + v q,p for every 1 ≤ p, q ≤ d.

We obtain that r 1 r 2 . . . r k is a balanced product of Γ-reflections hence by Lemma 3.2.2 it is an element of D(IET (Γ)) denoted w 2 . We also deduce that the product s 1 s 2 . . . s m is a balanced product of Γ-restricted rotations with type inside {(a, b) | a = b ∈ Γ+ }. Hence by Proposition 3.2.4 we obtain that it is also an element of D(IET (Γ)), denoted w 3 .

Finally we have f = w 1 hw 3 w 2 = δh with δ ∈ D(IET (Γ)) and h ∈ Ω Γ .

The next lemma gives an inclusion used to conclude in the general case: Lemma 3.2.18. For all Γ, A subgroups of R/Z such that A ⊂ Γ we have:

D(IET (A))Ω A ⊂ D(IET (Γ))Ω Γ Proof. The inclusion D(IET (A)) ⊂ D(IET (Γ)) is immediate. It is sufficient to show that Ω A ⊂ D(IET (Γ))Ω Γ . Let à be the preimage of A in R.
Let f be an element of Ω A . Then there exist n ∈ N and a 1 , a 2 , . . . , a n ∈ Ã 2 Ã and w 1 , w 2 , . . . , w n some Γ-reflections such that the type of w i is 2a i and there exist

g 1 , g 2 , . . . g n ∈ IET (A) such that f = n i=1 g i w i g -1 i . Let U := {i ∈ {1, 2, . . . , n} | a i ∈ Γ 2 Γ}. By definition we have {g i w i g -1 i | i ∈ U } ⊂ Ω Γ . Take V := {1, 2, . . . , n} U . As A is a subgroup of Γ we deduce that V = {j ∈ {1, 2, . . . , n} | a j ∈ 2 Γ}.
Thus for every j ∈ V the type of w j is in 4 Γ so we deduce that w j and g j w j g -1 j belong to D(IET (Γ)) (see Figure 1.3.1). We know that there exists h ∈ D(IET (Γ)) such that:

f = h j∈V g j w j g -1 j i∈U g i w i g -1 i Then f ∈ D(IET (Γ))Ω Γ .
We can prove the theorem for the general case:

Theorem 3.2.19. For any dense subgroup Γ of R/Z we have:

Ker(ε Γ ) = D(IET (Γ))Ω Γ
Proof. The inclusion from right to left is already proved. Let f ∈ Ker(ε Γ ), let n ∈ N and P := {I 1 , I 2 , . . . , I n } be a partition into Γ-intervals associated with f . We denote by L i the length of I i for every

1 ≤ i ≤ n. As ε Γ (f ) = 0 we know that there exist k ∈ N and a 1 , a 2 . . . , a k , b 1 , b 2 , . . . , b k ∈ Γ such that ω Γ (E f ) = 2 k i=1 a i ⊗ b i inside 2 Z Γ.
Let à be the subgroup of R generated by

{L i } i=1...n ∪ {1} ∪ {a i , b i } i=1...k . Then à contains Z, is finitely generated and is a subgroup of Γ. Let A be the image of à in R/Z.
The partition P is also a partition into A-intervals associated with f thus f belongs to IET (A). Furthermore we have ω A (E f ) = ω Γ (E f ). Hence:

ε A (f ) = [ω A (E f )] 2 Z A/2( 2 Z A) = [ω Γ (E f )] 2 Z A/2( 2 Z A) = [2 k i=1 a i ⊗ b i ] 2 Z A/2( 2 Z A) = 0
By Lemma 3.2.17 we deduce that f ∈ D(IET (A))Ω A and by Lemma 3.2.18 we deduce that f ∈ D(IET (Γ))Ω Γ .

The positive contribution

Thanks to Proposition 3.2.16, we know that every Γ-reflection of type 2 with ∈ Γ 2 Γ is conjugated to a Γ-restricted rotation of type ( , ). Also this is an element of IET + (Γ) which is not send on the trivial element by the morphism ε Γ . We use the notion of positive substitute in order to use the group homomorphism ε Γ to send such a reflection on a nontrivial element. For this we need to use the upward directed set M Γ and the Z-linearly independent finite subset S a of Γ+ for every a ∈ M Γ given by Corollary 2.1.2. We recall that for every a, b ∈ M Γ with a ≤ b we have Vect N (S a ) ⊂ Vect N (S b ) and Γ+ is equal to the direct limit lim -→ Vect N (S a ). For every a ∈ M Γ , we construct a map depending on S a . They are not group homomorphisms but satisfy the group homomorphism property on some products thanks to Proposition 3.1.7.

Some subsets of IET (Γ)

Let S be a finite set of Γ+ . We denote by G S the set of all f in IET (Γ) such that there exists a partition P into S-intervals associated with f . We remark that G S is not a group in general. We want to know how these sets and IET (Γ) are linked. Proof. Let f ∈ G S and let P be a partition into S-intervals associated with f . As S ⊂ Vect N (T ) each interval I ∈ P can be subdivided with intervals of length in T . After subdividing this way the intervals of P, we obtain a refinement

Q of P. We notice that Q is a partition into T -intervals associated with f so f ∈ G T .
As Γ+ is the direct limit of the Vect N (S a ) we deduce the next proposition: Proposition 3.3.2. For every finite subset S of Γ+ , there exists a ∈ M Γ such that for every b ≥ a we have S ⊂ Vect N (S b ).

From Propositions 3.3.1 and 3.3.2 we obtain:

Corollary 3.3.3. Let f ∈ IET (Γ). There exists a ∈ M Γ such that for every b ≥ a we have f ∈ G S b .
We also want to check that if we take any product of elements in IET (Γ) there will be a moment where we have a partition associated to the tuple of these elements.

Proposition 3.3.4. Let k ∈ N and f 1 , f 2 , . . . , f k ∈ IET (Γ). There exists a ∈ M Γ such that for every b ≥ a there exists P b a partition into S b -intervals associated with (f 1 , f 2 , . . . , f k ).
Proof. Thanks to Proposition 3.3.1 and Remark 3.1.5 we deduce that it is sufficient to find only one a ∈ M Γ such that there exists a partition into S a -intervals associated with

(f 1 , f 2 , . . . , f k ).
By Corollary 3.3.3 there exist a i ∈ M Γ and P i a partition into S a i intervals associated with f i for every 1 ≤ i ≤ k. We denote by V i the set of all the endpoints of the intervals in

P i . Let V = V 1 ∪ f -1 1 (V 2 ) ∪ . . . ∪ (f k-1 . . . f 1 ) -1 (V k ).
We know that V is finite thus there exist m ∈ N and v 0 , v 1 , . . . , v m ∈ Γ+ such that V = {v i } i=0...m . Up to change the order we can assume that

v 0 = 0 < v 1 < v 2 < . . . < v m-1 < v m = 1. Let I j be the interval [v j-1 , v j [
and j be the length of I j for every 1 ≤ j ≤ m. Let T = { j } j=1...m and P = {I j } j=1...m .

We have V 1 ⊂ V then P is a refinement of P 1 so P is a partition into T -intervals associated with f 1 . Similarly for every

2 ≤ i ≤ k we know that V i ⊂ f i-1 . . . f 1 (V ) thus f i-1 . . . f 1 (P) is a refinement of P i so f i-1 . . . f 1 (P) is a partition into T -intervals associated with f i . Hence P is a partition into T -intervals associated with (f 1 , f 2 , . . . , f k ).
Thanks to Proposition 3.3.2 there exists a ∈ M Γ such that for every b ≥ a we have T ⊂ Vect N (S b ) and by Proposition 3.3.1 and Remark 3.1.5 there exists a refinement Q of P which is a partition into S a -intervals associated with (f 1 , f 2 , . . . , f k ).

The S-map

Let S be a finite subset of Γ+ for all this subsection. We assume that S is free inside Vect Z (S). For every a, b ∈ Γ, we use the notation

[a ∧ b] 2 2 Z Γ = a ∧ b [mod 2].
Let f ∈ G S and P be a S-partition associated with f . We show that the value ε Γ (f + P ) [mod 2] does not depend on P, where ε Γ is the group homomorphism define in 2.3.9. Proposition 3.3.5. Let f ∈ IET (Γ) and let P, Q be two partitions into S-intervals associated with f . Then ε

Γ (f + P ) [mod 2] = ε Γ (f + Q ) [mod 2].
Proof. For this proof we denote by r I the I-reflection for every subinterval I of [0, 1[ and we recall that λ is the Lebesgue measure. First we reduce the case to a simpler one. Let f ∈ IET (Γ) and P, Q be two partitions into S-intervals associated with f . Let n, k ∈ N such that P = {I 1 , I 2 , . . . , I n } and Q = {J 1 , J 2 , . . . J k }. Up to change the index we can assume that the I i are consecutive intervals and so are the J i . We denote by M the unique partition into Γ-intervals associated with f that has the minimal number of intervals. Let m be this number and M := {M 1 , M 2 , . . . , M m } where the M i are consecutive intervals. As P, Q are also partitions into Γ-intervals we know that they are refinements of M .

Let 1 ≤ i ≤ m and let n 0 = k 0 = 0. There exist

n 1 < n 2 < . . . < n m , k 1 < k 2 < . . . < k m ∈ N such that M i = n i j=n i-1 +1 I j = k i j=k i-1 +1 J j . Hence λ(M i ) = n i j=n i-1 +1 λ(I i ) = k i j=k i-1 +1
λ(J j ). For every s ∈ S we define a s = Card({j

∈ {n i-1 + 1, . . . , n i } | λ(I j ) = s} and b s = Card({j ∈ {k i-1 + 1, . . . , k i } | λ(J j ) = s}. Then s∈S a s s = s∈S b s s. As S is free in Vect Z (S)
we deduce that a s = b s for every s ∈ S. This implies that n i = k i for every 1 ≤ i ≤ n and n = k. We also get the existence of a permutation σ i of the set

{n i-1 + 1, . . . , n i } such that λ(I j ) = λ(J σ i (j) ) for every j ∈ {n i-1 + 1, . . . , n i }.
We deduce that it is sufficient to show the result when only one permutation σ i is a transposition (a a + 1) with n i-1 + 1 ≤ a < n i . Let i 0 in {1, 2, . . . m} and a ∈ {n i 0 -1 + 1, . . . , n i 0 }. We assume now that σ i 0 is the transposition (a a + 1) and σ i = Id for every i = i 0 . Then I j = J j for every j ∈ {1, 2, . . . , n} {a, a + 1}. We also have I a ∪ I a+1 = J a ∪ J a+1 and λ(I a+1 ) = λ(J a ). In the case where f is order-preserving on M i 0 then f + P = f + Q . In the case where f is order-reversing on M i 0 we deduce that (f + Q ) -1 • f + P is equal to the square of the restricted rotation whose intervals associated are I a and I a+1 (see Figure 3.3.1).

In both cases we deduce ε

Γ (f + P ) [mod 2] = ε Γ (f + Q ) [mod 2]. I a I a+1 J a J a+1 f + P (f + Q ) 1 Figure 3.3.1 -Illustration of the fact that (f + Q ) -1 • f +
P is equal to the square of a restricted rotation in Proposition 3.3.5 Definition 3.3.6. We define the S-map as:

ψ S : IET (Γ) -→ 2 Z Γ/2 2 Z Γ f -→ ε Γ (f + P ) [mod 2] f ∈ G S 0 f / ∈ G S
Where P is a partition into S-intervals associated with f .

Thanks to Proposition 3.3.5, the map ψ S is well-defined. We check that ψ S satisfies the morphism condition when we have the existence of a partition in S-intervals associated with a tuple.

Proposition 3.3.7. Let n ∈ N and f 1 , f 2 , . . . , f n ∈ IET (Γ).
If there exists P a partition into S-intervals associated with (f 1 , f 2 , . . . , f n ) then:

ψ S (f n f n-1 . . . f 1 ) = n i=1 ψ S (f i )
Proof. Let n ∈ N and f 1 , f 2 , . . . , f n ∈ IET (Γ). The case n = 1 is trivial thus we assume n ≥ 2. Let P be a partition into S-intervals associated with (f 1 , f 2 , . . . , f n ). Let g 1 = (f 1 ) + P and for every 2 P) . By Proposition 3.1.7 we know that (f n f n-1 . . . f 1 ) + P = g n g n-1 . . . g 1 , then:

≤ i ≤ n let g i = (f i ) + f i-1 ...f 1 (
ψ S (f n f n-1 . . . f 1 ) = ε Γ ((f n f n-1 . . . f 1 ) + P ) [mod 2] = ε Γ (g n g n-1 . . . g 1 ) [mod 2]
Also ε Γ is a group homomorphism thus:

ψ S (f n f n-1 . . . f 1 ) = n i=1 ε Γ (g i ) [mod 2] = n i=1 ψ S (f i )

The group homomorphism

We define the group of germs at infinity of functions from

M Γ to 2 Z Γ/2 2 Z Γ as the quotient: Ξ Γ = ( 2 Z Γ/2 2 Z Γ) M Γ / Net 0 ,
where Net 0 is the normal subgroup of eventually zero functions defined by:

Net 0 := {(w a ) a∈M Γ | ∃a ∈ M Γ ∀b ∈ M Γ , b ≥ a ⇒ w b = 0}
We are now able to define a new group homomorphism from IET (Γ) to Ξ Γ . Definition 3.3.8. We define the positive contribution of IET (Γ) as the next map:

ψ Γ : IET (Γ) -→ Ξ Γ f -→ [(ψ Sa (f )) a∈M Γ ]
Proposition 3.3.9. The map ψ Γ is a group homomorphism.

Proof. Let f, g ∈ IET (Γ). By Proposition 3.3.4, there exists a 0 ∈ M Γ such that for every b ≥ a 0 , there exists a partition P b into S b -intervals associated with (f, g). Then by Proposition 3.3.7 we deduce that for every b ≥ a 0 we have

ψ S b (g • f ) = ψ S b (g) + ψ S b (f ).
We denote by w = (w a ) a∈M Γ the element of

( 2 Z Γ/2 2 Z Γ) M Γ defined by w a = ψ Sa (g • f ) -ψ Sa (g) -ψ Sa (f ) for every a ∈ M Γ . We remark that w b = 0 for every b ≥ a 0 thus w ∈ Net 0 . Then we have (ψ Sa (g • f )) a = (ψ Sa (g)) a + (ψ Sa (f )) a -w. So ψ Γ (g • f ) = ψ Γ (g) + ψ Γ (f ).
The following lemma gives the value of an element in IET(Γ). The proof is immediate from the definition of ψ Sa . Lemma 3.3.10. For every f ∈ IET(Γ), there exists a 0 ∈ M Γ such that for every b ∈ M Γ with b ≥ a 0 we have we have

ψ S b (f ) = ε Γ (f ) [mod 2]. In particular ψ Γ (f ) = [(ε Γ (f ) [mod 2]) a∈M Γ ].
Remark 3.3.11. In fact the definition of Ψ Sa gives us that for every a ∈ M Γ there exists a ∈ M Γ with a ≥ a such that for every b ≥ a we have

ψ S b (f ) = ε Γ (f ) [mod 2]
. This is a sufficient condition to obtain the wanted equality in Ξ Γ . It is also a necessary condition. Indeed, let us assume that there exists

a 0 ∈ M Γ such that for every b ∈ M Γ with b ≥ a 0 we have ψ S b (f ) = ε Γ (f ) [mod 2].
Then thanks to the property of M Γ to be an upward directed set we deduce that for every a ∈ M Γ there exists c a ∈ M Γ such that c a ≥ a and c a ≥ a 0 . Thus for every

b ∈ M Γ such that b ≥ c a we have ψ S b (f ) = ε Γ (f ) [mod 2].
The next proposition show that the set we used to define Ω Γ is not sent to the trivial element by ψ Γ ; thus this is not a subset of D(IET (Γ)). Proposition 3.3.12. Let ∈ Γ 2 Γ and r be a Γ-reflection of type 2 . Then ψ

Γ (r) = [( ∧ [mod 2]) a∈M Γ ] = 0.
Proof. We know that such a Γ-reflection is conjugate to a Γ-restricted rotation of type ( , ). We denote by s this Γ-restricted rotation. We have ε

Γ (s) = ∧ [mod 2]. Then by Lemma 3.3.10 we have ψ Γ (r) = ψ Γ (s) = [( ∧ [mod 2]) a ]. Thus if ψ Γ (r) = 0 then there exists (w a ) a∈M Γ ∈ Net 0 such that ( ∧ [mod 2]
) a + (w a ) a = 0. We deduce that ∧ [mod 2] = 0. Or this implies ∈ 2 Γ and this is a contradiction. Hence ψ Γ (r) = 0.

Description of IET (Γ) ab

With both morphisms ε Γ and ψ Γ we are now able to describe D(IET (Γ)). We recall that Ω Γ is the conjugate closure of the group generated by the set of all Γ-reflections of type 2 with ∈ Γ 2 Γ. Lemma 3.4.1. We have the inclusion:

Ω Γ ∩ Ker(ψ Γ ) ⊂ D(IET (Γ))
Proof. Let w ∈ Ω Γ ∩ Ker(ψ Γ ). There exist n ∈ N and g 1 , g 2 , . . . g n ∈ IET (Γ) and r 1 , r 2 , . . . , r n some Γ-reflections with type inside Γ 2 Γ such that w = n i=1 g i r i g -1 i . We know that every Γ-reflection is conjugate to a Γ-restricted rotation so there exist h 1 , h 2 , . . . h n ∈ IET (Γ) and s 1 , s 2 , . . . , s n some Γ-restricted rotations such that

s i = h i r i h -1 i for every 1 ≤ i ≤ n. Then w = n i=1 g i h -1 i s i h i g -1 i
. By Lemma 3.2.2 it is sufficient to show that w is a balanced product of Γ-reflections.

As for every 1 ≤ i ≤ n we have g i and g -1 i which appear the same number of time in w (the same is true for h and h -1 ) we deduce that it is sufficient to show that w := n i=1 s i is a balanced product of Γ-reflections. By Lemma 3.2.2 it is enough to show that w ∈ D(IET (Γ)).

We notice that ψ Γ (w) = ψ Γ (w ) thus by the assumption we deduce that ψ Γ (w ) = 0. Hence by Lemma 3.3.10 we deduce that ε Γ (w ) [mod 2] = 0. This equality stands in 2 Z Γ/2 2 Z Γ. There exists k ∈ N and for every 1 ≤ j ≤ k there exist a j , b j ∈ Γ+ with a j + b j < 1 and n j ∈ N and

ν j ∈ {-1, 1} such that ε Γ (w ) + k j=1 2n j ν j a j ∧ b j = 0. For every 1 ≤ j ≤ k let γ j be a Γ-restricted rotation of type (a j , b j ). Then the element w k j=1 (γ n j ν j j ) 2
is in IET(Γ) and satisfies:

ε Γ (w k j=1 (γ n j ν j j ) 2 ) = ε Γ (w ) + k j=1 2n j ν j a j ∧ b j = 0 By Theorem 2.3.24 the element w k j=1 (γ n j ν j j ) 2 is in D(IET(Γ)) ⊂ D(IET (Γ)). By Corollary 3.2.3 we know that k j=1 (γ n j ν j j ) 2 is in D(IET (Γ)). Then we deduce that w ∈ D(IET (Γ)). Hence w is in D(IET (Γ)). Theorem 3.4.2. We have D(IET (Γ)) = Ker(ε Γ ) ∩ Ker(ψ Γ ).
Proof. The inclusion from left to right is trivial.

Let f ∈ Ker(ε Γ )∩Ker(ψ Γ ). By Theorem 3.2.19 there exist g ∈ D(IET (Γ) and h ∈ Ω Γ such that f = gh. We deduce that ψ Γ (h) = ψ Γ (f ) = 0 so h ∈ Ker(ψ Γ ) ∩ Ω Γ . By Lemma 3.4.1 we obtain that h ∈ D(IET (Γ)), thus f ∈ D(IET (Γ)).

Corollary 3.4.3. We have D(IET (Γ)) = Ker(ψ Γ | Ker(ε Γ ) ). Lemma 3.4.4. The quotient Ker(ε Γ )/D(IET (Γ)) is isomorphic to the subgroup { ∧ [mod 2] | ∈ Γ} of 2 Z Γ/2 2 Z Γ.
Proof. By Corollary 3.4.3 we have:

Ker(ε Γ )/D(IET (Γ)) Im(ψ Γ | Ker(ε Γ ) ) = ψ Γ (Ker(ε Γ ))
By Theorem 3.2.19 and as ψ Γ is a group homomorphism we have the equality ψ Γ (Ker(ε Γ )) = ψ Γ (Ω Γ ). Furthermore Ω Γ is the normal closure of the group generated by all Γ-reflections of type 2 with ∈ Γ 2 Γ. Hence we deduce that:

ψ Γ (Ω Γ ) = {[( ∧ [mod 2]) n∈N ] | ∈ Γ 2 Γ = {[( ∧ [mod 2]) n∈N ] | ∈ Γ Thus ψ Γ (Ker(ε Γ )) { ∧ [mod 2] | ∈ Γ} .
Theorem 3.4.5. We have the following group isomorphisms:

IET (Γ) ab Im(ε Γ ) × Ker(ε Γ )/D(IET (Γ)) {a ⊗ a [mod 2] | a ∈ Γ} × { ∧ [mod 2] | ∈ Γ} ,
where the left term of the product is in 2 Z Γ/(2 2 Z Γ) and the right one is in

2 Z Γ/(2 2 Z Γ).
Proof. The second isomorphism is given by Corollary 3.2.14 and Lemma 3.4.4.

For the first isomorphism we recall that we have the following exact sequence:

1 → Ker(ε Γ )/D(IET (Γ)) → IET (Γ) ab → IET (Γ)/ Ker(ε Γ ) → 1
Each group in this exact sequence has exponent 2. Then they are also F 2 -vectorial spaces. We deduce that this exact sequence is an exact sequence of F 2 -vectorial spaces, thus it splits and gives the result. Remark 3.4.7. The inclusion of IET(Γ) in IET (Γ) induces a group morphism ι from IET(Γ) ab /2(IET(Γ) ab ) to IET (Γ) ab . By Theorem 1.2.8 we know that IET(Γ) is generated by Γ-restricted rotations thus we deduce that the image of ι is the subgroup

{p ⊗ q + q ⊗ p [mod 2] | p, q ∈ Γ} × {a ∧ a [mod 2]} of 2 Z Γ/(2 2 Z Γ) × 2 Z Γ/(2 2 Z Γ). This is isomorphic to 2 Z Γ/(2 2 Z Γ) and if Γ has dimension d then its dimension is d(d+1)
2 as F 2vector space. In this case ι is not surjective and its cokernel has dimension d over F 2 . In the case where Γ has infinite dimension over Z we deduce that Im(ι) also has infinite dimension over F 2 . By Proposition 2.3.18 we deduce that the group {p ⊗ q

+ q ⊗ p [mod 2] | p, q ∈ Γ} is equal to the group {a ⊗ a [mod 2] | a ∈ Γ} if and only if Γ = 2 Γ. Then ι is surjective if and only if Γ = 2 Γ. Proposition 3.4.8. The group homomorphism ι is injective. Proof. For every f ∈ IET(Γ) we denote by [f ] its image in IET(Γ) ab . Thanks to Theorem 2.3.24 we know that [f ] ∈ 2 IET(Γ) ab if and only if ε Γ (f ) ∈ 2 2
Z Γ. Hence to prove the statement it is enough to prove that for every f ∈ IET(Γ) such that ε Γ (f ) = 0 and proj Ker(ε Γ ) (f ) = 0 we have ε Γ (f ) ∈ 2 2 Z Γ. We use notations of inversions defined in Definition 3.2.5. By Corollary 2.1.3 there exists n ∈ N and a Z-linearly independent family {l 1 , l 2 , . . . , l n } of Γ + and n i,j ∈ Z such that E f,1 = i,j n i,j l i ⊗ l j . The equality proj Ker(ε Γ ) (f ) = 0 gives us that n i,i = 0. We have ε Γ (f ) = i =j (n i,j +n j,i )l i ⊗l j [mod 2] = 0.

We deduce that 2 divides (n i,j + n j,i ) for every 1 ≤ i = j ≤ n. We obtain that :

ε Γ (f ) = i =j n i,j l i ∧ l j = i<j (n i,j -n j,i )l i ∧ l j = i<j (n i,j + n j,i )l i ∧ l j -2 i<j n j,i l i ∧ l j
We deduce that 2 divides ε Γ (f ) and this gives the result.

Introduction

The beginning of the study of the symmetric group of an infinite set X, denoted by S(X), date back at least in 1915 with an article of Vitali [START_REF] Vitali | Sostituzioni sopra un'infinita numerabile di elementi[END_REF] and one of Andreoli [START_REF] Andreoli | Sui gruppi di sostituzioni che operano su infiniti elementi[END_REF]. Even if he does not use the language of group theory, Vitali proves that every element of S(Z) is a finite product of squares. A direct consequence is the following: Theorem 4.1.1 (Vitali [Vit15]). The only group homomorphism from S(X) into Z/2Z is the trivial one.

A natural subgroup to consider is the one consisting of all finitely supported permutations, denoted by S fin (X). It is a normal subgroup and with the same proof as in finite permutation group, we can define a surjective group homomorphism ε fin (X) onto (Z/2Z, +) called the classical signature. We denote it by A fin (X) the kernel of the classical signature. It is called the alternating subgroup and it is a subgroup of index two of S fin (X) and also a normal subgroup of S(X). In 1929, Onefri [START_REF] Onofri | Teoria delle sostituzioni che operano su una infinità numerabile di elementi[END_REF] proves that S fin (X) and A fin (X) are the only proper normal subgroups of S(X) when X is a countable set. This result was extended to every infinite set X by Baer [START_REF] Baer | Die kompositionsreihe der gruppe aller eineindeutigen abbildungen einer unendlichen menge auf sich[END_REF]. With these groups we obtain the following exact sequence:

0 → Z/2Z = S fin (X)/A fin (X) → S(X)/A fin (X) → S(X)/S fin (X) → 1
It is a central extension, thus it gives a canonical element of the cohomological group H 2 (S(X)/S fin (X), Z/2Z), called the Kapoudjian class of X. It appears in the work of Kapoudjian and Sergiescu [Kap02,[START_REF] Kapoudjian | An extension of the Burau representation to a mapping class group associated to Thompson's group T , Geometry and dynamics[END_REF], and also in the work of Cornulier [Cor19a] in the context of near actions.

Here we work with the interval [0, 1[ and we recall that S fin = S fin ([0, 1[). We denote also ε fin = ε fin ([0, 1[). For every subgroup G of S([0, 1[)/S fin we denote by G its preimage in S([0, 1[). Then, we have the following exact sequence:

0 → Z/2Z = S fin /A fin → G/A fin → G → 1
As before, it is a central extension, thus it gives a canonical element of the cohomological group H 2 (G, Z/2Z), called the Kapoudjian class of G. For such a G, Cornulier [START_REF]Realizations of groups of piecewise continuous transformations of the circle[END_REF] asks when its Kapoudjian class vanishes, this implies in particular that G/A fin is isomorphic to the direct product G × Z/2Z.

The vanishing of this class means that the above exact sequence splits; this means that there exists a group homomorphism from G onto Z/2Z which extends the signature ε fin . Theorem 4.1.1 implies that the Kapoudjian class of [0, 1[, which is equal to the Kapoudjian class of S([0, 1[)/S fin , does not vanish.

In this chapter we prove the following theorem: The idea of the proof of Theorem 4.1.2 is to associate for every f ∈ PC and every finite partition P of [0, 1[ into intervals associated with f , two numbers. The first is the number of interval of P where f is order-reversing and the second is the signature of a particular finitely supported permutation. The next step is to prove that the sum modulo 2 of this two numbers is independent from the choice of partition. Then we show that it is enough to prove that ε| IET is a group homomorphism. For this we show that it is additive when we look at the composition of two elements of IET by calculate the value of the signature with a particular partition.

In Section 4.3, we apply these results to the study of normal subgroups of PC and certain subgroups. More specifically we prove: Theorem 4.1.4. Let G be a subgroup of PC which is simple nonabelian and let G be its preimage in S. Then G has exactly 5 normal subgroups, which constitute the following diagram:

G S fin
Ker(ε| G )

A fin

{1}

Let AIET be the subgroup of PC consisting of elements that are piecewise affine, and let AIET + be its subgroup of piecewise orientation-preserving elements. N. Guelman and I. Liousse [START_REF] Guelman | Bounded simplicity of affine interval exchange transformations and interval exchange transformations[END_REF] prove that AIET + is uniformly perfect and uniformly simple. Then with Theorem 1.3.4 we deduce that AIET is also perfect and so simple (see Corollary 1.3.3). Hence with Theorem 1.3.4 and Corollary 1.3.5 we deduce that PC , PC + , AIET , AIET + , IET are examples of groups that satisfy conditions of Theorem 4.1.4.

For every real interval I we denote by I • its interior in R and if I = [0, t[ we agree that its interior is ]0, t[. We use the notion of partition associated defined in Section 1.2.1. We recall that for every f ∈ PC , we denote by Π f the set of all partitions into intervals associated with f . We recall that P min f is the unique partition in Π f such that every partition Q ∈ Π f is a refinement of P min f .

Construction of the signature homomorphism

Definitions

The main decomposition we use here is the one given by Proposition 3. It is important to notice that we will use here partitions associated with an element of PC and not all essential partitions. We also give a definition more dynamical for the default of pseudo right-continuity: Definition 4.2.3. Let f be an element of PC , n ∈ N and P = {I 1 , I 2 , . . . , I n } ∈ Π f . For every 1 ≤ j ≤ n, let α j be the left endpoint of I j and β j be the left endpoint of f (I • j ). We define the default of pseudo right continuity for f about P denoted σ ( f,P) as the finitely supported permutation which sends f (α j ) to β j for every 1 ≤ j ≤ n (this is well-defined because the set of all f (α j ) is equal to the set of all β j ).

The equivalence between the two definitions is given by the right-continuity of f + P .

Definition 4.2.4. For f ∈ PC and P ∈ Π f , define ε( f, P) ∈ Z/2Z as R( f, P) +

ε fin (σ ( f,P) ) [mod 2]. We define also ε( f ) = ε( f, P fin f ).
Proposition 4.2.5. For every τ ∈ S fin and every P ∈ Π τ we have ε(τ, P) = ε fin (τ ).

Proof. It is clear that for every τ ∈ S fin and every partition P associated with τ we have R(τ, P) = 0 and σ (τ,P) = τ .

We deduce that ε extends the classical signature ε fin . Thus we will write ε instead of ε fin . Proposition 4.2.6. Every right-continuous element f of PC + satisfies ε( f, P) = 0 for every P ∈ Π f . Proof. In this case, for every partition P into intervals associated with f we always have R( f, P) = 0 and σ ( f,P) = Id.

Proof of Theorem 4.1.2

In order to prove that ε is a group homomorphism, it is useful to calculate ε( f ) thanks to ε( f, P) for every f ∈ PC and P ∈ Π f . Lemma 4.2.7. For every f ∈ PC and every P ∈ Π h we have ε( f ) = ε( f, P).

Proof. Let f and P be as in the statement. By minimality of P min f , in term of refinement, we deduce that there exist n ∈ N and P 1 , P 2 , . . . , P n ∈ Π f such that:

1. P 1 = P min f ; 2. P n = P;
3. for every 2 ≤ i ≤ n the partition P i is a refinement of the partition P i-1 where only one interval of P i-1 is cut into two.

Hence it is enough to show

ε( f, Q) = ε( f, Q ) where Q, Q ∈ Π f such that there exist consecutive intervals I, J ∈ Q with I ∪ J ∈ Q and Q {I ∪ J} = Q {I, J}.
Let α be the left endpoint of I and let x be the right endpoint of I (x is also the left endpoint of J). There are only two cases but in both cases, we know that σ ( f,Q) = σ ( f,Q ) except maybe on f (α) and f (x):

1. The first case is when f is order-preserving on (I ∪ J) • . Then as

Q {I, J} = Q {I ∪ J} we get R( f, Q) = R( f, Q ).
As f is order-preserving on the interior of

I ∪ J we know that σ ( f,Q ) ( f (α)) is the left endpoint of f ((I ∪ J) • ) which is the left endpoint of f (I • ) thus equals to σ ( f,Q) ( f (α)).
With the same reasoning we deduce

that σ ( f,Q ) ( f (x)) = σ ( f,Q) ( f (x)) hence σ ( f,Q) = σ ( f,Q ) . Thus in Z/2Z we have R( f, Q ) + ε(σ ( f,Q ) ) = R( f, Q) + ε(σ ( f,Q) ).
2. The second case is when f is order-reversing on (I ∪ J)

• . Then we get R( f, Q) = R( f, Q ) + 1. This time σ ( f,Q ) ( f (α)) is still the left endpoint of f ((I ∪ J) • ) which is the left endpoint of f (J • ) thus equals to σ ( f,Q) ( f (x)).
With the same reasoning

we deduce that σ ( f,Q ) ( f (x)) = σ ( f,Q) ( f (α)).
Then by denoting τ the transposition

( f (x) σ ( f,Q ) ( f (α))), we obtain σ ( f,Q) = τ • σ ( f,Q ) .
We must notice that the transposition is not the identity because

f -1 (σ ( f,Q ) ( f (α))
) is an endpoint of one of the intervals of Q and x is not.

In conclusion in Z/2Z we have:

R( f, Q ) + ε(σ ( f,Q ) ) = R( f, Q ) + 1 + 1 + ε(σ ( f,Q ) ) = R( f, Q) + ε(σ ( f,Q) ) I J I J α α x x h(x) h(x) σ (h;Q 0 ) (h(α)) σ (h;Q)
( f,Q) ( f (x)) = σ ( f,Q ) ( f (x))
. Right: Case where f is order-reversing on (I ∪ J) • and we see that

σ ( f,Q) ( f (x)) = ( f (x) σ ( f,Q ) ( f (α))) • σ ( f,Q ) ( f (x)).
If φ ∈ Homeo + ([0, 1[), then it follows from Proposition 4.2.6 that ε(φ) = 0. We show that ε is invariant by the action of Homeo + ([0, 1[) on PC . Proof. Let f ∈ PC and φ ∈ Homeo + ([0, 1[) be as in the statement. Let n ∈ N and P := {I 1 , I 2 , . . . , I n } ∈ Π f . Then Q := {φ -1 (I 1 ), φ -1 (I 2 ), . . . , φ -1 (I n )} is in Π fφ . We know that φ is order preserving then for every 1, ≤ i ≤ n, fφ preserves (reverses respectively) the order on φ -1 (I • i ) if and only if f preserves (reverses respectively) the order on I • i , so R( f, P) = R( fφ, Q). We can notice that the left endpoint of φ -1 (I • i ) (denoted by α i ) is

Introduction

We recall that the group IET is the group consisting of all permutations of [0, 1[ continuous outside a finite set, right-continuous and piecewise a translation. We study a generalization of IET in higher dimension. Let d ≥ 1 be an integer. We denote by X = Historically, H. Haller [START_REF] Haller | Rectangle exchange transformations[END_REF] introduced 2-rectangle exchange transformations in 1981 and it is mainly ergodic properties of a single 2-rectangle exchange transformation which are studied. More generally, dynamics of piecewise isometries on polytopes are studied, in particular by A. Goetz [START_REF] Goetz | Dynamics of piecewise isometries[END_REF], however the group itself is rarely considered. In contrast, the group of piecewise affine self-homeomorphisms of some manifolds are recently considered in particular by D. Calegari and D. Rolfsen [START_REF] Calegari | Groups of PL homeomorphisms of cubes[END_REF].

Definition 5.1.1. Let f ∈ Rec d and P be a partition of X into rectangles (resp. a gridpattern). We said that P is a partition associated with f (resp. a grid-pattern associated with f ) if for every K ∈ P the restriction of f to K is a translation. Then the set f (P) := {f (K) | K ∈ P} is a new partition of X into rectangles called the arrival partition of f with P. We denote by Π f the set of all partitions associated with f . From now on every partition is assumed to be finite. Remark 5.1.2. The fact that Rec d is a group under composition is immediate. One can see that if f, g ∈ Rec d and P ∈ Π f , Q ∈ Π g , then there exists a partition R into d-rectangles that refines both f (P) and Q. Thus f -1 (R) is a partition into d-rectangles such that g • f acts on every d-rectangle of f -1 (R) by translation.

In the following, the "d" of d-rectangle may be omitted whenever there is no possible confusion.

We introduce two kinds of special elements in Rec d .

Definition 5.1.3. A restricted shuffle in direction i is an element σ R,s,i of Rec d where R is a (d -1)-subrectangle of e ⊥ i and s is a restricted rotation, defined by:

1. if pr ⊥ i (x) / ∈ R, σ R,s,i (x) = x; 2. if pr ⊥ i (x) ∈ R: (a) for j = i, σ R,s,i (x) j = x j ; (b) σ R,s,i (x) i = s(x i ).
For disjoint translation-isometric rectangles P, Q ⊂ [0, 1[ d , define the Rec d -transposition τ P,Q as the element of Rec d defined as the identity outside P ∪ Q, and as a translation on each of P, Q, exchanging them. The set of all Rec d -transpositions is denoted by T d . Notation 5.1.4. If I and J are the two intervals associated with s then the d-rectangles P 1 and P 2 , defined by pr i (P 1 ) = I, pr i (P 2 ) = J and pr ⊥ i (P 1 ) = pr ⊥ i (P 2 ) = R, are two rectangles which partitioned the support of f and where f is continuous on both of them. We say that f shuffles this two rectangles. We know that the set of all restricted rotation is a generating subset of IET, see Theorem 1.2.8. Restricted shuffles are a natural generalization of these elements in Rec d . It is natural to ask if they define a generating subset for Rec d and the next theorem answers it positively. For d = 1, it essentially amounts to seeing an IET as a permutation of n intervals, and arguing that the symmetric group S n is generated by transpositions (i, i+1). For d ≥ 2 there is no such simple combinatorial argument and much more work is required; it is done in Section 5.4. Theorem 5.1.5. The set of all restricted shuffles is a generating subset of Rec d .

For d = 2 a variant of the proof, providing a combinatorial refinement of Theorem 5.1.5, is performed in Section 5.5

The group IET d acting coordinate-wise is a subgroup of Rec d which for d ≥ 2 is a proper subgroup.

One main difficulty we have to dealt with for d ≥ 2 is that there does not always exist a grid that is mapped to another grid by translation of all components.

In the case where d ≥ 2, if we take two disjoint d-rectangles which are translationisometric such that their orthogonal projections on Vect(e i ) are not [0, 1[ for every 1 ≤ i ≤ d, then the Rec d -transposition which permutes these two rectangles is not in IET d .

Thanks to Theorem 5.1.5 we obtain that D(Rec d ) is generated by conjugates of commutators of two restricted shuffles. With this result we prove the following theorem: In Section 5.7 we extend Theorem 1.3.6 which identifies the abelianization of IET thanks to the SAF-homomorphism. We denote by R ⊗k the kth tensor power of R over Q. In the case of Rec d we prove that it is enough to consider an analogue of this group homomorphism in every direction to obtain the following result: This theorem is done by Arnoux-Fathi-Sah in dimension 1 and thanks to Theorem 5.1.5, the proof is similar in dimension d.

Grid-patterns, setwise Q-freeness

We fix d ≥ 1; in a first reading, one can assume d = 2. We introduce the notion of partition associated with an element of Rec d and also the one of grid-pattern. Definition 5.2.1. A partition P of X into rectangles is called a grid-pattern if for every

1 ≤ i ≤ d, there exists a partition Q i of [0, 1[ into half-open intervals such that P = Q 1 × Q 2 × . . . × Q d .
Definition 5.2.2. Let n ∈ N and P = {P 1 , P 2 , . . . , P n } be a partition of X into rectangles. For every 1 ≤ i ≤ d, let H i = {pr i (P j )} 1≤j≤n and let E i := {a ∈ [0, 1[ : ∃I ∈ H i such that a is an endpoint of I}. Let k i be the cardinal of E i and let a 1 < a 2 < . . . < a k i be its elements. We define

Q i := {[a j , a j+1 [} 1≤j≤k i -1 , it is a finite partition of [0, 1[ into intervals. Then Q = Q 1 × Q 2 × . . . × Q d is a grid-pattern called the grid-pattern refinement of P.
In fact we will need some rigidity on partitions associated with an element of Rec d . For this we want to have some objects to be Q-free. Definition 5.2.3. Let P be a partition into rectangles of [0, 1[ d . For every 1 ≤ i ≤ d we denote by F i the set {λ(pr i (K)) | K ∈ P}. If for every 1 ≤ i ≤ d the set F i is Q-linearly independent then we say that P is a setwise Q-free partition.

Warning. The Q-independence required is that of the set {λ(pr i (K)) | K ∈ P}, and not the family (λ(pr i (K))) K∈P }. So the set-wise freeness condition says, roughly speaking, that the only Q-linear dependence relations among the λ(pr i (K)), for K ∈ P (for each fixed K) are equalities. Proposition 5.2.4. Let Q be a grid-pattern. There exists a setwise Q-free grid-pattern Q that refines Q.

Proof. Thanks to Corollary 2.1.3 we know that for every finite subset S of positive real numbers there exists a finite Q-free subset of positive real numbers S such that every element of S is a linear combination of elements in S with nonnegative integer coefficient. This statement is Lemma 4.1 of Vorobets in [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF]. Here let

Q = Q 1 × . . . × Q d where Q i is a partition into intervals of [0, 1[ and let F i := {λ(I) | I ∈ Q i }.
By applying the previous fact for every F i we obtain a new finite set of length F i and we know we can refines every A grid-pattern that is not setwise Q-free. Right: A setwise Q-free grid-pattern which refines the left grid-pattern.

Q i into a finite partition Q i into intervals of [0, 1[ such that for every I ∈ Q i we have λ(I) ∈ F i . Then Q := Q 1 × . . . × Q d is a setwise Q-free grid-pattern which refines Q.
It will be important to have some rigidity on the number of pieces which are translationisometric inside a partition. The next lemma explains how there exists a bijection between two partitions which share the same combinatorial conditions such that every piece and its image by this bijection are translation-isomorphic. The following corollary is the main result to use the induction property. Lemma 5.2.5. For every 1 ≤ i ≤ d, let F i be a setwise Q-free subset of R + . Let P and P be two partitions into d-rectangles of [0, 1[ d such that for every K ∈ P ∪ P we have λ(pr i (P )) ∈ F i . Then, there exists a bijection δ between P and P such that for every K ∈ P we have K and δ(K) which are translation-isometric. If K ∈ P ∩ P we can also ask δ(K) = K. By setwise Q-freeness of F i we deduce that F = {a 1 ⊗ . . . ⊗ a d | (a 1 , . . . , a j ) ∈ F 1 × . . . × F j } is a setwise Q-free subset of A ⊗d . By definition of P and P we have:

μ d ([0, 1[ d ) = μ d ( K∈P K) = a∈F Card({K ∈ P | μ d (K) = a})a = μ d ( K∈P K) = a∈F Card({K ∈ P | μ d (K) = a})a
Then by setwise Q-freeness of F we deduce that for every a ∈ F we have Card({K

∈ P | μ d (K) = a}) = Card({K ∈ P | μ d (K) = a}).
From this we deduce that there exists a bijection δ between P and P such that for every K ∈ P we have K and δ(K) which are translation-isometric and for every K ∈ P ∩ P we can have δ(K) = K. Corollary 5.2.6. In particular, there exists f ∈ Rec d such that P ∈ Π f and f (P) = P and for every K ∈ P ∩P we have f (K) = K. More precisely the support of f is partitioned by P (P ∩ P ) and for every K ∈ P, the restriction of f to K is a translation.

First observations

We establish some easy particular cases of Theorem 5.1.5, which asserts that Rec d is generated by restricted shuffles.

Generation by restricted shuffles

The proof of Theorem 5.1.5 is by induction on the dimension d and the case of the dimension 1 is already known to be true.

Let f ∈ Rec d and Q be a grid-pattern associated with f . Thanks to Proposition 5.2.4 we can assume that Q is a setwise Q-free grid-pattern.

Let d ≥ 2 be the ambient dimension and assume Theorem 5.1.5 true for Rec d-1 . Here the height is represented by the d-th coordinate. For every illustration in dimension 2 we use the following element f test of Rec 2 . The following partition P test (on the left of the following picture) is associated with f test , and is understood to be setwise Q-free. We denote by P test = f test (P test ) (on the right of the following picture).

Grd(P) = {K ∈ P | 0 ∈ pr d (K)}.
Let K 0 be an element of Grd(P). A tower above K 0 is a subset T of P such that :

1. K 0 ∈ T ; 2. ∀K ∈ T, pr ⊥ d (K) = pr ⊥ d (K 0 ); 3. The set K∈T pr d (K) is a subinterval of [0, 1[. The element K of T which satisfies sup(pr d (K)) = sup K∈T pr d (K) is called the top of
the tower T , denoted by Top(T ). The highest tower above K 0 , denoted by T (K 0 ), is the maximal tower above K 0 according to the inclusion order.

Definition 5.4.2. A city of P is a subset of P containing Grd(P), and which is a union of towers. The highest city of P, denoted by City(P), is the union of all highest towers above an element of the ground Grd(P). The top of a city V ⊂ P is the set of Top(T ) when T ranges over maximal towers in V. The sky of P, denoted by Sky(P), is the complement of City(P) in P.

C (P) = {min(pr d (K)) | K ∈ Sky(P)}.
The set C (P) is empty if and only if P = City(P). Otherwise, the minimum of the set C (P) is called the working height of P denoted by C min (P).

The idea is to move pieces of City(P) with restricted shuffles in direction in {1, 2, . . . , d-1} such that the new partition P obtained satisfies C (P ) ⊂ C (P) {C min (P)}. For this we describe more precisely how and where we move pieces. | K ∈ P} is setwise Q-free, thus for every a ∈ [0, 1[, the number of rectangles K such that λ(pr d (K)) = a is the same in every tower T ⊂ City(P). Then by using only restricted shuffles in direction d, we can move pieces inside every tower T ⊂ City(P) to order them according to the length of their projection on Vect(e d ). The image of P by the product of these restricted shuffles is a grid-pattern. Lemma 5.4.6. Let P be a setwise Q-free partition such that C (P) = ∅. There exists a product g of restricted shuffles in direction inside {1, 2, . . . , d -1} such that P ∈ π g and:

C (g(P)) ⊂ C (P) {C min (P)}. Proof. For every 1 ≤ i ≤ d let F i = {λ(pr i (K)) | K ∈ P}; it is a setwise Q-free subset of R + . By definition we know that Ω -= {pr ⊥ d (K) | K ∈ Top(City)(P)} and Ω + = {pr 2 (K) | K ∈ Work + ∪ Top(City(P)) Work -} are two partitions of [0, 1[ d-1 such that for every K ∈ Ω -∪ Ω + and every 1 ≤ i ≤ d we have λ(pr i (K)) ∈ F i .
Then, by Corollary 5.2.6 we deduce that there exists δ ∈ Rec d-1 such that Ω -∈ Π δ (for every element K of Ω -, the restriction of δ to K is a translation) and δ(Ω -) = Ω + and for every K ∈ Ω -∩ Ω + we have δ(K) = K. As we assumed Theorem 5.1.5 in dimension d -1, we know that δ can be written as the product of restricted shuffles of Rec d-1 . Then we define g ∈ Rec d such that:

g(x) = (δ × Id)(x) if pr 2 (x) < C min (P) x else.
From this definition we obtain that g is the product of restricted shuffles in Rec d with direction in {1, 2 . . . , d-1}. Also by definition of δ we obtain that for every K ∈ Grd(P) we have g(T (K)) ⊂ T (g(K)) and g(Sky(P)) = Sky(P). This implies C (g(P)) ⊂ C (P). Also as δ(Ω -) = Ω + we deduce that for every K ∈ Sky(P) such that min(pr d (K)) = C min (P) there exists

Q K ∈ Grd(P) such that δ(pr ⊥ d (Q K ) = pr ⊥ d (K). Hence we have K ∈ T (g(Q K )
) and this implies that C min (P) / ∈ C (g(P)).

Then by induction on the cardinal of the complexity we deduce the following proposition:

A refinement for Rec 2

Here we establish a more precise and concrete statement in dimension 2. Theorem 5.1.5 says every element f in Rec d can be obtained as a composition of restricted shuffles. It is tempting to improve this statement by fixing a setwise Q-free partition P ∈ Π f , and then shuffling rectangles in f (P) without changing the partition. The proof seems at first sight to provide this, but the induction step forces to change the partition. In dimension 2, we can avoid this, see Theorem 5.5.2 below.

In this case we can be more precise than Theorem 5.1.5. Remark 5.5.3. To motivate the setwise Q-free property, we illustrate with a partition which is a in the image by Rec d of a grid-pattern Q which is not setwise Q-free. Indeed if we do not allowed to cut pieces of Q then for every sequence of restricted shuffles on Q, the image of Q by this sequence is always Q. Lemma 5.5.5. Let P be a setwise Q-free partition such that C (P) = ∅. Then there exists a sequence of restricted shuffles on P such that the image of P by this sequence is a grid-pattern.

With these two results, the proof of Theorem 5.5.2 is the same as the one of Theorem 5.1.5 until Lemma 5.4.6, where we proved the following refinement in dimension 2: Lemma 5.5.6. Suppose d = 2. Let P be a setwise Q-free partition. There exists a product g of restricted shuffles in direction inside {1, 2, . . . , d -1} such that P ∈ π g and: C (g(P)) ⊂ C (P) {C min (P)}.

Also there exists a sequence (r 1 , . . . , r n ) of restricted shuffles on P such that g = r n •. . .•r 1 .

Proof. First we rearrange every tower of City(P) such that pieces of every tower is ordered by increasing order about their length of their 2-projection.

We recall that Ω -= {pr ⊥ d (K) | K ∈ Top(City(P))} and Ω + = {pr ⊥ d (K) | K ∈ Work + ∪ Top(City(P)) Work -}. by Corollary 5.2.6 we deduce that there exists δ ∈ Rec d-1 such that Ω -∈ Π δ (for every element K of Ω -, the restriction of δ to K is a translation) and δ(Ω -) = Ω + and for every K ∈ Ω -∩ Ω + we have δ(K) = K. The main argument is that every connected component C of Site(P) = We can see that g C only moves towers of City(P). And as these towers are rearrange such that pieces of every tower is ordered by increasing order about their length of their 2-projection. We deduce that there exists a sequence (r 1 , . . . , r n ) of restricted shuffle on P such that g C = r n • . . . • r 1 . Let g be the product of every g C where C ranges over the set of all connected components of Site(P). It satisfies the statement of the lemma.

s permute P 2 and f (P 1 ) by translations, Q 2 and f (Q 1 ) by translations, and be identity elsewhere. Then s[f, τ P 1 ,Q 1 ]s -1 = τ P,Q . Hence τ P,Q ∈ N .

Abelianization of Rec d

In this section we generalize the abelianization group homomorphism of IET given by Arnoux-Fathi and Sah in [Arn81a, Sah81] called the SAF-homomorphism. The idea is to define, in each direction, a homomorphism, analogous to the 1-dimensional SAFhomomorphism.

For every group G, we denote by D(G) its derived subgroup and by G ab := G/D(G) its abelianization. For every f ∈ Rec d we denote by f its image in (Rec d ) ab . For every α ∈ R we denote by {α} its fractional part. We recall that R ⊗d is the d-th tensor power of R over Q.

Tensor-valued measures

With notation in the proof of Lemma 5. where λ is the Lebesgue measure on R. In order to have more homogeneous results for every direction we need to use variants of the μ d that place some coordinates at the end: Definition 5.7.1. For every 1 ≤ i ≤ d we define a Boolean algebra measure on A ⊗d by μ d,i = σ i • μ i : where σ i is the linear automorphism of R ⊗d defined by:

σ i (x 1 ⊗ • • • ⊗ x d ) = (x 1 ⊗ • • • ⊗ x i-1 ⊗ x i+1 ⊗ • • • ⊗ x d ⊗ x i ).
Thus:

μ d,i : A ⊗d -→ R ⊗d I 1 × . . . × I d -→ μ d (I 1 × . . . × I i-1 × I i+1 × . . . × I d × I i ).
For every restricted rotation we exhibit a useful representative of its conjugacy class. Definition 5.7.2. Let 1 ≤ i ≤ d and let a j ∈ ]0, 1] for every j ∈ {1, . . . , d} {i}; let a = (a 1 , . . . , a i-1 , a i+1 , . . . , a d ). Let α ∈ ]0, 1] and β ∈ [0, α[. Let P and Q be the two rectangles such that pr j (P ) = pr j (Q) = [0, a j [ for every j = i and pr i (P ) = [0, αβ[ and pr i (Q) = [αβ, α[. We call the standard shuffle of parameters a, i, α, β the restricted shuffle in direction i that shuffles P and Q; we denote it by R a,i,α,β . For every f ∈ Rec d and 1 ≤ i ≤ d, we define ν i (f ) : [0, 1[ d → R by ν i (f )(x) = pr i (f (x))-pr i (x). One can notice that for every g ∈ Rec d we have ν i (f •g) = ν i (f )•g+ν i (g). Also by definition of f it is immediate that for every x ∈ R, the inverse image ν i (f ) -1 (x) belongs to A ⊗d and ν i : [0, 1[ d → R takes finitely many values. Definition 5.7.3. For every 1 ≤ i ≤ d we define the map ξ i as the following:

The generalized SAF-homomorphism

ξ i : Rec d -→ R ⊗(d+1) f -→ α∈R μ d,i ((ν i (f )) -1 ({α})) ⊗ α
We define ξ i as the projection of ξ i in R ⊗(d-1) ⊗ ( 2 Q R).

Example 5.7.4. Let 1 ≤ i ≤ d and a = (a 1 , . . . , a d-1 ) ∈ ]0, 1] d-1 . For every α ∈ ]0, 1] and every β ∈ [0, α[ we have:

ξ i (R a,i,α,β ) = a 1 ⊗ . . . ⊗ a d-1 ⊗ (α -β) ⊗ β + a 1 ⊗ . . . ⊗ a d-1 ⊗ β ⊗ (-(α -β)).
Then we deduce that:

ξ i (R a,i,α,β ) = 2(a 1 ⊗ . . . ⊗ a d-1 ⊗ (α ∧ β)) = a 1 ⊗ . . . ⊗ a d-1 ⊗ (α ∧ 2β).
For every 1 ≤ j = i ≤ d we have ξ j (R a,i,α,β ) = 0 because ν j (R a,i,α,β )(x) = 0 for every x ∈ [0, 1[ d . for every f ∈ Rec d , is called the generalized SAF-homomorphism.

The map ξ i is an analogue of the SAF-homomorphism given by Arnoux-Fathi and Sah in [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF]. Then the proof that the SAF-homomorphism is a group homomorphism works for ξ i with immediate changes; we give the proof for the sake of completeness. Proposition 5.7.6. The map ξ = (ξ 1 , . . . , ξ d ) is a surjective group homomorphism.

Proof. For every f, g ∈ Rec d we have the following equalities:

ξ i (f • g) = α∈R μ d,i ((ν i (f • g)) -1 ({α})) ⊗ α = α∈R μ d,i ((ν i (f ) • g + ν i (g)) -1 ({α})) ⊗ α = β,γ∈R μ d,i (ν i (f ) • g) -1 ({β}) ∩ (ν i (g)) -1 ({γ}) ⊗ (β + γ) = β∈R μ d,i ((ν i (f ) • g) -1 ({β}) ⊗ β + γ∈R d μ d,i ((ν i (g)) -1 ({γ}) ⊗ γ = ξ i (f ) + ξ i (g)
The last equality is given by the fact that λ i is invariant by an element of Rec d , so that g * μ d,i = μ d,i . We deduce that ξ i is a group homomorphism and this implies that ξ i is also a group homomorphism.

To show surjectivity, it is enough to show that {0} i-1 ⊕ (R ⊗(d-1) ⊗ Λ 2 Q R) ⊕ {0} d-i is contained in the image. One can notice that the group R ⊗(d-1) ⊗ ( 2 Q R) is generated by the subset {a 1 ⊗ . . . ⊗ a d-1 ⊗ (α ∧ β) | a j , α, β ∈ [0, 1 2 ]}. Then it is sufficient to remark that for every a 1 , . . . , a d-1 , α, β ∈ [0, 1 2 ] the element a 1 ⊗ . . . ⊗ a d-1 ⊗ (α ∧ β) is the image by ξ i of the standard shuffle of parameters (a 1 , . . . , a d-1 ), i, α, β/2 (see Example 5.7.4). The second equality is the same proof by using Lemma 5.7.13 instead of Lemma 5.7.12.

For every u ∈ R we denote by u the ceiling of u. For every a ∈ R d-1 we denote by a = ( Also for α, α ∈ R + such that αα ∈ R + we have:

|a i | ) = Ψ[i, a,
Ψ i (a, α, β) = Ψ i (a, αα , β) + Ψ i (a, α , β).

In conclusion we obtain that Ψ i is linear in the variable α.

The linearity of Ψ i in the variable a j for 1 ≤ j ≤ d is the same proof as the linearity in the variable α using Lemma 5.7.13 instead of Lemma 5.7.12.

Invariants de certains groupes d'origine dynamique

Résumé. Le but de cette thèse est d'étudier des groupes agissant par isométries par morceaux sur un intervalle en s'intéressant à leur abélianisé. Le cas du groupe des échanges d'intervalles IET a été traité par Arnoux-Fathi et Sah. Tout d'abord, pour tous sous-groupe Γ de R/Z, on identifie l'abélianisé de tous les sous-groupes IET(Γ) du groupe IET ; ainsi que celui de tous les sous-groupes IET (Γ) du groupe des échanges d'intervalles avec renversements IET . L'idée est adapter le morphisme signature sur les groupes de permutations finis en essayant de mesurer l'ensemble des inversions d'un élément.

Par ailleurs on démontre aussi que la signature des groupes des permutations finis s'étend au groupe des permutations de de [0, 1[ qui sont continus en dehors d'un nombre fini de points. Cela a pour conséquence que la classe de Kapoudjian, un élément du second groupe de cohomologie, s'annule.

Ensuite, on se place en dimension d ≥ 1 et on considère le groupe Rec d des permutations du rectangle [0, 1[ d qui bougent un nombre fini de sous-rectangles par translations et qui sont l'identité ailleurs. On démontre que la généralisation naturel des rotations restreintes (qui forment un système de générateur du groupe IET) forme un système générateur de Rec d . Puis on identifie son abélianisé en généralisant le travail fait pour le groupe IET par Arnoux-Fathi et Sah.

Mots-clés : abélianisé, échanges d'intervalles, échanges de rectangles, groupes continus par morceaux, signature.

Invariants of some groups with dynamical origin

Abstract. In this thesis we study groups piecewise acting by isometries on an interval by identifying their abelianization. The case of the Interval Exchange Transformations (IET) group has been done by Arnoux-Fathi and Sah. First, for every subgroup Γ of R/Z, we identify the abelianization of every subgroup IET(Γ) of the group IET and also every subgroup IET (Γ) of the Interval Exchange Transformations group with flips IET . The idea is to adapt the group homomorphism signature on finite permutation groups by measuring the set of inversions of an element.

Also we prove that the group homomorphism signature on finite permutation groups can be extended to the group consisting of permutations of [0, 1[ which are continuous outside a finite number of points. A consequence is the vanishing of an element of the second cohomology group called the Kapoudjian class.

Then, we deal with higher dimension. Let d ≥ 1 and let Rec d be the group of all permutations of [0, 1[ d which are a translation on a finite number of subrectangles and fix the rest of [0, 1[ d . We prove that the natural generalization of restricted rotations (which define a generating subset of IET) define a generating subset of Rec d . Next we identify the abelianization of Rec d by extending the work of Arnoux-Fathi and Sah on IET.

Keywords: abelianization, interval exchanges, rectangle exchanges, piecewise continuous groups, signature.

Image de couverture : Description de l'image de couverture

8

  Un élément de Rec 2 .

0 1 1

 1 Left: A graph of an element of IET (without specifying values at discontinuity points). Right: Its "action" on [0, 1[.

  Left: A graph of an element of IET that contains flips (without specifying values at discontinuity points). Right: Its "action" on [0, 1[.

Γ 2 Z

 2 which is generally different from the second exterior power 2 Z Γ. In Chapter 2 we prove the following: Theorem 2 (See Theorem 2.3.24). There exists an explicit surjective group homomorphism ε Γ : IET(Γ) → Γ whose kernel is equal to the derived subgroup D(IET(Γ)).

Theorem 5 (

 5 see Theorem 4.1.4). Let G be a subgroup of PC which is simple nonabelian and let G be its preimage in S. Then G has exactly 5 normal subgroups, which constitute the following diagram: exchanges A way to generalize the group IET in dimension d for every d ≥ 1, is to replace the interval [0, 1[ by the rectangle [0, 1[ d . Then we can consider the group Rec d consisting of all bijections f of [0, 1[ d such that there exists a finite partition into subrectangles of [0, 1[ d such that f is a translation on every rectangle of this partition. Then for d = 1 we have the equality Rec 1 = IET.

An element of Rec 2 .

 2 In Chapter 5, we begin the study of Rec d by giving an analogue of a restricted rotation called restricted shuffle (see Definition 5.1.3) and proving the following: Theorem 6 (See Theorem 5.1.5). The set of all restricted shuffles is a generating subset of Rec d .

  (see Definition 5.1.3); we denote by T the set of all rectangle transpositions and we prove the following theorem: Theorem 0.0.1 (See theorem 5.1.6). The derived subgroup D(Rec d ) is simple and is generated by its subset T d . It is contained in every nontrivial normal subgroup of Rec d .

Figure 1 . 2 . 1 -

 121 Figure 1.2.1 -Left: Graph of a restricted rotation. Right: representation of a restricted rotation in pictures.

Theorem 1 . 2 . 8 .

 128 Let f ∈ IET and let P be a partition associated with f . Let S = {λ(I) | I ∈ P}. Then f is a product of restricted rotations with type inside S × S.

Figure 1 . 2 . 2 -

 122 Figure 1.2.2 -Left: Graph of a transposition. Right: representation of a transposition in pictures.

Figure 1 . 2 . 3 -

 123 Figure 1.2.3 -Left: Case where min(a, b) = b. Right: Case where min(a, b) = a.

Proposition 1 . 2 .

 12 18. For every f ∈ PC there exist φ, ψ ∈ Homeo([0, 1[) such that f • φ and ψ • f are two elements of IET .

Corollary 1 . 3 . 5 .

 135 The group PC is perfect and thus simple.

Figure 1 2 Z 2 Z

 122 Figure 1.3.1 -How a reflection can be seen as a commutator.

Figure 1 . 3 . 2 -

 132 Figure 1.3.2 -Illustration for the value of the SAF-homomorphism in Proposition 1.3.9

2 ZR

 2 and this gives the conclusion. Lemma 2.2.5. Any transformation f in Ker(ϕ Γ ) can be written as a balanced product of Γ-restricted rotations.

Figure 2 . 2 . 1 -

 221 Figure 2.2.1 -Proof of Lemma 2.2.6 in the case where f and g have non-overlapping support.

  Proposition 1.2.13 give us the next theorem. It is proved by Y.Vorobets in the case Γ = R/Z in [Vor17]. Theorem 2.2.8. Every balanced product of Γ-restricted rotations can be written as a product of Γ-transpositions. Proof. Let (f 1 , f 2 , . . . , f n ) be a balanced tuple of restricted rotations. The proof is by strong induction on the length n of the tuple. If n = 1 then (f 1 ) is a balanced tuple of Γ-restricted rotations, thus f 1 is a Γ-restricted rotation of type (a, a) with a ∈ Γ so it is also a Γ-transposition. For the general case, let (a, b) be the type of f 1 . If a = b then f 1 is a Γ-transposition and (f 2 , f 3 , . . . , f n ) is a balanced tuple of restricted rotations. By the induction assumption we obtain the result. If a = b then there exists

  Such a μ is called a Boolean algebra measure for A in G. Notation 2.3.2. We recall that Itv(Γ) is the set of all intervals [a, b[ with a and b in Γ and 0 ≤ a < b ≤ 1. Let A Γ be the Boolean algebra of subsets of [0, 1[ generated by Itv(Γ). Then A Γ is a Boolean subalgebra of {0, 1} [0,1[ . By noting λ the Lebesgue measure on [0, 1[ we get that λ is a Boolean measure for A Γ in Γ.

Proposition 2 . 3 . 3 .

 233 Let X and Y be two sets, let A be a Boolean subalgebra of {0, 1} X and let B be a Boolean subalgebra of {0, 1} Y . Let G and H be two abelian groups, let μ : A → G be a Boolean algebra measure for A in G and ν : B → H be a Boolean algebra measure for B in H. Let C := A ⊗ B be the Boolean algebra product (generated by subsets of the form a × b with a in A and b in B). Then there exists a unique Boolean algebra measure ω : C → G ⊗ H for C in G ⊗ H such that for every a in A and b in B we have ω(a × b) = μ(a) ⊗ ν(b). Proof. Let ω 1 and ω 2 be two such Boolean algebra measures, thus they are equal on every a × b for a ∈ A and b ∈ B. Let c be an element of C, then there exist k in N and a 1 , . . . , a k in A and b 1 , . . . , b k in B such that c = k i=1

  where D 1 and D 2 are two finite Boolean subalgebras of C then by noting D the Boolean subalgebra generated by D 1 and D 2 we get that D is a finite Boolean subalgebra of C containing c. Thus m D | D 1 is a Boolean measure for D 1 in G ⊗ H which satisfies m D | D 1 (a × b) = μ(a) ⊗ ν(b) for every a in A and b in B with a × b in D 1 . By unicity we get m D | D 1 = m D 1 and the same argument gives m D | D 2 = m D 2 thus m D 1 (c) = m D (c) = m D 2 (c). So by putting ω(c) = m D (c), the map ω is well-defined. also if we take two disjoint elements c and c in C. Then by taking any finite Boolean subalgebra D of C which contains c and c we get m D (c + c ) = m D (c) + m D (c ) = ω(c) + ω(c ) and the value does not depend on D. Thus ω is the wanted Boolean algebra measure. Let now D be a finite Boolean subalgebra of C. Then there exist k, ∈ N and a 1 , . . . , a k ∈ A and b 1 , . . . , b ∈ B such that every d ∈ D is a Boolean combination of a i × b j .

2 Z

 2 finitely additive and satisfies m D (a × b) = μ(a) ⊗ ν(b) for every a ∈ A, b ∈ B such that a × b ∈ D . Notation 2.3.4. By applying the previous proposition with X = Y = [0, 1[ and A = B = A Γ and μ = ν = λ, there exists a unique Boolean algebra measure ω Γ : A Γ ⊗ A Γ → Γ such that for every a, b, c and d in Γ+ with a < b ≤ 1 and c < d ≤ 1 we have ω Γ ([a, b[ × [c, d[) = (ba) ⊗ (dc).

Figure 2 . 3 . 1 -

 231 Figure 2.3.1 -Illustration of how to calculate the value of εΓ on a Γ-transposition f in Proposition 2.3.12. Left: The graph of f . Right: The set E f .

  Then by Proposition 2.3.18 there exists b ∈ Γ such that a = 2b. Thus if we define g as the unique Γ-transposition of type b that swaps [u, u + b[ and [v, v + b[ and h as the unique IET(Γ) that swaps [u, u + b[ with [u + b, u + a[ and [v, v + b[ with [v + b, v + a[. Then f = ghgh and as g 2 = h 2 = Id we deduce that f ∈ D(IET(Γ)).

Lemma 2 . 3 .

 23 20. Let k ∈ N and τ 1 , τ 2 , . . . , τ k be Γ-transpositions with pairwise disjoint support. Then τ 1 τ 2 .

  are the intervals swapped by τ 1 . Let g and h be Γ-transpositions such that g swaps the intervals[u, u + a 2 [ and [v, v + a 2 [ and h swaps the intervals [u + a 2 , u + a 1 [ and [v + a 2 , v + a 1 [. Thus τ 1 = hg.Moreover g and τ 2 are two Γ-transpositions with same type and non-overlapping support. Then by the previous case, the product gτ 2 is in D(IET(Γ)) and its support does not intersect the support of h. Then f = h(gτ 2 ) is the wanted decomposition.

Corollary 2 .

 2 3.22. Let τ and τ be two Γ-transpositions. There exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that τ τ = σh and the support of σ and h are included in the union of the support of τ and the one of τ .Proof. By Lemma 1.2.30 we deduce that f := τ τ has finite order. Hence by Lemma 2.3.21 we obtain the result. Lemma 2.3.23. Let k in N and τ 1 , τ 2 , . . . , τ k be some Γ-transpositions. Then there exist a Γ-transposition σ and h ∈ D(IET(Γ)) such that τ 1 τ 2 . . . τ k = σh.

CHAPTER 3

 3 Abelianization of IET (Γ)Résumé en français. Pour tout sous-groupe IET (Γ) on établit un isomorphisme naturel entre son abélianisé et le 2-groupe {a⊗a[mod 2] | a ∈ Γ} × { ∧ [mod 2] | ∈ Γ} .Ce groupe est un sous-groupe abélien élémentaire de2

Corollary 3 . 1 . 2 .

 312 is an element of PC + and thus it exists a unique finitely supported permutation σ ( f,P) such that σ ( f,P) • ( I∈A( f,P) r I ) • f is right-continuous. Let f ∈ PC (resp. IET ) and P be a partition associated with f . There exists a unique subset A(f, P) ⊂ f (P) such that ( I∈A(f,P) r I ) • f belongs to PC (resp. IET). Definition 3.1.3. The finitely supported permutation σ ( f,P) is called the default of pseudo right-continuity of f according to P. The composition σ ( f,P) • ( I∈A( f,P) r I ) • f is equal to ( I∈A(f,P) r I ) • f . We denote it by f + P and it is called the positive P-substitute of f and f . The dependance on the partition is really important as we can see on Figures 3.1.1 and 3.1.2.

Figure 3 . 1 . 1 -Figure 3 . 1 . 2 -

 311312 Figure 3.1.1 -Positive substitute for a reflection in the case S = {x, a, 1 -(x + a)}.

Figure 3 . 2 . 1 -

 321 Figure 3.2.1 -Left: Set of inversions for a restricted rotation. Right: Set of inversions for a I-reflection

  Figure 3.2.2 we obtain the following: Proposition 3.2.16. Every Γ-reflections of type 2 with ∈ Γ 2 Γ is conjugated to a Γ-restricted rotation of type ( , ).

  Figure 3.2.2

Proposition 3 . 3 . 1 .

 331 Let S and T be two finite subsets of Γ+ . If S ⊂ Vect N (T ) then G S ⊂ G T . More precisely for every partition P into S-intervals of [0, 1[ there exists a refinement Q of P which is a partition into T -intervals of [0, 1[.

Remark 3 . 2 , as F 2 - 2 over F 2 .

 32222 4.6. If Γ has dimension d then {a ⊗ a [mod 2] | a ∈ Γ} has dimension d(d+1) vector space, and { ∧ [mod 2] | ∈ Γ} has dimension d, as F 2 -vector space, so IET (Γ) ab has dimension d(d+3)

Theorem 4. 1 . 2 .

 12 There exists a group homomorphism ε : PC → Z/2Z that extends the classical signature on S fin . Corollary 4.1.3. Let G be a subgroup of PC . Then the Kapoudjian class of G is zero.

Figure 4 . 2 . 1 -

 421 Figure 4.2.1 -Illustrations of the two cases appearing in Lemma 4.2.7. Left: Case where f is orderpreserving on (I ∪J) • and we see that σ( f,Q) ( f (x)) = σ ( f,Q ) ( f (x)). Right: Case where f is order-reversing

Lemma 4. 2 . 8 .

 28 For every f ∈ PC and everyφ ∈ Homeo + ([0, 1[) we have ε( fφ) = ε( f ) = ε(φ f ).

  [0, 1[ d the left half-open square of dimension d. Let B = {e 1 , e 2 , . . . , e d } be the canonical basis of R d and we denote by λ the Lebesgue measure on R. For 1 ≤ i ≤ d let pr i be the orthogonal projection on Vect(e i ) and pr ⊥ i be the orthogonal projection on the hyperplane e ⊥ i . For an element x ∈ R d we use the notation x i = pr i (x). A natural way to generalize left halfopen intervals is to consider elements of the form I 1 × . . . × I d where I i is a left half-open subinterval of [0, 1[. They are called left half-open d-rectangles. In the following, every d-rectangle is supposed to be left half-open. We define the rectangle exchange transformations group of dimension d, denoted by Rec d , as the set of all permutations f of [0, 1[ d such that there exists a finite partition of [0, 1[ d into d-rectangles such that f is a translation on each of these d-rectangles. Elements of Rec d are called d-rectangle exchange transformations.

Figure 5 . 1 . 1 -

 511 Figure 5.1.1 -Left: Examples of restricted shuffles in dimension 2 with different directions. Right: Example of a Rec 2 -transposition.

Theorem 5 . 1 . 6 .

 516 The derived subgroup D(Rec d ) is simple and generated by its subset T d of Rec d -transpositions. It is contained in every in every nontrivial normal subgroup of Rec d .

Theorem 5 . 1 . 7 .

 517 There exists a natural surjective group homomorphism from Rec d onto (R ⊗(d-1) ⊗(2Q R)) d , called the generalized SAF-homomorphism, whose kernel is the derived subgroup D(Rec d ).

Finally in Section 5

 5 .7.4 we consider the subgroup GtG d of Rec d generated by the subset IET d ∪T d . While for d = 1 it is obviously equal to the whole group IET = Rec 1 , a consequence of Theorem 5.1.7 for d ≥ 2 is: Corollary 5.1.8. The group GtG d is a proper normal subgroup of Rec d , which strictly contains D(Rec d ).

Figure 5 . 2 . 1 -

 521 Figure5.2.1 -We assume that {a, b} is a setwise Q-free subset of R as well as {c, d}. Left: A grid-pattern that is not setwise Q-free. Right: A setwise Q-free grid-pattern which refines the left grid-pattern.

  Proof. Let A be the Boolean algebra of subsets of [0, 1[ generated by the set of all left half-open subintervals of [0, 1[. We denote by A ⊗d the Boolean algebra tensor product. It is generated by subsets of the form I 1 × . . . × I d with I i in A for every 1 ≤ i ≤ d. We recall that λ is the Lebesgue measure on R and we denote by R ⊗d the dth tensor power of R over Q. Thanks to Proposition 2.3.3 we know that there exists a Boolean algebra measure μ d for A ⊗d in R ⊗d such that μ d (I 1 × . . . × I d ) = λ(I 1 ) ⊗ . . . ⊗ λ(I d ).

Figure 5 . 4 . 1 -

 541 Figure 5.4.1 -Definition of f test , P test and P test . We now introduce a number of simple definitions in this setting, which for the above test example P test are illustrated in Figures 5.4.2, 5.4.5, 5.4.6 Definition 5.4.1. Let P be a setwise Q-free rectangle partition of [0, 1[ d . The ground of P is the following subset of P:

Figure 5 .

 5 Figure 5.4.2 -Hatched pieces compose the Ground of P test , it is also a city of P test . All grey pieces (hatched or not) compose City(P test ). Full white pieces represent the sky of P test .

Definition 5 .Figure 5 .4. 3 -

 553 Figure 5.4.3 -The set of all grey pieces represents Top(City(P test )) and the set of all hatched pieces represents Work + (P).

Figure 5 .

 5 Figure 5.4.4 -In dimension 3, illustration of a city of a partition where the hatched space represents the site of the partition.

Proposition 5 .

 5 4.7. Let Q be a setwise Q-free grid-pattern of [0, 1[ d . For every f ∈ Rec d such that Q ∈ Π f , there exists a finite product r f of restricted shuffles such that f (Q) ∈ Π r f and C (r f (f (Q))) = ∅.Thanks to Proposition 5.4.7 and Proposition 5.4.5 we deduce Theorem 5.1.5.

Figure 5 .4. 5 -

 55 Figure 5.4.5 -Iterations to find a product r of restricted shuffles such that P test ∈ Π r and Sky(r(P test )) = ∅. On every left pictures, all grey pieces represent the highest city, all grey hatched pieces represent towers whose top's height is the complexity of the partition and all white hatched pieces represent pieces of sky of the partition which are also in the upper work.

Figure 5 .

 5 Figure 5.4.6 -Illustration of what looks like a setwise Q-free partition with an empty sky and how moving pieces inside each tower can lead to a setwise Q-free grid-pattern.

Definition 5 . 5 . 1 . 2 Theorem 5 . 5 . 2 .

 5512552 Let P be a partition into rectangles of [0, 1[ d . A restricted shuffle on P is a restricted shuffle which shuffles two rectangles of P. For n ∈ N * , a n-sequence of restricted shuffles on P is a a sequence (r 1 , . . . , r n ) of restricted shuffles such that for every 1 ≤ i ≤ n the element r i is a restricted shuffle on r i-1 • . . . • r 1 (P). The partition r n • . . . • r 1 (P) is called the image of P by this sequence.Here is the refined version of Theorem 5.1.5, in dimension Suppose d = 2. For every f ∈ Rec d and for every setwise Q-free partition P ∈ Π f , there exists a sequence of restricted shuffles (r 1 , . . . , r n ) on P such that f = r n • . . . • r 1 .

Figure 5 . 5 . 1 -

 551 Figure 5.5.1 -Left: A grid-pattern Q which is not setwise Q-free. Right: A rearrangement of Q which is not the image of Q by a sequence of restricted shuffles on Q.

K

  is a left half-open interval and there exists Ω - C ⊂ Ω -which partitions C. Similarly we can define the subset Ω + C of Ω + which partitions C. Then by Q-freeness we can also ask δ to send Ω - C on Ω + C . Then we define g C ∈ Rec 2 such that: g C (x) = (δ × Id)(x) if pr 2 (x) < C min (P) and pr 1 (x) ∈ C x else

  2.5, we recall that A is the Boolean algebra of subset of [0, 1[ generated by the set of all left half-open subinterval of [0, 1[, and μ d : A ⊗d → R ⊗d is the Boolean algebra measure such that μ d (I 1 × . . . × I d ) = λ(I 1 ) ⊗ . . . ⊗ λ(I d )

Every f ∈

 f Rec d is a piecewise translation; thus we directly see that the pushforward measure f * (μ d ) is equal to μ d . Hence for every 1 ≤ k ≤ d the measure μ d,i (see Definition 5.7.1) also satisfies f * (μ d,i ) = μ d,i .

Definition 5 . 7 . 5 .

 575 The map ξ :Rec d → R ⊗(d-1) ⊗ ( 2 Q R) d defined by: ξ(f ) = (ξ 1 (f ), ξ 2 (f ), . . . , ξ d (f ))

  , a, α](nβ) = Ψ[i, a, m α m ](nβ) = mΨ[i, a, α m ](nβ). Hence: Ψ[i, a, qα](β) = Ψ[i, a, α m ](nβ) = 1 m Ψ[i, a, α](nβ) = Ψ[i, a, α](qb).

α

  |α+α | ](β |α + α | d-1 j=1 |a i | ) +Ψ[i, a, α |α+α | ](β |α + α | α |α| × |α| |α+α | .Then by Lemma 5.7.15 we deduce that:Ψ[i, a, α |α+α | ](β |α + α | d-1 j=1 |a i | ) = Ψ[i, a, α |α| ](β |α + α | × |α| |α+α | d-1 j=1 |a i | ) = Ψ i (a 1 , . . . , a d-1 , α, β).A similar computation can be done to obtain:Ψ i, a, α |α + α | β |α + α | d-1 j=1 |a i | ) = Ψ i (a 1 , . . . , a d-1 , α , β .Hence we have:Ψ i (a 1 , . . . , a d-1 , α + α , β) = Ψ i (a 1 , . . . , a d-1 , α, β) + Ψ i (a 1 , . . . , a d-1 , α , β).Now let assume α + α ∈ R. If α + α is negative then by Definition 5.7.14 we have:Ψ i (a, α + α , β) = Ψ i (a, -(α + α ), -β).

  Minimal and finite order elements in IET . . . . . . . . . . . . . 21 1.2.3.1 Decomposition into elements with disjoint support . . . 21 1.2.3.2 Finite order elements . . . . . . . . . . . . . . . . . . . 22 1.3 Derived subgroup and abelianization . . . . . . . . . . . . . . . . 24 1.3.1 Simplicity of the derived subgroup . . . . . . . . . . . . . . . . . 24 We denote by PC (X) the subgroup of S([0, 1[ consisting of all elements that are continuous outside a finite set. Its subgroup consisting of all piecewise orientation-preserving transformations is denoted PC + .

	1.1 Definitions
	1.1.1 Construction
	The group of interval exchanges IET can be easily define as the group of all bijections of
	[0, 1[ that are continuous outside a finite set, right-continuous and piecewise a translation.
	This implies that we always send a right-open and left-closed interval to another. However
	if we want to allow flips then a right-open and left-closed interval should be sent onto a
	right-closed and left-open interval. Thus the definition of IET is not immediate. One way
	to deal with this is to considered such elements up to a finitely supported permutation.
	The following is the formal construction to do this.
	We denote by S([0, 1[) the group of bijections of [0, 1[ to [0, 1[. This group contains a
	natural normal subgroup denoted by S fin consisting of all finitely supported permutations.
	For every subgroup G of S([0, 1[)/S fin , we denote by G its preimage in S([0, 1[). We use
	this notation because our interests is on subgroups of S([0, 1[)/S fin , also it is easier to
	define subgroups of S([0, 1[).
	Definition 1.1.1.
	1.3.2 Abelianization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
	1.3.2.1 Simple groups . . . . . . . . . . . . . . . . . . . . . . . 25
	1.3.2.2 The second skew-symmetric power . . . . . . . . . . . . 26
	1.3.2.3 Abelianization of IET . . . . . . . . . . . . . . . . . . . 26

Contents 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Topological-full groups . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Subfamilies and tools . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1 Partition associated and combinatorial description . . . . . . . . 16 1.2.2 Subfamilies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2.1 In IET . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.2.2 In IET and IET . . . . . . . . . . . . . . . . . . . . 20 1.2.2.3 In PC . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.3

.

  Contents 2.1 Positive cone of Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Kernel of the SAF-homomorphism's restriction to IET(Γ) . . . 30 2.2.1 Balanced product of Γ-restricted rotations . . . . . . . . . . . . . 30

2.2.2 Ker(ϕ Γ ) is generated by Γ-transpositions . . . . . . . . . . . . . 32 2.3 Description of the abelianization of IET(Γ) . . . . . . . . . . . . 33 2.3.1 Boolean measures . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Creation of a signature . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.3 Description of Ker(ε Γ ) . . . . . . . . . . . . . . . . . . . . . . . . 38 2.1 Positive cone of

  Proposition 3.1.1. Let f ∈ PC (resp. IET ) and let P be a partition associated with f . Then there exist a unique subset A( f, P) ⊂ f (P) and a unique finitely supported per-

mutation σ ( f,P) such that σ ( f,P) • ( I∈A( f,P) r I ) • f belongs to PC + (resp. IET + ) and is right-continuous.

  a 1 |a 1 | , . . . , a d-1 |a d-1 | ). Definition 5.7.16. For every 1 ≤ i ≤ d we define : , . . . , a d-1 , α, β) -→ Ψ[i, (a 1 , . . . , a d-1 ), α |α| ](β |α| Proposition 5.7.17. For every 1 ≤ i ≤ d the map Ψ i is (d + 1)-multilinear over Z. Proof. Let a = (a 1 , . . . , a d-1 ) ∈ R d-1 . The linearity in the variable β is given by Lemma 5.7.11. Let α, α ∈ R + . We have α+α |α+α | ≤ 1 so by Lemma 5.7.12 we obtain:

	Ψ i :	R d+1	-→	(Rec d ) ab
		(a 1 d-1	|a i | ).
				j=1

Ψ[i, a, α+α |α+α | ](β |α + α | d-1 j=1

Z Γ/(2 2 Z Γ).

Proposition 3.1.7. Let n ∈ N and f 1 , f 2 , . . . , f n ∈ IET (Γ). Let P be a partition into Sintervals associated with (f 1 , f 2 , . . . , f n ). Let g 1 = (f 1 ) + P and g i = (f i f i-1 . . . f 1 )

for every 2 ≤ i ≤ n. Then (f n f n-1 . . . f 1 ) + P = g n g n-1 . . . g 1 . Proof. For this proof we denote by r I the I-reflection for every subinterval I of [0, 1[. By iteration it is sufficient to show the result for n = 2. Let f, g ∈ IET (Γ) and P be a partition into S-intervals associated with (f, g). Let n ∈ N such that P = {I 1 , I 2 , . . . , I n } and f (P) = {J 1 , J 2 , . . . , J n } and f + P (I i ) = J i for 1 ≤ i ≤ n. Let 1 ≤ i ≤ n, we notice that

There are 4 cases: 1. If f is order-preserving on I i and g is order-preserving on J i then g • f is orderpreserving on I i so :

If f is order-preserving on I i and g is order-reversing on J i then g•f is order-reversing on I i so :

If f is order-reversing on I i and g is order-preserving on J i then g•f is order-reversing on I i so :

4. If f is order-reversing on I i and g is order-reversing on J i then g•f is order-preserving on I i so :

2 Analogue of the signature

Balanced product of reflections

We give here a first description of D(IET ). It is inspired of the work done for IET with balanced product restricted rotations.

Definition 3.2.1. Let n ∈ N and r 1 , r 2 , . . . , r n be some Γ-reflections. For every ∈ Γ+ let n be the number of Γ-reflections of type among these elements. The tuple (r 1 , r 2 , . . . , r n ) is a balanced tuple of Γ-reflections if 2 divides n for every ∈ Γ+ . We say that a product of Γ-reflections is a a balanced product of Γ-reflections if it can be written as a product of a balanced tuple of Γ-reflections.

Lemma 3.2.2. The set of all balanced products of Γ-reflections is a generating subset of D(IET (Γ)).

Proof. As any element of IET (Γ) is a finite product of Γ-reflections (see Proposition 1.2.16) and as a reflection has order 2 we deduce that every element of D(IET (Γ)) is a balanced product of reflections. Let r and s be two reflections with the same type. Let t be the transposition that swaps the support of r with the support of s, then s = trt -1 . As a reflection has order 2, the product rs is a commutator.

Let n ∈ N and r 1 , r 2 , . . . r n be Γ-reflections such that r 1 r 2 . . . r n is a balanced product of reflections. Then n is even and up to compose with an element of D(IET ) we can assume that r 2i-1 and r 2i have the same type for every 1 ≤ i ≤ n 2 . Thus by the previous case we deduce that r 1 r 2 . . . r n is in D(IET ).

By Proposition 1.2.16 and Lemma 3.2.2 we deduce the following:

Proof. Let a, b ∈ Γ+ . Let r be a restricted rotation of type (a, b) an s be a restricted rotation of type (b, a). Let I and J be the two consecutive intervals permuted by r. Then r is the composition of the I-reflection, the J-reflection and the I ∪ J-reflection thus a product of a reflections of type a with one of type b and one of type a + b. The same is true for s thus we obtain that rs is the product of two reflections of type a, two of type b and two of type a + b; so rs is a balanced product of Γ-reflections.

The group homomorphism

Here we start with the work done in Section 2.3. We remark that if we denote A R the Boolean algebra of subsets of [0, 1[ generated by the set of all intervals [a, b], A R the one generated by the set of all intervals [a, b[ and A fin the one generated by all the singletons {x} then A R is isomorphic to A * R := A R /A fin . This is why we do not make a difference between A Γ and its image in A * R . The notion of inversions as defined in 2.3.6 is no longer relevant because for every reflection r we have E r / ∈ A Γ ⊗ A Γ ; indeed E r is a triangle. We need to be more precise:

The other equality has a similar proof. We denote by f (P) the arrival partition of f associated with P. We know that φ is continuous thus f (P) is in Π φ and we deduce that P ∈ Π φ f . Also φ is order-preserving then R( f, P) = R(φ f, P)). We know that σ (φ, f (P)) = Id then we can notice that φ • σ ( f,P) • f sends the left endpoint of I i to the left endpoint of φ f (I • i ). Then σ (φ f,P) = φσ ( f,P) φ -1 and we deduce that ε(σ

Thanks to Proposition 1.2.18 it is enough to prove that ε| IET is a group homomorphism.

Lemma 4.2.9. The map ε| IET is a group homomorphism.

Proof. Let f, g ∈ IET . Let P ∈ Π f and Q ∈ Π g . For every I ∈ Q (resp. J ∈ P) we denote by α I (resp. β J ) the left endpoint of I (resp. J). Up to refine P and Q we can assume that P = g(Q) thus g({α

Corollary 4.2.10. The map ε is a group homomorphism.

Normal subgroups of PC and some subgroups

Here we present some corollaries of Theorem 4.1.2. For every group G we denote by D(G) its derived subgroup. Definition 4.3.1. For every group H, we define J 3 (H) as the subgroup generated by elements of order 3.

Let G be a subgroup of PC containing S fin . We denote by G its projection on PC . We recall that A fin is a normal subgroup of G, and has a trivial centraliser. We deduce that for every nontrivial normal subgroup H of G contains A fin .

From the short exact sequence:

we deduce the next short exact sequence which is a central extension:

This short exact sequence splits because the signature

Remark 4.3.5. In the context of topological-full groups, the group J 3 (G) appears naturally (with some mild assumptions) and is denoted by A(G) by Nekrashevych in [START_REF] Nekrashevych | Simple groups of dynamical origin, Ergodic Theory Dynam[END_REF]. In some case of topological-full groups of minimal groupoids (see [START_REF]Topological full groups of one-sided shifts of finite type[END_REF]) we have the equality A(G) = D(G) thanks to the simplicity of D(G). In spite of the analogy, it is not clear that the corollary can be obtained as particular case of this result.

CHAPTER 5

Rectangle exchange transformations Here is a second elementary particular case of Theorem 5.1.5, which will also be needed.

Proposition 5.3.2. For all disjoint translation-isometric P, Q rectangles, the Rec d -transposition τ P,Q is a product of restricted shuffles.

Proof. We first prove this in the special case when there exists

In this case we obtain it is a product of two restricted shuffles. Indeed, this is a consequence of the fact that this lemma is true when . Let r 1 be the restricted shuffle in direction i that shuffles P with R (this one send P on Q) and r 2 be the restricted shuffle in direction i that permutes P with S. Then the composition r -1 2 r 1 is equals to the rectangle transposition that permutes P with Q. Now let us prove the general case. Let P and Q be two rectangles which are translationisometric such that P ∩ Q = ∅. Let P i := pr i (P ) and

and by the special case above, we know that t i is a product of two restricted shuffles in direction i. Then s is a finite product of restricted shuffles.

We now consider another special case, of an element of Rec d mapping grid to grid by translating pieces. Recall from Remark 5.3.4 that not every element of Rec d has this form.

Proposition 5.3.3. Every element f ∈ Rec d such that there exists a setwise Q-free gridpattern Q such that f (Q) is a grid-pattern can be written as a finite product of restricted shuffles.

Hence there exists an element g of IET d such that g(f (Q)) = Q. By Proposition 5.3.1 we know that g is a finite product of restricted shuffles. Also as g • f send Q on itself we deduce that g • f is a permutation on every maximal subset of translation-isometric rectangles of Q. Hence it is a product of Rec d -transpositions and by Proposition 5.3.2 we deduce that f is a finite product of restricted shuffles. Remark 5.3.4. For an element of Rec d there does not always exist a grid-pattern associated which is sent to another grid-pattern. For example this is the case for every restricted shuffle σ R,s,i of infinite order such that R = [0, 1[ d-1 .

At this point we are unable to prove Theorem 5.5.2 for arbitrary d. Here are some possible step towards a proof. Definition 5.5.7. Define (S δ ) as the following statement. For every R be finite union of rectangles in [0, 1[ δ . Let P, Q be rectangle partitions of R. Suppose that for each i there is a Q-free subset F i of ]0, 1[ such that for every K ∈ P ∪ Q, we have λ(pr i (K)) ∈ F i . Then one can change Q into P by a finite sequence of shuffles.

Then the statement S d-1 implies Theorem 5.5.2 in dimension d, the argument being an immediate adaptation of the above one.

Indeed, we know that S 1 holds. Here R is just a disjoint union of intervals, and the difficulty is that components of R can have complicated shapes in general. Note that proving (S δ ) immediately reduces to the case when R is connected; however it sounds convenient not to assume R connected in order to set up a proof (e.g., by induction on the number of the rectangles).

The derived subgroup

We recall that T d is the subset of all Rec d -transpositions where d ≥ 1 is the ambient dimension. In this subsection, we prove Theorem 5.1.6, namely that T d is a generating subset of D(Rec d ). We start with some preliminary observations. Lemma 5.6.1. Every element of order 2 in Rec d is a product of Rec d -transpositions with pairwise disjoint support.

For v positive, choose a finite partition W v of X v into rectangles, and let W be the union, for v positive, of all W v . Then f is the (disjoint support) product of all τ K,f (K) for K ranging over W. Proof. (a) Let f ∈ T d and P and R be the two rectangles switched by f . We can decompose P = P 1 P 2 such that P 1 and P 2 are translation-isometric. Let f 1 be the element that switches P 1 with f (P 1 ) and let f 2 be the element that switches P 1 with P 2 and f (P 1 ) with f (P 2 ). Then we have

(b) From Lemma 5.6.1, it follows that the subgroup N generated by T d coincides with the subgroup generated by elements of order 2. By (a),

For d=1, simplicity of D(Rec d ) was proved by Sah [START_REF] Sah | Scissors congruences of the interval[END_REF] and it follows that D(Rec d ) is generated by Rec d -transpositions. Vorobets [START_REF] Yaroslav | On the commutator of the group of interval exchanges[END_REF] more recently reproved simplicity of D(Rec 1 ), by first proving that it is generated by transpositions. Our approach for arbitrary d ≥ 1 is inspired by the latter. Definition 5.6.3. For every ε > 0 we define T ε d as the set of all rectangle transpositions τ K,L such that K ∪ L is contained in a square of length ε. Proof. (a) From usual commutator formulas it follows that in a group, every commutator

Then thanks to Theorem 5.1.5 we deduce that every commutator of elements in Rec d can be written as the product of conjugates of commutators of restricted shuffles. Hence thanks to Lemma 5.6.1 we deduce that it is enough to prove that every commutator of restricted shuffles is a product of elements of order 2. We already saw that this statement is true in dimension 1. Let i, j ∈ {1, 2 . . . , d} and s, s be two restricted rotations and R, R be two (d -1)-subrectangles of [0, 1[ d-1 . We have different cases:

(a) If i = j then for every x ∈ [0, 1[ d and for every k ∈ {1, . . . , d} with k = i, we have

Then by using the result in dimension 1 we deduce that [σ R,s,i , σ R ,s ,i ] is a product of elements of order 2.

Then by using again the equality between commutators we deduce that it is enough to show that the commutator [σ R,s,i , σ R ,s ,i ] is a product of elements of order 2, where R and R are as small as we want. In particular as i = j we can assume that R and R are small enough such that for every

Then in this case the commutator [σ R,s,i , σ R ,s ,i ] permutes cyclically three disjoint rectangles by translations. Hence it is a product of two rectangle transpositions.

(b) Let f be a rectangle transposition. It can be written as disjoint support product of rectangle transpositions, each exchanging two rectangles K, L each of lengths ≤ ε 3 . We can then construct disjoint rectangles K = K 0 , K 1 . . . , K n = L, each congruent to K, such that K i ∪ K i+1 is contained in a square of size ≤ ε for each i. Since τ K,L is contained in the subgroup generated by the τ K i ,K i+1 , this completes the proof.

(c) We can assume that U is a square of size ε. Then each generator as in (b) is conjugate, by a rectangle transposition, to an element supported by this square, and the result follows.

We deduce the simplicity of the derived subgroup D(Rec d ): Proof. Let N be a nontrivial normal subgroup of Rec d normalized by Rec d . Let f be a non-identity element of N . For some ε, there exists a square K of length ε, such that f is a translation on K and such that K and f (K) are disjoint. Let us prove that every rectangle transposition τ P,Q with P ∪ Q ⊂ K belongs to N . By Proposition 5.6.4 (c) this yields the conclusion.

Cut P and Q in two equal halves according to the d-coordinate: let P 1 and Q 1 be their lower halves, and P 2 , Q 2 their upper halves. Then [f, τ P 1 ,Q 1 ] permutes P 1 and Q 1 by translations, permutes f (P 1 ) and f (Q 1 ) by translations, and is identity elsewhere. Let

An inverse of the generalized SAF-homomorphism

The group (R

In order to show that the kernel Ker(ξ) is equal to the derived subgroup D(Rec d ), we construct an inverse of ξ in a similar way Arnoux-Fathi and Sah did in [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF][START_REF] Sah | Scissors congruences of the interval[END_REF].

In the following we use additive notations for (Rec d ) ab . We begin by giving a generating subset of (Rec d ) ab , for that, we see that we can relate every restricted shuffle to a standard shuffle by conjugating with a rectangle transposition: Proposition 5.7.7. Let 1 ≤ i ≤ d and let a j ∈ ]0, 1] for every j ∈ {1, . . . , d} {i} and let α, β ∈ ]0, 1]. Let r be a restricted shuffle in direction i that shuffles two rectangles P and Q such that λ(pr j (P )) = λ(pr j (Q)) = a j for every j = i and λ(pr i (P )) = α and λ(pr i (Q)) = β. We assume that sup(pr i (P )) = inf(pr i (Q)). Then r is conjugate to the standard shuffle of parameters (a 1 , . . . , a i-1 , a i+1 , . . . , a d ), i, α + β, β.

Proof. All maps considered move only in direction i, and hence this is an immediate consequence of the case d = 1 observed in [START_REF] Arnoux | Échanges d'intervalles et flots sur les surfaces, Ergodic theory[END_REF] From Theorem 5.1.5 and Proposition 5.7.7 we deduce the following corollary: Corollary 5.7.8. The group (Rec d ) ab is generated by the subset

In the following, we use a lot of parameters. For the sake of clarity, parameters will be inside brackets and not in index. Definition 5.7.10. Let a ∈ ]0, 1] d-1 and 1 ≤ i ≤ d. For every α ∈ ]0, 1] we define:

We define Ψ[i, a, α] = 0 whenever α = 0 or a j = 0 for at least one 1 ≤ j ≤ d -1. Proof. One can see that R a,i,α,β is the rotation of angle β on P ∪ Q when we look at the projection on Vect(e i ). We recall that for every b ∈ R we have that α{b/α} is equal to b modulo α. Hence for every c ∈ R we have α{(b + c)/α} = α{b/α} + α{c/α} [mod α]. We deduce that for every b, c ∈ R we have R a,i,α,α{(b+c)/α} = R a,i,α,α{b/α} • R a,i,α,α{c/α} , and this concludes.

In the following we prove that Ψ is linear on some other variable for small values.

Lemma 5.7.12. Let a = (a 1 , . . . , a d-1 ) ∈ [0, 1] d-1 and

Proof. The case where α = 0 or α = 0 is immediate. We assume α = 0 and α

. By Lemma 5.7.9 there exist g, h ∈ Rec such that:

We know that R a,i,2β ,β is in the derived subgroup D(Rec) thanks to Theorem 5.1.6.

Hence R a,i,α+α ,β = R a,i,α,β + R a,i,α ,β . We remark that α{β /α} = α {β /α } = β so we have:

Then we can multiply by n this equality and by linearity on the variable β we deduce

, we have:

. Let b = a + ue j and c = a + (ua j )e j . One can notice that for every β ∈ [0, α[ we have:

Thus R b,i,α,β = R a,i,α,β + R c,i,α,β and this concludes.

We extend the definition of

Also for every 1 ≤ j ≤ d -1 such that |qa j | ≤ 1, let b = (a 1 , . . . , a j-1 , qa j , a j+1 , . . . , a d-1 ), we have:

Proof. Let n ∈ Z and m ∈ N * such that q = n m . As |qα| ≤ 1 we have | 1 m |a ≤ 1. By Lemma 5.7.11 and Lemma 5.7.12 we deduce that:

We deduce that Ψ i can be seen as a group homomorphism from R ⊗(d+1) onto (Rec d ) ab .

Lemma 5.7.18. For every 1 ≤ i ≤ d and every a

Proof. Let a = (a 1 , . . . , a d-1 ). By Q-linearity of Ψ it is enough to show the lemma for the case where a 1 , . . . , a d-1 , α, β ∈ [0, 1 2 [ and such that α > β > αβ. Then we remark that β{α/β} = αβ. Then by Lemma 5.7.9 we have:

= R a,i,α,β + R a,i,β,α-β = R a,i,α,β + R a,i,α,α-β + R a,i,2(α-β),α-β = 0.

Thanks to Lemma 5.7.18 and Corollary 5.7.8, we deduce that Ψ i induces a group homomorphism from R ⊗(d-1) ⊗ ( 2 Q R) onto (Rec d ) ab . It is also denoted by Ψ i . Definition 5.7.19. We define the generalized SAF-inverse as the following group homomorphism: Proof. We recall that the group IET is generated by its subset of restricted rotations, see Theorem 1.2.8. For every 1 ≤ i ≤ d and every ε > 0, we define: