Keywords: groups . . . . Mapping class group, Unitary representation, Measured foliation, Almost invariant vector, Boundary representation, Irreducibility

Mapping class groups of closed surfaces with punctures play important roles as prototypes of current research in geometric group theory. The representation theory of a group is a way to understand both the group structure and dynamic properties of that group. While there are massive literatures on finite dimensional (projective) unitary representations of mapping class groups, not so much on infinite dimensional ones. The aim of this thesis is to investigate mapping class groups from the perspective of infinite dimensional unitary representations based on current understanding of mapping class groups in the context of geometric group theory. It has two parts.

In the first part, for a surface, we introduce a family of unitary representations of its mapping class group based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As an application, we obtain an inequality concerning the action of the mapping class group on the Teichmüller space. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-regular representations with respect to geometrical subgroups.

In the second part, for a closed hyperbolic surface, we show that the boundary representation of its mapping class group is ergodic, which generalizes the classical result of Masur on ergodicity of the action of the mapping class group on the projective measured foliation space of the surface. As a corollary, we show that the boundary representation of the mapping class group is irreducible. This confirms a conjecture of Bader-Muchnik in the case of mapping class groups with respect to Thurston measure classes.

Résumé

Les groupes modulaires de surfaces fermées à points masqués jouent un rôle important comme prototypes par la recherche moderne en théorie géométriques des groupes. La théorie des représentations d'un groupe est un moyen de comprendre à la fois la structure du groupe et ses propriétés dynamiques. Bien qu'il existe beaucoup de littérature sur les représentations unitaires (projectives) de dimension finie des groupes modulaires, il y en a beaucoup moins sur celles de dimension infinie. Le but de cette thèse est d'étudier le groupe modulaire du point de vue des représentations unitaires de dimension infinie, dans le contexte de la théorie géométrique des groupes. Ce mémoire comporte deux parties.

Dans la première partie, nous introduisons pour une surface une famille de représentations unitaires de son groupe modulaire, basée sur l'espace des feuillages mesurés. Pour cette famille de représentations, nous montrons qu'aucune d'elles n'a de vecteurs presque invariants. En corollaire, nous obtenons une inégalité concernant l'action du groupe modulaire sur l'espace de Teichmüller. Nous classifions aussi, à équivalence faible près, les représentations unitaires quasi-régulières par rapport à ses sous-groupes géométriques.

Dans la seconde partie, pour une surface hyperbolique fermée, nous montrons que la représentation au bord de son groupe modulaire est ergodique, ce qui généralise un résultat classique de Masur sur l'ergodicité de l'action du groupe modulaire sur l'espaces projectif des feuillages mesurés de la surface. En corollaire, nous montrons que la représentation au bord du groupe de modulaire est irréductible, ce qui démontre une conjecture de Bader-Muchnik dans le cas du groupe modulaire par rapport à la classe des mesures de Thurston.

Mot clés: Groupe modulaire, Représentation unitaire, Feuillage mesuré, Vecteur presque invariant, Représentation au bord, Irréductibilité

Introduction in English

The main goal of this thesis is to understand some infinite dimensional unitary representations of mapping class groups of a surface. For finite dimensional (projective) unitary representations of mapping class groups, one could refer to, for instance, Roberts [START_REF] Roberts | Skeins and mapping class groups[END_REF] or Blanchet-Habegger-Masbaum-Vogel [START_REF] Blanchet | Three-manifold invariants derived from the Kauffman bracket[END_REF] and recent related developments.

Let S = S g,n be a closed, connected, orientable surface of genus g with n punctures.

The mapping class group Mod(S) of S is the group of isotopy classes of orientationpreserving homeomorphisms of S. Mapping class groups play important roles in understanding low-dimensional manifolds. For instance, according to the recent solution of the virtually fibered conjecture (see Agol [START_REF] Agol | The virtual Haken conjecture[END_REF], Wise [START_REF] Daniel | Research announcement: the structure of groups with a quasiconvex hierarchy[END_REF] and Thurston's work [START_REF] Otal | Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3[END_REF]), mapping class groups essentially allow us to construct all closed hyperbolic manifolds in dimension three. Associated to S, there are two well-known spaces that are equipped with a Mod(S)action. First, the space of measured foliations MF(S) of S which is the set of equivalence classes of measured foliations on S and second, the projective measured foliation space PMF(S) which is the quotient of MF(S) by the positive reals R + . Both spaces are important to understand the group structure of Mod(S), see for example [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF]. Our task is to investigate the unitary representations of Mod(S) associated to MF(S) and PMF(S) with respect to certain natural measures.

Almost invariant vectors and mapping class groups

There is a family of measures on MF(S) so that Mod(S) acts on MF(S) ergodically with respect to these measures. Namely the measures classified in Lindenstrauss-Mirzakhani [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF] and Hamenstädt [START_REF]Invariant Radon measures on measured lamination space[END_REF] (see also Section 1.3.1), generalizing the Thurston measure on MF(S) (see Masur [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF],Masur [START_REF]Ergodic actions of the mapping class group[END_REF]). We will call these measures generalized Thurston measures. One then obtains a family of unitary representations of Mod(S) by considering the induced action of Mod(S) on the associated L 2 -space with respect to these measures. One can check that the family of unitary representations considered in Paris [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF] is a special subfamily.

The first chapter investigates these unitary representations of mapping class groups.

Recall that a locally compact group G has Kazhdan's Property (T) if every unitary representation of G that has almost invariant vectors also has a non-zero invariant vector (see Definition 1.1.1 for almost invariant vectors). The following question is still widely considered to be open for surfaces of genus at least 3 (for genus at most 2, see Freedman-Krushkal [START_REF] Freedman | On the asymptotics of quantum SU(2) representations of mapping class groups[END_REF] and Taherkhani [START_REF] Taherkhani | The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups[END_REF]). Question 0.1.1 (Ivanov [START_REF]Fifteen problems about the mapping class groups, Problems on mapping class groups and related topics[END_REF]). Does Mod(S) have Kazhdan's Property (T)?

Note that, becuse of ergodicity, none of these unitary representations defined by generalized Thurston measures on MF(S) can have non-zero invariant vectors. Inspired by Ivanov's question, one can ask whether these representations have almost invariant vectors. The first main result of Chapter 1 indicates that they don't. Namely, Theorem 0.1.2 (Theorem 1.4.1). For a compact surface S = S g,n with 3g+n ≥ 4 and each generalized Thurston measure µ, the associated representation (π µ , L 2 (MF(S), µ)) of Mod(S) does not have almost invariant vectors. This theorem has two applications both for first cohomology and the action of the mapping class group Mod(S) on the Teichmüller space Teich(S) of S, see Corollary 1.4.2 and Corollary 1.4.3.

The proof of Theorem 1.4.1 also enables us to give a classification for a family of quasi-regular unitary representations, which is a stronger version of Corollary 5.5 in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF].

Theorem 0.1.3 (Theorem 1.5.4). Let S = S g,n be a compact surface with 3g + n ≥ 4. Let γ = k i=1 γ i and δ = l i=1 δ i , where {γ i } and {δ i } are two collections of pairwise disjoint, distinct isotopy classes of essential simple closed curves on S.

1. If at least one of k and l is not 3g -3 + n, then the associated unitary representations π γ and π δ are weakly equivalent if and only if γ and δ are of the same topological type (that is, there is a mapping class f so that γ = f (δ)).

2. Suppose that S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If k = 3g -3 + n, then π γ is weakly equivalent to the regular representation λ S .

3. Suppose that S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If k = 3g -3 + n, then π γ is not weakly contained in λ S .

Boundary representations of mapping class groups

We now assume that S = S g and g ≥ 2. As PMF(S) is a quotient of MF(S), the Thurston measure on MF(S) induces a measure ν on PMF(S) by considering a section of the quotient map MF(S) -→ PMF(S) and the coned-off construction of the measure (Dowdall-Duchin-Masur [START_REF] Dowdall | Statistical hyperbolicity in Teichmüller space[END_REF]) (See also Chapter 2 for details). The measure class [ν] is then Mod(S)-invariant. By a standard construction (Bekka-de la Harpe-Valette [START_REF] Bekka | Kazhdan's property (T)[END_REF]), one can construct a unitary representation π = π ν , called the boundary representation of Mod(S). The main result of Chapter 2 is the following Theorem 0.2.1 (Corollary 2.2.15). Let S = S g be a closed surface of genus g ≥ 2.

The boundary representation of Mod(S) on L 2 (PMF(S), ν) is irreducible.

This theorem confirms the following conjecture in the case of Mod(S) with respect to the Thurston measure class [ν] on PMF(S).

Conjecture 0.2.2 (Bader-Muchnik [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF]). Let G be a locally compact group and µ a spread-out probability measure on G. The quasi-regular representation associated to the µ-Poisson boundary of G is irreducible.

In fact, we show the following more general ergodic-type theorem. See Definition 2.2.1 for ergodicity of representations.

Theorem 0.2.3 (Theorem 2.2.14). Let P r o be the radial projection from the Teichmüller space Teich(S) -{o} to the Teichmüller boundary PMF(S). Then there exists a sequence of finite subsets E n ⊂ Mod(S) such that the associated quasi-regular representation π ν is ergodic with respect to (E n , P r o ) and bounded Borel funtions on PMF(S).

Questions

We end the introduction by some loose ends. One corollary of Theorem 1.4.1 is that, for any generalized Thurston measure µ on MF(S), H 1 (Mod(S), π µ ) = H 1 (Mod(S), π µ ).

As one could characterize the Kazhdan's Property (T) by the reduced first cohomology (see Shalom [START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]), one could ask if H 1 (Mod(S), π µ ) = 0 for all generalized Thurston measure µ on MF(S)? Or more generally, Question 0.3.1. For p large enough and a generalized Thurston measure µ, does Mod(S) act on L p (MF(S), µ) via isometries properly?

The above question in fact relates to the following question asked by Hamenstädt [START_REF]Actions of the mappings class group[END_REF] in her ICM talk in 2010:

Question 0.3.2. Does Mod(S) have the Haagerup property?

One motivation for Question 0.3.1 is that, according to a result of Bourdon [START_REF] Bourdon | Cohomologie et actions isométriques propres sur les espaces L p , Geometry, topology, and dynamics in negative curvature[END_REF], for large p, the L p -representation of a finitely generated hyperbolic group G given by the right action on G gives a proper affine isometric action of G. Meanwhile, a special type of generalized Thurston measures is given by the set of vertices in curve graph which is hyperbolic as a graph.

Based on Chapter 2, one could also ask Question 0.3.3. For S = S g (g ≥ 2) and the Thurston measure µ on MF(S), is the associated representation reducible?

The answer to this question in the case of g = 1 is negative due to the linear structure of Mod(S 1 ) = SL(2, Z) and one can not hope it to be true for all generalized Thurston Les deux espaces sont importants pour comprendre la structure de groupe de Mod(S), voir par exemple [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF]. Nous étudions dans ce mémoire les représentations unitaires de 10 Mod(S) associées à MF(S) et PMF(S) par rapport à certaines mesures naturelles.

Vecteurs presque invariants et groupes modulaires

Il existe une famille de mesures sur MF(S) par lesquelles l'action de Mod(S) sur MF(S) est ergodiquement. Ce sont les mesures étudiées dans [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF] et [START_REF]Invariant Radon measures on measured lamination space[END_REF] (voir aussi Section 1.3.1), généralisant la mesure de Thurston sur MF(S) (voir [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF], [START_REF]Ergodic actions of the mapping class group[END_REF]), nous les appellerons mesures de Thurston généralisées. On obtient alors une famille de représentations unitaires de Mod(S) en considérant l'action induite de Mod(S) sur l'espace des fonctions de carré intégrable. On peut vérifies que la famille de représentations unitaires considérée dans [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF] est une sous-famille particulière.

Le premier chapitre étudie ces représentations unitaires du groupe modulaire. Rappelons qu'un groupe localement compact G a la propriété (T) de Kazhdan si toute représentation unitaire de G qui a des vecteurs presque invariants possède un vecteur invariant non nul (voir Définition 1.1.1 pour la notion de vecteurs presque invariants).

La question suivante est encore considérée comme ouverte pour des surfaces de genre de supéreur ou égal à 3 (pour genre au plus 2, voir [START_REF] Freedman | On the asymptotics of quantum SU(2) representations of mapping class groups[END_REF] et [START_REF] Taherkhani | The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups[END_REF]).

Question 0.4.1 (Ivanov [31]). Est-ce que Mod(S) a la propriété (T) de Kazhdan ?

Par ergodicité, aucune de représentations unitaires définies par les mesures de

Representations au bord du groupes modulaires

Nous supposons maintenant que S = S g et g ≥ 2. Comme PMF(S) est un quotient de MF(S), la mesure de Thurston sur MF(S) induit une mesure ν sur PMF(S)

en considérant une section de la projection MF(S) -→ PMF(S) et de la construction naturelle [START_REF] Dowdall | Statistical hyperbolicity in Teichmüller space[END_REF]. La classe de la mesure [ν] est alors Mod(S)-invariante.

Par une construction standard [START_REF] Bekka | Kazhdan's property (T)[END_REF], on peut construire une représentation unitaire

(π = π ν , L

Questions ouvertes

Nous terminons l'introduction par quelques questions connexes. Un corollaire du Théorème 1.4.1 est que, pour toute mesure de Thurston généralisée µ sur MF(S),

On a

H 1 (Mod(S), π µ ) = H 1 (Mod(S), π µ ).
Comme on peut caractériser la propriété (T) de Kazhdan par l'annulation de la première cohomologie réduite (voir [START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]), il est naturel de se demander si H 1 (Mod(S), π µ ) = 0 pour tout Thurston généralisé mesure µ sur MF(S)? Ou plus généralement, Question 0. Il est bien connu que Mod(S) agit ergodiquement sur X par rapport à µ [START_REF] William | Ergodic theory on moduli spaces[END_REF]. La question est de savoir si l'action de Mod(S) sur X est fortement ergodique [START_REF] Schmidt | Kazhdan's property T , strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynam[END_REF],

ou encore, si la mesure µ, qui est construite algébriquement, est l'unique moyenne invariante sur l'action de Mod(S) sur L ∞ (X, µ). Lorsque g = 1, le résultat est connu par [START_REF] Schmidt | Kazhdan's property T , strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynam[END_REF], mais il est ouvert dans les autres cas.

Chapter 1

Almost invariant vectors and mapping class groups

This chapter is taken from [START_REF] Ma | On a family of unitary representations of mapping class groups[END_REF].

Introduction

Let S = S g,n be a compact, connected, orientable surface of genus g with n boundaries, the mapping class group Mod(S) of S is defined to be the group of isotopy classes of orientation-preserving homeomorphisms of S which preserving each boundary components (without the assumption that it should fix each boundary pointwise).

Throughout this paper, (g, n) is assumed to satisfy 3g + n ≥ 4 and a subsurface of S is allowed to be disconnected.

Given a discrete group G, a unitary representation is a pair (π, V ) where V is a Hilbert space and π : G → U (V ) is a homomorphism from G to the group of all unitary operators of V [START_REF] Bekka | Kazhdan's property (T)[END_REF]. Infinite dimensional unitary representations of mapping class groups Mod(S) received a lot of attention recently. In [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF], the author considers unitary representations given by the action of Mod(S) on the curve complex associated to S. See [START_REF] Ellegaard | Cohomology of mapping class groups and the abelian moduli space[END_REF], [START_REF] William M Goldman | Mapping class group dynamics on surface group representations[END_REF] for more topics in this direction.

The group Mod(S) acts on the space of measured foliations MF(S), which is defined as the set of equivalence classes of non-zero measured foliations on S. As the action is ergodic with respect to generalized Thurston measures µ [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF], [START_REF]Ergodic actions of the mapping class group[END_REF], [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF], [START_REF]Invariant Radon measures on measured lamination space[END_REF] (see Section 1.3.1 for a brief description of the measures), one obtains a family of unitary representations by considering the induced action of Mod(S) on the space L 2 (MF(S), µ). It is quite easy to see that the family of unitary representations considered in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF] is a special subfamily. However, unlike representations studied in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF],

Example 1.3.4 will show that some of representations considered here are reducible.

Definition 1.1.1. Let (π, V ) be a unitary representation of a discrete group G. The representation π is said to have almost invariant vectors if for every finite set K ⊆ G and every > 0, there exists v ∈ V such that

max g∈K π(g)v -v < v .
The main result of this paper is about the existence of almost invariant vectors for the representation π µ associated to the action of Mod(S) on L 2 (MF(S), µ). The existence of such vectors for other representations of mapping class group has been discussed in [START_REF] Ellegaard | Mapping class groups do not have kazhdan's property (t)[END_REF].

Theorem 1.1.1 (Theorem 1.4.1). For a compact surface S = S g,n with 3g + n ≥ 4

and each generalized Thurston measure µ, the associated representation π µ of Mod(S)

does not have almost invariant vectors.

The first direct application of this theorem is the following:

Corollary 1.1.2 (Corollary 1.4.2). Let S = S g,n be a compact surface with 3g +n ≥ 4

and µ be a generalized Thurston measure, then

H 1 (Mod(S), π µ ) = H 1 (Mod(S), π µ ),
where π µ is the associated representation of Mod(S).

For the second application, we will obtain a geometric inequality of independent interest concerning the action of Mod(S) on the Teichmüller space Teich(S) of S.

Corollary 1.1.3 (Corollary 1.4.3). Let S = S g,n be a compact surface with 3g +n ≥ 4

and γ be the isotopy class of an essential simple closed curve on S. Then there exists a finite subset {φ 1 , ..., φ n } of Mod(S) consisting of pseudo-Anosov mapping classes and a constant > 0, such that, for every point X in Teich(S), we have:

max i∈{1,2,...,n}    α∈Mod(S).γ e -2 X (α) (e ∆ φ i X (α) -1) 2    ≥ α∈Mod(S).γ e -2 X (α) ,
where ∆ φ i X (α) = X (α)φ i .X (α) and X (α) is the geodesic length of α.

For unitary representations associated to discrete measures on the space of measured foliations, some of them are irreducible and some are reducible. We will discuss irreducible decompositions (See Proposition 1.5.1). We will also use the same method as in the proof of the main theorem, combined with recent results in [START_REF] Bridson | Mapping class groups and outer automorphism groups of free groups are C * -simple[END_REF], [START_REF] Breuillard | C * -simplicity and the unique trace property for discrete groups[END_REF], [START_REF] Bekka | Quasi-regular representations of discrete groups and associated C * -algebras[END_REF], to give a classification for a family of quasi-regular unitary representations (with respect to subgroups), which is a stronger version of Corollary 5.5 in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF]. Recall that, given two unitary representations (π, H) and (φ, K) of a discrete group G, π is weakly contained in φ if for every ξ in H, every finite subset Q of G and > 0, there exist

η 1 , ..., η n in K such that max g∈Q π(g)ξ, ξ - n i=1 φ(g)η i , η i < .
If π is weakly contained in φ and φ is weakly contained in π, then φ and π are said to be weakly equivalent. By Proposition F.1.7 in [START_REF] Bekka | Kazhdan's property (T)[END_REF], Definition 1.1.1 is equivalent to say that the trivial representation is weakly contained in the representation π. We then have the following theorem.

Theorem 1.1.4 (Theorem 1.5.4). Let S = S g,n be a compact surface with 3g + n ≥ 4.

Let γ = k i=1 γ i and δ = l i=1 δ i , where {γ i } and {δ i } are two collections of pairwise disjoint, distinct isotopy classes of essential simple closed curves on S.

1. If at least one of k and l is not 3g -3 + n, then the associated unitary representations π γ and π δ are weakly equivalent if and only if γ and δ are of the same topological type (that is, there is a mapping class f so that γ = f (δ)).

2. Suppose S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If k = 3g-3+n, then π γ is weakly equivalent to the regular representation λ S .

3. Suppose S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If k = 3g -3 + n, then π γ is not weakly contained in λ S .

This paper is organized as follows. Section 1.2 is devoted to preliminary for group cohomology with coefficients in unitary representations. The proof of the main theorem is given in Section 1.4. The proof is divided into two general lemmas: Lemma 1.4.4 and Lemma 1.4.5, and concluded by a technical statement, namely Proposition 1.3.7, concerning actions of subgroups of mapping class groups on MF(S). Section 1.3 is mainly devoted to this proposition and Section 1.5 is for irreducible decompositions and the classification up to weak equivalence.

Cohomology with coefficients in representations

Cohomology and reduced cohomology. For a discrete group G and a unitary representation (V, π), one can talk about both cohomology and reduced cohomology group of G with coefficients in π. Definitions of cohomology and reduced cohomology of discrete groups with coefficients in a representation π are standard, so we refer to [START_REF] Martin | On the first L p -cohomology of discrete groups[END_REF], [START_REF] Ellegaard | Cohomology of mapping class groups and the abelian moduli space[END_REF], [START_REF] Bekka | Kazhdan's property (T)[END_REF]. We briefly recall that one defines the following vector spaces for a unitary representation (V, π):

Z 1 (G, π) . = {b : G → V |b(gh) = b(g) + π(g)b(h), for all g, h ∈ G} ; B 1 (G, π) . ={b ∈ Z 1 (G, π)|there exists v ∈ V, such that for all g ∈ G, b(g) = π(g)v -v}; H 1 (G, π) . = Z 1 (G, π)/B 1 (G, π); H 1 (G, π) . = Z 1 (G, π)/B 1 (G, π),
where the closure in the last one is for uniform convergence. The vector space

H 1 (G, π)(resp. H 1 (G, π))
is the first (resp. reduced) cohomology group with coefficients in π.

Almost invariant vectors.

The following Guichardet's theorem provides a way [START_REF] Martin | On the first L p -cohomology of discrete groups[END_REF]). Let G be a finitely generated discrete group and (V, π) be a unitary representation without nonzero invariant vectors. Then the following two are equivalent:

to determine if H 1 (G) = H 1 (G). Theorem 1.2.1 ([
1. The associated first reduced cohomology is the same as the first cohomology, that is,

H 1 (G, π) = H 1 (G, π);
2. The representation π does not have almost invariant vectors.

One observation is that not having almost invariant vectors is closed under taking limit, more precisely, we have the following lemma. Proof. Suppose that the pair (K, ), where K is a finite subset of G and > 0, is given by the condition that π| W does not have almost invariant vector. Given any element ξ ∈ V -W , there is a sequence of elements {ξ n } ⊆ W such that ξ n → ξ as n → ∞. Then, for n large enough , we have:

max g∈K π(g)ξ -ξ = max g∈K π(g)ξ -π(g)ξ n + π(g)ξ n -ξ n + ξ n -ξ ≥ max g∈K π(g)ξ n -ξ n -2 max g∈K ξ n -ξ ≥ ξ -δ.
V = i α i g i : i |α i | 2 < ∞, α i ∈ C, g i ∈ F 2 .

Generalized Thurston measures and dynamics on measured foliation spaces

In this section we will describe the integral theory on the space of measured foliations and the action of subgroups of mapping class groups on the space of measured foliations. A subgroup of Mod(S) in which all elements except the identity are pseudo-Anosov mapping classes will be called a pseudo-Anosov subgroup.

Measures and L 2 -theory on MF(S).

Generalized Thurston measures on MF(S).

The space of measured foliations MF(S) of a surface S is the set of equivalence classes of transversely measured (singular) foliations on S. Using train tracks, one can show that MF(S) has a piecewise linear integral structure such that Mod(S) acts on it as automorphisms (that is, preserves this piecewise linear integral structure) [START_REF] William P Thurston | The geometry and topology of three-manifolds[END_REF].

Therefore, in such local PL coordinates, Mod(S) acts as linear transformations.

A consequence of this PL structure is that MF(S) can be equipped with a Mod(S)-invariant measure µ T h , called the Thurston measure on MF(S). Moreover, this measure can be generalized to obtain a family of locally finite, ergodic Mod(S)-invariant measures µ

[(R,γ)] T h
on MF(S) for complete pairs (R, γ), which will be called generalized Thurston measures. We present a brief summary of the construction of generalized Thurston measures µ

[(R,γ)] T h
according to [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF].

Let γ = i c i γ i , c i > 0 be a multi-curve on S, that is, γ is a collection of isotopy classes of pairwise distinct, pairwise disjoint essential simple closed curves {γ i } on S so that each curve has been weighted by c i > 0. After fixing a hyperbolic structure on S, one can think a multi-curve γ = i c i γ i , c i > 0 as a collection of simple closed geodesics { γ i } on S with γ i labeled by a positive real number c i , where γ i is the unique geodesic representative in γ i . We will use γ to denote both the formal sum i c i γ i and the subset γ i of S. Cutting S along γ, one obtains a decomposition into a disjoint union

S -γ = T i ,
where {T i } is a collection of subsurfaces of S with boundary smoothly embedded in

S. For R = S i
with {S i } ⊆ {T i }, the pair (R, γ) will be called a complete pair. For a complete pair

(R = S i , γ), define MF(R) = i MF * (S i )
where MF * (S i ) = MF(S i ) 0 S i in which 0 S i is the zero foliation on S i . The space MF(R) can be Mod(R, γ)-embeded on MF(S) via enlarging boundary curves [See [START_REF] Fathi | Thurston's work on surfaces[END_REF], Exposé 6.6 for enlarging curves]. Denote by M(R) the image of this embedding.

This set is endowed with the product measure µ R = µ i T h , where µ i T h is the Thurston measure of S i . Define also

M (R, γ) = {F + γ : F ∈ M(R)} ⊆ MF(S).
The inclusion induces a measure on MF(S), denoted by µ

[(R,γ)] T h
and supported on the set of Mod(S)-orbits of M (R, γ), from the product measure µ R . Special cases are when R = ∅ and γ is the isotopy class of a non-separating curve, or when R = S and γ = ∅. The corresponding measure in the case of R = ∅ is a discrete measure, denoted by µ γ and supported on Mod(S).γ which is regarded as a subset of MF(S), while in the case of γ = ∅ it is exactly the Thurston measure µ T h on MF(S).

The following remarkable theorem indicates that generalized Thurston measures µ

[(R,γ)] T h
are exactly the set of all locally finite, Mod(S)-invariant, ergodic measures on MF(S).

Theorem 1.3.1 (Hamenstädt [START_REF]Invariant Radon measures on measured lamination space[END_REF],Lindenstrauss-Mirzakhani [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF]). Any locally finite Mod(S)-invariant ergodic measure on MF(S), up to a constant multiple, is in the

form of µ [(R,γ)] T h
, where (R, γ) is a complete pair.

Associated L 2 -theory over MF(S).
The case of discrete measures. Recall that when R = ∅, µ

[(R,γ)] T h
is the discrete measure supported on the set Mod(S).γ, where Mod(S).γ is regarded as a subset of MF(S). We will first deal with the case that γ is the isotopy class of an essential simple closed curve on S and denote the measure by µ γ .

Let X γ = C 0 γ (S) be the subset of vertices of the curve complex consisting of Mod(S)•γ. By considering the Dirac measure supported on X γ , one can define the Hilbert space

2 (X γ ). It is clear that 2 (X γ ) is Mod(S)-equivariantly isomorphic to L 2 (MF(S), µ γ ).
On the other hand, let G γ = Mod(S, γ) =Stab γ (Mod(S)) be the set of all elements in Mod(S) that fix γ, then 2 (X γ ) can be further Mod(S)-equivariantly identified with 2 (Mod(S)/G γ ). These two spaces give the same unitary representation of Mod(S), actually we have Theorem 1.3.2 (Paris [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF]). The infinite dimensional unitary representation of Mod(S)

given by 2 (Mod(S)/G γ ) is irreducible.

Remark 1.3.1. This theorem was proved in a more general setting for 1-multi-curves on S, that is, γ = c i γ i with c i = 1 for all i.

Thus, in particular, this representation does not have non-zero invariant vectors.

Meanwhile, the irreducibility also allows us to describe 2 (Mod(S)/G γ ) more geometrically.

The first description of 2

(Mod(S)/G γ ) is classical. For f ∈ 2 (X γ ), let Supp(f ) = {v ∈ X γ : f (v) = 0}. The function f has compactly support if the cardinality of Supp(f ) is finite. Define the subspace W of 2 (X γ ) as the set of elements in 2 (X γ )
which have compact support. As X γ is discrete, the following notation will be used to

represent f ∈ W : f = n i=1 k i α i . Note that W is Mod(S)-invariant and the closure W of W in 2 (X γ ) is then 2 (X γ ) itself
. This description will be used in the proof of the main theorem in the case of discrete measures.

The second description of 2 (Mod(S)/G γ ) needs more explanations. Let Teich(S) be the Teichmüller space of S, and for each point X of Teich(S), define a function on

X γ by f X (α) = e -X (α) , α ∈ X γ
where X (α) is the length of the unique geodesic in the isotopy class α.

Proposition 1.3.3. The function defined above belongs to 2 (X γ ).

Proof. It amounts to say

α∈Xγ e -2 X (α) < ∞.
Thus this proposition is a corollary of the result of Birman-Series [8] or Mirzakhani [START_REF] Mirzakhani | Growth of the number of simple closed geodesics on hyperbolic surfaces[END_REF] about the polynomial growth of simple closed geodesics.

Let W be the subspace of 2 (X γ ) which consisting of finite linear combinations of elements in {f X : X ∈ Teich(S)}. It is also true to see that this subspace is Mod(S)-invariant. Also by irreducibility, the closure W of W is 2 (X γ ).

Remark 1.3.2. The second description gives rise to a parametrization for 2 (X γ ) via the Teichmüller space, thus it can be viewed as a reply to Problem 2.5 in [START_REF] William M Goldman | Mapping class group dynamics on surface group representations[END_REF] for representations under consideration.

For the case of R = ∅ and γ is a general integral multicurve γ = k i γ i with k i ∈ N, Obviously, there is a mapping class s that permutes the γ i 's. Denote H = Mod(S, γ) and H = Mod(S, δ), then we have the exact sequence:

1 → H → H → Z 2 → 1.
That is, H is a normal subgroup of H of index 2. This exact sequence allows us to define a self-map of the left cosets {f H} as follows. Write H as H sH. There are two Mod(S)-invariant bijections:

Mod(S) • γ ↔ {[g] = gH}, Mod(S) • δ ↔ {[f ] = f H }.
As f H = f H f sH, the set {gH} can be rewritten as {f H, f sH}, this reformulation induces a well-defined inversion i :

f H = [f ] → [f s] = f sH. A function φ on G/H = {gH} is called even if for every [g] ∈ G/H, φ([g]) = φ(i([g])) and a function ϕ on G/H is called odd if for every [g] ∈ G/H, ϕ([g]) = -φ(i([g])).
Define V 1 to be the subset of 2 (G/H) consisting of even functions and V 2 to be the subset of 2 (G/H) consisting of odd functions. It is easy to see that such two vector spaces are non-empty, closed and Mod(S)-invariant subspaces of 2 (G/H).

Remark 1.3.3. For any discrete measure mentioned above, the associated unitary representation has no nonzero invariant vectors.

The case of non-discrete measures.

For general measures, we mention one remark.

Remark 1.3.4. If R is nontrivial, ergodicity of the action shows that the associated unitary representation has no nonzero invariant vectors.

Actions of subgroups of Mod(S) on MF(S).

Train tracks and a construction of pseudo-Anosov mapping classes. For later use, we first recall some facts about train tracks and a construction of pseudo-Anosov mapping classes by Thurston. All discussions here are standard and wellknown, we refer to [START_REF] Penner | Combinatorics of train tracks[END_REF], [START_REF] Farb | A primer on mapping class groups[END_REF],[ [START_REF] Fathi | Thurston's work on surfaces[END_REF], Exposé 13], [START_REF]On the geometry and dynamics of diffeomorphisms of surfaces[END_REF] for more details.

A train track τ in a surface S is an embedded smooth graph with extra conditions on vertices. A train track is called recurrent if it supports a positive transverse measure, that is, a measure assigns a positive number to every edge. A transversely recurrent train track is a train track such that every edge has a nontrivial essential transverse intersection with a simple closed curve. A birecurrent train track is thus a train track that both recurrent and transversely recurrent. A maximal birecurrent train track is a birecurrent train track that cannot be a proper subtrack of any other train track.

Any measured foliation is carried by a maximal train track. We only remark here that, for a maximal birecurrent train track τ , the set E(τ ) of all positive transverse measures on τ is a positive linear submanifolds, that is, a subset of some Euclidean space defined by a family of linear equations with the condition that all parameters are positive. For the torus T , the set MF(T ) of linear measured foliations can be covered by four affine charts E(τ i ) associated to four maximal birecurrent train tracks.

We fix these four types of train tracks as blocks and denote them by {τ 1 , τ 2 , τ 3 , τ 4 }.

See [ [START_REF] Penner | Combinatorics of train tracks[END_REF], Section 2.6, Figure 2.6.1] for such four train tracks in the annulus, thus in the torus.

We now sketch a construction of pseudo-Anosov mapping classes given by Thurston Therefore there are four types of train tracks, denoted also by {τ 1 , τ 2 , τ 3 , τ 4 }, so that

L(S, σ) ⊆ 4 i=1 E(τ i ).
A direct computation shows that linear measured foliations on S induced by this flat structure are determined by weights on two edges of τ i 2 1 , thus each L(S, σ) E(τ i ) is parameterized by two free independent parameters. Lemma 1.3.5. Let S = S g,n be a compact surface with 3g + n ≥ 5 and α, β be two curves as above, then each τ i is birecurrent and the set L(S, σ) of linear measured foliations with respect to a flat structure σ constructed as described above is of null

µ T h -measure.
Proof. It is obvious that each τ i is birecurrent. We divide the proof of the rest into two cases according to whether τ i is maximal or not. If τ i is not maximal, then any measured foliation carried by τ i is not maximal [START_REF] Penner | Combinatorics of train tracks[END_REF]. By [ [START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF], Lemma 2.3], E(τ i ) has null µ T h -measure. If τ i is maximal, then, as τ i is a birecurrent train track, E(τ i ) is an open subset of MF(S) and thus every point in E(τ i ) should be determined by weights on 6g -6 + 2n edges of τ i . As remarked above that E(τ i ) L(S, σ) is determined by weights on two edges of τ i 2 1 which can be extended to obtain 6g -6 + 2n free parameters of E(τ i ). That is to say, E(τ i ) L(S, σ) is locally given by x 3 = x 4 = ... = x 6g-6+2n = 0 in R 6g-6+2n whose coordinates is given by {x 1 , ..., x 6g-6+2n }. Therefore,

E(τ i ) L(S, σ) is a null set. Since L(S, σ) ⊆ 4 i=1 E(τ i ), hence L(S, σ) is a null set as well.
Almost properly discontinuous action.

We introduce a concept for a group action on a Borel space (that is, a topological space endowed with a Radon measure) which is weaker than usual properly discontinuous action.

Y = P r -1 (Λ(H))
, where P r : MF(S) → PMF(S) is the projection, the action of

H on MF(S) is thus almost properly discontinuous and µ T h (Y ) = 0.
For other S, we deduce this lemma by first passing to PMF(S) and then using the result of McCarthy-Papadopoulos [START_REF] Mccarthy | Dynamics on Thurston's sphere of projective measured foliations[END_REF] on limit sets. Let φ and ψ be two independent pseudo-Anosov mapping classes obtained by Thurston's construction. By the pingpong lemma, one can construct a free pseudo-Anosov subgroup H generated by some powers of φ and ψ. As remarked before that stable and unstable measured foliations of pseudo-Anosov elements in H are linear measured foliations and L(S, σ) is a closed subset, therefore, by Lemma 1.3.5, the limit set Λ(H) of H, which is defined to be the closure of the set of fixed points of non-trivial elements of H with respect to the action on PMF(S), has the property that

µ T H (P r -1 (Λ(H))) = 0.
On the other hand, one can define the zero set Z(Λ(H))(⊆ PMF(S)) of Λ(H) [START_REF] Mccarthy | Dynamics on Thurston's sphere of projective measured foliations[END_REF].

By combining with facts [See [START_REF] Mccarthy | Dynamics on Thurston's sphere of projective measured foliations[END_REF]

MF(S) = lim L 1 →0,L 2 →∞ B L 1 L 2 (X),
where

B L 1 L 2 (X) = {ν ∈ MF(S) : X (ν) ∈ [L 1 , L 2 ]} is a compact set and, as pointed out in the proof of [[36], Lemma 3.1], B L 1 L 2 (X) ( g∈Mod(S) g.M (R, γ)) is equal to B L 1 L 2 (X) ( n i=1 g i .M (R, γ)
), for some finite set {g 1 , ..., g n } ⊂ Mod(S). Fix a free pseudo-Anosov subgroup H of Mod(S) and take any compact subset K ⊆ g∈Mod(S) g.M (R, γ). Taking L 1 small enough and L 2 large enough, one can assume

K ⊆ B L 1 L 2 (X). We now claim that |{h ∈ H : h.K K = ∅}| < ∞.
Let Z = Mod(S).γ and X : Z → R + . We first claim that there is a finite set

J ⊆ Z such that {h ∈ H : h.K K = ∅} ⊆ {h ∈ H : h.J J = ∅}.
For every element in K can be written as γ + ν such that X (γ) is bounded. If h.K K = ∅, then h(γ) also has bounded X -length and all bounds can be chosen

to be uniform on K, say [a, b]. Since X is a proper map on Z (that is, the inverse of compact set is also compact), J = -1 X ([a, b]
) is then a finite subset of Z containing both h(γ) and γ. So one has {h ∈ H : h.K K = ∅} ⊆ {h ∈ H : h.J J = ∅}.

By the discussion of the case R = ∅, the set {h ∈ H : h.J J = ∅} is finite which implies that the finiteness of |{h ∈ H : h.K K = ∅}|. Now taking the measure zero set to be Y = MF(S) -g∈Mod(S) g.M (R, γ) completes the proof.

H-related cover.

B 1 = {X k ∈ A 1 |X k K = ∅} as well as C 1 = {Y k |Y k = X k K, X k ∈ B 1 }. Delete Y k ∈C 1 X k from K and denote the resulting compact set by K 1 . Then for K 1 , there is a family B 2 = {X k ∈ A 2 |X k K 1 = ∅} as well as C 2 = {Y k |Y k = X k K 1 , X k ∈ B 2 }.
Delete Y k ∈C 2 X k from K 2 and denote the resulting compact set by K 3 . Continuing this process, there is a cover of K which can be written in the following formula:

K ⊆ n k=1 Y i ∈C k Y i .
So K can be covered by finitely many pairwise disjoint µ-measurable sets (we allow some of them to be null sets). This will be called an H-related cover of K , since, for each k, C k is a family of disjoint sets that lie inside the H-orbit of some set.

Nonexistence of almost invariant vectors

Let H(µ) = L 2 (MF(S), µ), where µ = µ

[(R,γ)] T h
is a generalized Thurston measure explained in Section 3.1.1, and π µ be the associated unitary representation of Mod(S).

The main result of this section is the following: Let γ be the isotopy class of an essential simple closed curve on S, X = Mod(S).γ and X be a point in the Teichmüller space Teich(S) of S.

Denoting ∆ φ i X (α) = X (α) - φ i .X (α)
, where α ∈ X, and using the description of 2 (X) via Teich(S) in Section 1.3.1, the following inequality is easy to show: Corollary 1.4.3. Let S = S g,n be a compact surface with 3g + n ≥ 4 and γ be the isotopy class of an essential simple closed curve on S. Then there exists a finite subset {φ 1 , ..., φ n } of Mod(S) consisting of pseudo-Anosov mapping classes and a constant > 0, such that, for every point X in Teich(S), we have:

max i∈{1,2,...,n}    α∈Mod(S).γ e -2 X (α) (e ∆ φ i X (α) -1) 2    ≥ α∈Mod(S).γ e -2 X (α) .
We divide the proof of Theorem 1.4.1 into two lemmas. First we prove a lemma used for discrete measures.

Lemma 1.4.4. Let G be a discrete countable group and X be a discrete set equipped with a G-action. Suppose that there is a rank 2 free subgroup H of G such that H acts on X freely. Then the unitary representation π = 2 (X) of G associated to the action of G on X does not have almost invariant vectors.

Remark 1.4.1. This lemma is well-known, we give an elementary proof here mainly for heuristic purposes.

Definition 1.4.1. Let H be a rank 2 free group and X be a space on which H acts.

Suppose x ∈ X such that the stabilizer Stab H (x) of x is trivial. The image of H under the orbit map H → X, h → h.x is called the 2-tree based at x (with respect to (H, X)).

Proof of Lemma 1.4.4. By Lemma 1.2.3, we can pass to subgroups. For the action of the group H on the space X and any point p ∈ X, consider the 2-tree based at p with respect to (H, X).

Let W be the subspace of 2 (X) consisting of functions with finite support. As W is G-invariant and dense, by Lemma 1.2.2, it is enough to show that (π| W , W ) does not have almost invariant vectors. That is, we have to find (K, ) with the property that

max g∈K π(g)f -f 2 ≥ f 2 , for all f ∈ W.
Since H ∼ = F 2 , as mentioned in Remark 1.2.1, the left regular representation 2 (H)

does not have almost invariant vectors, thus such a pair (K, ) exists for the regular representation. Fix such pair (K, ) for the rest of the proof. Here are two facts.

Facts:

1. For every 2-tree T based at a point, 2 (T) is H-equivariantly isomorphic to 2 (H).

2. Different 2-trees are disjoint and thus, if the support A 1 of f 1 ∈ 2 (X) and the support A 2 of f 2 ∈ 2 (X) are located in different 2-trees, then f 1 and f 2 are orthogonal.

These two facts imply that we only need to deal with 2 -functions on X whose finite support contained in a single 2-tree. In fact, for every f ∈ W , if we decompose its support K f as

K f = n i=1 K f i ,
where K f i lie in different 2-trees and f i is defined to be the restriction of f on such different 2-trees, then

f = n i=1 f i , π(g)f -f 2 = n i=1 π(g)f i -f i 2
, for all g ∈ K.

Note that K ⊆ H is fixed. If the support of f i is contained in a 2-tree T i , by Remark 1.2.1, there exists

g i ∈ K such that π(g i )f i -f i 2 ≥ f i 2 .
Now for every f i , let g i be an element satisfying the above inequality. If two 2trees f i , f j correspond to the same g i = g j , then f i + f j also satisfies that inequality.

As K is finite, denote K = m and so f can be further decomposed, that is, f =

f 1 + f 2 + • • • + f s (s ≤ m) such that f k = j f jk ,
where f jk ∈ {f 1 , ..., f n } and {f jk } j correspond to the same g k ∈ K. We claim that there exists g l ∈ K such that

π(g l )f -f 2 ≥ s f 2 ≥ m f 2 .
Otherwise, since for all g i selected, we have

π(g i )f -f 2 ≥ π(g i )f i -f i 2 ≥ f i 2 , (1.4.1)
then

f 2 = m i=1 m f 2 > m i=1 π(g i )f -f 2 ≥ s i=1 π(g i )f -f 2 ≥ s i=1 f i 2 = f 2 .
The second inequality is the assumption and the last inequality is inequality (1.4.1).

Thus there exists a pair (K, η = K ) such that

max g∈K π(g)f -f 2 ≥ η f 2 , for all f ∈ W.
So the proof of the lemma is completed.

Then we prove a lemma used for non-discrete measures.

Lemma 1.4.5. Let G be a discrete countable group and (X, µ) be a Borel space.

Suppose that G acts on X by measure-preserving homeomorphisms. If there exists a rank 2 free subgroup H of G such that H acts on X almost properly discontinuously and freely, then the unitary representation π = L 2 (X, µ) of G associated to the action of G on X does not have almost invariant vectors.

Proof of Lemma 1.4.5. Also by Lemma 1.2.3, we can pass to subgroups. Fix a null subset Y of X such that H acts on X -Y properly discontinuously. For any point p ∈ X, consider the image of H under the orbit map, given by h -→ h.p.

Since the stabilizer Stab p (H) is trivial, this map is injective. This is the 2-tree based at p with respect to (H, X). Define W to be the G-invariant subspace of L 2 (X, µ)

consisting functions f ∈ L 2 (X, µ) that compactly supported on X -Y . Thus W = L 2 (X, µ) as µ is a Radon measure. So as before, we only need to prove the theorem in the case of (W, π| W ). For each f ∈ W supported on one H-orbit of a measurable set U , that is,

K f ⊆ h∈H h.U,
where K f is the compact support of f and the union is disjoint indexed by H, fix a point p in U and associate an element A f ∈ 2 (T), where T is the 2-tree based on p,

via A f (h.p) = h.U |f | 2 dµ 1 2 . Define K = g ∈ H| g or g -1 ∈ K ,
where K is the same finite subset of H as in Lemma 1.4.4. For f , one has:

K f |π(g)f -f | 2 dµ = h∈H h.U |π(g)f -f | 2 dµ ≥ h∈H h.U |π(g)f | 2 dµ 1 2 - h.U |f | 2 dµ 1 2 2 = h∈H A π(g)f (h.p) -A f (h.p) 2 = h∈H π(g -1 )A f (h.p) -A f (h.p) 2 ,
where the second inequality is the triangle inequality. By Lemma 1.4.4,

max g∈K π(g)f -f 2 ≥ max g∈K h∈H |(π(g)A f )(h.p) -A f (h.p)| 2 = max g∈K π(g)A f -A f 2 ≥ η A f 2 = f 2 ,
where is a multiple of the constant η in Lemma 1.4.4, as in this case we have (1) If the index of G γ in G γ is one, then the associated representation in2 (Mod(S)/G γ ) of Mod(S) is irreducible.

K = 2 K. If the compact set K f is not contained in one H-orbit,
(2) If the index of G γ in G γ is n > 1, then the associated representation of Mod(S) in 2 (Mod(S)/G γ ) is reducible. Proof. (1) is obvious, since the representation 2 (Mod(S)/G γ ) is 2 (Mod(S)/G γ )
which is irreducible by Remark 1.3.1.

Now assume that [G

γ : G γ ] = n > 1. Let X γ = Mod(S)
.γ and Y γ = Mod(S).γ, then X γ is a Mod(S)-equivariant discrete covering space of Y γ of degree n. So every a proper closed Mod(S)-invariant subspace of 2 (X γ ), which implies the reducibility.

Classification up to weak containment.

We first fix some notations. Fix a hyperbolic structure on S. Denote by γ = k i=1 γ i and δ = k i=1 δ i , that is, multi-curves on S with coefficients all of 1s. Such multicurves will be called 1-multi-curves. For any 1-multi-curve γ = k i=1 γ i on S, we will call the union of geodesic representatives of γ a geometric multi-curve and, for any i, the representative α i a geometric component. Denote by G γ (G δ , resp.) the corresponding subgroup of Mod(S), and by π γ (π δ , resp) the associated unitary representation on 2 (Mod(S)/G γ )( 2 (Mod(S)/G δ ), resp.). Let λ S be the regular representation of the mapping class group Mod(S) of S on 2 (Mod(S)). We first recall some definitions which can be found in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF], [START_REF] Bekka | Kazhdan's property (T)[END_REF], [START_REF] Bekka | Quasi-regular representations of discrete groups and associated C * -algebras[END_REF].

Let G be a countable discrete group and H be a subgroup of G, the commensurator of H is defined to be

Com G (H) = g ∈ G : gHg -1 H has finite index in H and gHg -1 .
A discrete group is said to be C*-simple if every unitary representation, which is weakly contained in the regular representation of G, is weakly equivalent to the regular representation. Let γ and δ be geometric multi-curves, then γ and δ are of the same type if there is an element f in Mod(S) such that f (γ) = δ. We say a subgroup H of G has the spectral gap property if the unitary representation 2 (X) associated to the action H X = G/H -{H} does not have almost invariant vectors. In this section, we give a classification for unitary representations of Mod(S) associated to discrete measures.

Lemma 1.5.2. Given a 1-multi-curve γ on S and let m be the number of its geometric components.

1. If m = 3g -3 + n, then G γ is amenable.

2. If 1 ≤ m < 3g -3 + n, then G γ has the spectral gap property.

Proof. If m = 3g -3 + n, then G γ is virtually abelian, thus it is amenable. For other cases, as m < 3g -3 + n, one can cut S along geometric components so that the resulting surface has at least one connected component that admits two pseudo-Anosov mapping classes generating a rank 2 pseudo-Anosov subgroup. Assume components admitting pseudo-Anosov mapping classes are labelled as T 1 , ..., T k , two pseudo-Anosov mapping classes in each Mod(T i ) and the associated rank 2 pseudo-Anosov subgroup are also denoted by ϕ i , ψ i , H i , respectively. Note that pseudo-Anosov homeomorphisms fix boundaries. Then define two maps ϕ and ψ on S (thus their isotopy classes) by extending ϕ = i ϕ i and ψ = i ψ i . Hence the subgroup H generated by ϕ and ψ is a rank 2 free group. Moreover the action of H on the set X γ -{γ} has trivial stabilizers. Otherwise, if an element φ in H fix δ ∈ X γ -{γ}, then by the construction of H, the geometric intersection number of δ and γ is nonzero and thus it intersects one of T i . We cut S along γ so that δ becomes a family of isotopy classes of arcs. Since φ fixes δ, up to some powers of φ, it fixes each resulting isotopy class of arcs. But then it can be shown that, for some i, there is an element in H i that fixes the isotopy class of an essential simple closed curve, which contradicts the assumption that H i is a pseudo-Anosov subgroup. By Lemma 1.4.4, we can conclude that G γ has the spectral gap property.

Lemma 1.5.3 (Theorem A in [START_REF] Bekka | Quasi-regular representations of discrete groups and associated C * -algebras[END_REF]). Let G be a countable discrete group and H be a subgroup of G that has the spectral gap property. Let L be a subgroup of G satisfying λ S . However, for other types of γ, we don't know if π γ is weakly contained in λ S .

And we don't know what can be said about unitary representations corresponding to non-discrete measures on the space of measured foliations.

Chapter 2

Boundary representations of mapping class groups

Introduction

Let S = S g be a closed, connected, orientable surface of genus g. Recall that the mapping class group Mod(S) of S is defined to be the group of isotopy classes of orientation-preserving homeomorphisms of S. Throughout this chapter, the genus g is assumed to be at least One motivation of this thesis is to use geometric objects, such as MF(S) and PMF(S) to understand unitary representations of Mod(S) (see also the first chapter for related topics). Recall that, for a probability measure class-preserving action of G on (X, [ν]), one defines a unitary representation of G on L 2 (X, ν), called a quasiregular representation (see Section 2.2.1 for more details and the reader should not confuse this terminology with the one in the first chapter). Hence, for a probability measure class-preserving ergodic action, it is natural to ask that whether the quasi-regular representation is irreducible. Notice that this is not true for a measurepreserving ergodic action as it always has C1 X as a nontrivial closed invariant subspace. For the ergodic action of Mod(S) on PMF(S) with respect to [ν], we prove: on L 2 (PMF(S), ν), the space of square integrable functions on PMF(S) with respect to the Thurston measure ν, is irreducible.

In fact, we prove an ergodic-type theorem, namely Theorem 2.2.14, for this quasiregular representation and as a by-product of this ergodic-type theorem, we have the classical result of Masur [START_REF]Ergodic actions of the mapping class group[END_REF] on ergodicity of the action (see Remark 2.2.1 for the relation between ergocidity of an action and the associated quasi-regular representation). However, since our work uses Masur's result implicitly, so we don't give a new proof to his result.

The main theorem is related to a question of Bader-Muchnik in the context of random walks on groups. Namely, let G be a discrete group and µ be a probability measure on G. Let (∂G, ν) be the Poisson boundary of G associated to the µ-random walk on G. Then the measure class [ν] is G-invariant, hence defines a quasi-regular representation of G on L 2 (∂G, ν). In [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF], inspired by the cases of free groups and lattices in Lie groups, Bader-Muchnik proposed the following conjecture:

Conjecture 2.1.2 (Bader-Muchnik [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF]). For a locally compact group G and a spreadout probability measure µ on G, the quasi-regular representation associated to the µ-Poisson boundary of G is irreducible.

Before returning to mapping class groups, we first mention briefly some progress on this conjecture. As mentioned above, this conjecture is true for certain random walks on free groups and lattices in Lie groups (see [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF] and references therein). Hence it is true for the mapping class group Mod(S) = SL(2, Z) of closed surface of genus one acting on PMF(S) = S 1 with respect to the Lebesgue measure which is also identified with the Thurston measure on PMF(S). All identifications are Mod(S)-equvariant.

For lattices in Lie groups, one can also deduce the irreducibility from ergodicity of the associated quasi-regular representation (see [START_REF] Boyer | Ergodic boundary representations[END_REF]). The conjecture is then verified in [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF] for the fundamental group of compact negatively curved manifolds with respect to the Patterson-Sullivan measure by Bader-Muchnik. Their result has been further generalized to hyperbolic groups [START_REF] Garncarek | Boundary representations of hyperbolic groups[END_REF] with respect to the Patterson-Sullivan measure by Garncarek and some discrete subgroups of the group of isometries of a CAT (-1) space with non-arithmetic spectrum by Boyer [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF]. Note that in all cases above, the Patterson-Sullivan measure on the Gromov boundary coincides with the Poisson boundary of (G, µ) for some probability measure µ on G. However, Björklund-Hartman-Oppelmayer [START_REF] Björklund | Random walks on dense subgroups of locally compact groups[END_REF] recently showed that there are random walks on some Lamplighter groups and solvable Baumslag-Solitar groups providing counterexamples to this conjecture.

The relationship between the main theorem and above progress is the following. On the one hand, there is a long history on exploiting similarities between mapping class groups and hyperbolic groups which is quite fruitful. To name very few among massive literatures, we mention [START_REF] Masur | Teichmüller space is not Gromov hyperbolic[END_REF], [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF] and [START_REF]Geometry of the mapping class groups. I. Boundary amenability[END_REF]. On the other hand, by [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF], the Thurston measure on PMF(S) is the Patterson-Sullivan measure on the Teichmüller boundary of the Teichmüller space of S which is in the similar situation with the previous known cases. We also mention that Rafi recently announced that PMF(S) equipped the Thurston measure class is the µ-boundary of some random walk on Mod(S).

Outline of the proof. By regarding PMF(S) as the Teichmüller boundary of the Teichmüller space Teich(S) of S, we follow the approach in Boyer-Pittet-Link [START_REF] Boyer | Ergodic boundary representations[END_REF]: irreducibility of a representation is deduced from the ergodicity of the representation (see Section 2.2.1 for definitions), namely Theorem 2.2.14. To prove Theorem 2.2.14, we adapt a criterion (Theorem 2.2.13) in [START_REF] Boyer | Ergodic boundary representations[END_REF]. The bulk of the work is to construct a family of finite subsets of Mod(S) and show that they satisfy the conditions in Theorem 2.2.13. The family of subsets is constructed by carefully choosing elements in Mod(S) with enough hyperbolicity so that the cardinality of these subsets goes to infinity. Actually, we even need the growth to be exponential. The subsets are described before Lemma 2.2.11 relying on Dowdall-Duchin-Masur [START_REF] Dowdall | Statistical hyperbolicity in Teichmüller space[END_REF]. Then by results on counting lattices in Athreya-Bufetov-Eskin-Mirzakhani [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF], one can show that these sets have exponential growth. Then the next step is to verify the convergence and uniform boundedness in Theorem 2.2.13. As the action of Mod(S) on Teich(S)

is not homogeneous, the approach in Boyer-Pittet-Link [START_REF] Boyer | Ergodic boundary representations[END_REF] no longer works. We use ideas in Bader-Muchnik [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF], Boyer [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF]: we first prove the Harish-Chandra estimates in Section 2.4 as in [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF] and then use it to deduce convergence and uniform boundedness (Section 2.5.1). Note that the proof of convergence in [START_REF] Boyer | Ergodic boundary representations[END_REF] is purely Lie theoretic and our proof for mapping class groups is based on the Harish-Chandra estimates. The proof of the Harish-Chandra estimates (Section 2.4) is the novelty in this paper. Instead of doing estimations directly, we first relate it to integration on intersection numbers and then use the map considered in Masur-Minsky [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF] which relates Teich(S) to the curve complex of S to simplify integrations.

Erratum. The last part of the proof of Lemma 2.3.5 is not correct since by our choice of E n , [ζ] might be non uniquely ergodic. But this lemma is correct and Tiozzo-Yang has a proof for it. We also point out that Duchin's proof of Theorem 2.2.7 is unpulished, therefore our proof of Lemma 2.5.4 is not complete. In this section, we will recall ergodic quasi-regular representations and a criterion for showing ergodicity of representations. The reader is referred to [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF], [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF] and [START_REF] Boyer | Ergodic boundary representations[END_REF] for more details.

Quasi-regular unitary representations.

Let G be a locally compact secondcountable group and X be a second-countable Hausdorff topological space. Let ν be a probability Borel measure on X. Assume that G acts on X as homeomorphisms and G preserves the measure class [ν] of ν, namely, G preserves null ν-measure sets . Choose ν ∈ [ν], thus for every γ ∈ G, the measure γ * ν is absolutely continuous with respect to ν and ν is absolutely continuous with respect to γ * ν. Denote the corresponding Radon-Nikodym derivative by c(γ, ν) = dγ * ν dν . One can construct a unitary representation π ν of G on L 2 (X, ν) as follows: For every f ∈ L 2 (X, ν), every

x ∈ X and every γ ∈ G, π ν (γ)f (x) is defined to be π ν (γ)f (x) = f (γ -1 x)c(γ, ν)

1 2 (x).
The representation π ν will be called a quasi-regular (unitary) representation of G. We remark that if ν, µ are in the same measure class, then π ν , π µ are unitary equivalent.

Assume that c(γ, ν) 1 2 is integrable for each γ ∈ G with respect to ν. The Harish-Chandra function Φ associated to π ν is then defined to be the integral

Φ(γ) = π ν (γ)1 X , 1 X L 2 (X,ν) = X c(γ, ν) 1 2 (x)dν(x).
Ergodic quasi-regular representations. From now on, we always assume that G is a discrete group. Let (X, ν), π ν as above and B(L 2 (X, ν)) be the Banach space of bounded operators on L 2 (X, ν). Let e K : K -→ X be a map from a finite subset K of G to X and f : X -→ C be a bounded Borel function. Consider the following elements in B(L 2 (X, ν)):

M f (K,e K ) : L 2 (X, ν) -→ L 2 (X, ν), φ → 1 |K| γ∈K f (e K (g)) π ν (γ)φ Φ(γ) , P 1 X : L 2 (X, ν) -→ L 2 (X, ν), φ → X φdν1 X , m(f ) : L 2 (X, ν) -→ L 2 (X, ν), φ → f φ.
We now introduce an ergodicity for quasi-regular representations which generalizes the usual ergodicity for measure class-preserving group actions. Recall that a sequence

F n ∈ B(L 2 (X, ν)) converges to F ∈ B(L 2 (X, ν)), written as F n → F , in the weak operator topology if, for every φ, ψ ∈ L 2 (X, ν), lim n→∞ F n (φ), ψ L 2 =< F (φ), ψ > L 2 .
Definition 2.2.1 (Boyer-Link-Pittet [START_REF] Boyer | Ergodic boundary representations[END_REF]). Let G, (X, ν), π ν , f as above. Suppose that for every n ∈ N, there is a pair

(K n , e n : K n -→ X) such that K n is a finite subset of G and such that |K n | → ∞ as n → ∞.
The representation π ν is called ergodic with respect to (K n , e n ) and f , if we have the following convergence in the weak operator topology:

M f (Kn,en) → m(f )P 1 X .
Remark 2.2.1. It is easy to see that the ergodicity of a measure class-preserving group action is weaker than the ergodicity of the associated quasi-regular representation.

One could refer to [[12], Proposition 2.5] for its proof.

The following criterion for the ergodicity of a quasi-regular representation is essentially contained in [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF] and summarized in [START_REF] Boyer | Ergodic boundary representations[END_REF].

Theorem 2.2.1 ([12] Theorem 2.2). Let G, (X, ν) as above and π ν be the associated quasi-regular representation of G on L 2 (X, ν). Let L be a length function on G and let (X, d) be a metric space inducing the topology of X. For every n ∈ N, let E n be a symmetric finite subset of G, that is E n = E -1 n , and e n : E n -→ X be a map. Assume that the following conditions hold:

(1) for every g ∈ G, π ν

(g)1 X L ∞ (X,ν) < ∞, (2) lim n→∞ |E n | = ∞, (3) for all Borel subsets W, V ⊂ X such that ν(∂W ) = ν(∂V ) = 0, lim sup n→∞ 1 |E n | | γ ∈ E n : e n (γ -1 ) ∈ W and e n (γ) ∈ V | ≤ ν(W )ν(V ), (4)
for every r ≥ 0, there is a non-increasing function

h r : [0, ∞) -→ [0, ∞) such that lim s→∞ h r (s) = 0 and such that ∀n ∈ N, ∀γ ∈ E n , π ν (γ)1 X , 1 {x∈X:d(x,en(γ))≥r} L 2 Φ(γ) ≤ h r (L(γ)), (5)
sup n M 1 X En 1 X L ∞ (X,ν) < ∞.
Then the quasi-regular representation π ν is ergodic with respect to (E n , e n ) and any

f ∈ H L ∞ (X,ν)
, where H is a vector space generated by {1 U : ν(∂U ) = 0 and U is a Borel subset of X}.

Remark 2.2.2. Thanks to condition (1), the Harish-Chandra function exists for each γ in G.

Proposition 2.2.2 (Bader-Muchnik [START_REF] Bader | Boundary unitary representationsirreducibility and rigidity[END_REF]). Under the assumptions in the above theorem, if moreover ν is a Radon measure, then π ν is irreducible.

Quasi-regular representations of mapping class groups

We now consider quasi-regular representations of mapping class groups and state our main theorem. For more on mapping class groups and Teichmüller theory, we refer a point o ∈ Teich(S) that is considered to be a Riemann surface X via uniformization.

A holomorphic quadratic differential q ∈ H 0 (X, Ω ⊗2 X ) on X is locally of the form q(z)dz 2 such that q(z) is a holomorphic function. Define a norm on q by q = X |q(z)|dxdy and consider the unit open ball B 1 (X) with respect to • . The set QD(X) of holomorphic quadratic differentials is a vector space and can be identified with the cotangent space of Teich(S) at o. There is a homeomorphism π :

B 1 (X) -→ Teich(S)
sending each open unit ray in QD(X) starting at the origin to a Teichmüller geodesic starting at o. The Teichmüller compactification is then the visual compactification by adding ending points in the unit sphere of QD(X) to each ray. The Teichmüller compactification will be denoted by Teich(S). Thus, the boundary ∂Teich(S) of Teich(S) is the unit sphere QD 1 (X).

One could give a geometric description of ∂Teich(S) via projective measured foliations. A measured foliation on S is a singular foliation of S endowed with a transverse measure. The space MF(S) of measured foliations is then the set of equivalent classes of measured foliations where the equivalence is given by Whitehead moves and isotopy. The space MF(S), endowed with the weak topology on measures, is homeomorphic to R 6g-6 . The quotient, called the projective measured foliation space PMF(S) of S, of MF(S) by the nature action of R + is homeomorphic to the 6g -7 sphere S 6g-7 . Both MF(S) and PMF(S) are equipped with a Mod(S)-action.

There is a deep relation between MF(S) and QD(X). Namely, for each holomorphic quadratic differential q, the vertical measured foliation V(q) of q = q(z)dz 2 is the foliation given by the integral curves of the holomorphic tangent vector field on S such that each vector has a value in negative real numbers under q, where the transverse measure is given by integration of |Re √ q|. By a theorem of Hubbard-Masur, the map V that assigns each holomorphic quadratic differential q on X to V(q) is a homeomorphism from QD(X) -{0} onto MF(S). The composition π • V of the map V : QD(X) -{0} -→ MF(S) and the quotient map π : MF(S) -→ PMF(S)

gives the identification of QD 1 (X) with PMF(S). Thus, we will regard PMF(S)

as the boundary of the Teichmüller compactification of Teich(S). The equivalent class of ξ ∈ MF(S) in PMF(S) will be denoted by [ξ]. Any q ∈ QD 1 (X) (hence

[V(q)] ∈ PMF(S)
) determines a Teichmüller geodesic ray g t starting from o, hence, by abuse of terminologies, we will call q and [V(q)] the direction of g t and sometimes write g t as g q t or V(q)(t).

Any isotopy class γ of essential simple closed curves on S defines a (topological) foliation λ(γ). Hence, any weighted isotopy class cγ of essential simple closed curves defines a foliation λ ∈ MF(S). The measured foliation λ, as topological foliation, is the same as λ(γ), but the transverse measure is given by c. Therefore, let C(S) denote the set of isotopy classes of essential simple closed curves, there is an embedding of

C(S) × R + into MF(S).
The image is dense (See Thurston [START_REF]On the geometry and dynamics of diffeomorphisms of surfaces[END_REF]). This embedding enable us to define three functions that we will use. The first one is the intersection number on MF(S). The intersection number i : MF(S) × MF(S) -→ R + is the unique continuous function on MF(S) × MF(S) that extends the geometric intersection number of two essential simple closed curves and satisfies i(cλ, ξ) = ci(λ, ξ)

for every c > 0 (See [START_REF] Rees | An alternative approach to the ergodic theory of measured foliations on surfaces[END_REF], Corollary 1.11). The second one is the extremal length. Let

o = [(X, φ)] ∈ Teich(S)
where X is a Riemann surface. Let γ be the isotopy class of an essential simple closed curve. The extremal length Ext X (γ) of γ in X is defined to be

Ext X (γ) = sup ρ ρ (γ) 2 ,
where ρ runs over all metrics with unit area in the conformal class of X and ρ (γ)

is the infimum of ρ-length of simple closed curves in γ. Then the extremal length Ext X : MF(S) -→ R + is the unique continuous function on MF(S) that extends the extremal length of C(S) and satisfies Ext X (cλ) = c 2 Ext X (λ) for c ∈ R + (See [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF],

Proposition 3). Note that the extremal length in fact is defined on Teich(S)×MF(S),

namely, if [(X, φ)] = [(Y, ψ)] ∈ Teich(S), then Ext X (•) = Ext Y (•). So we will write Ext o (•) rather than Ext X (•) for o = [(X, φ)].
The third one is the hyperbolic length o (γ) which is defined to be the X-length of unique X-hyperbolic geodesic γ in the isotopy class γ. The function o (•) can be uniquely extended as well to MF(S) to obtain a continuous function o on MF(S) [START_REF]Earthquakes are analytic[END_REF]. We will use the following relation:

given a point o in Teich(S), then there exists a constant C = C(o), depending on o,

such that ∀ξ ∈ MF(S), 1 C o (ξ) ≤ Ext o (ξ) ≤ C o (ξ).
Recall that a measured foliation λ is called minimal if it has no simple closed leaves.

Two measured foliations are said to be topologically equivalent if they, as topological foliations, are differ by isotopies and Whitehead moves. A measured foliation ξ is called uniquely ergodic if it is minimal and any measured foliation ζ that topologically

equivalent to ξ is measure equivalent to ξ, that is, [ξ] = [ζ].
When ξ is uniquely ergodic, we will call [ξ] uniquely ergodic. The following two lemmas are essential to our approach using intersection numbers.

Lemma 2.2.3. (Rees [START_REF] Rees | An alternative approach to the ergodic theory of measured foliations on surfaces[END_REF], Theorem 1.12 or Masur [START_REF]Two boundaries of Teichmüller space[END_REF]) Let λ be a uniquely ergodic measured foliation and η be any measured foliation. Then i(λ, η) = 0 if and only if

[λ] = [η].
Lemma 2.2.4 (Masur's criterion [START_REF]Hausdorff dimension of the set of nonergodic foliations of a quadratic differential[END_REF]). Given > 0. If a Teichmüller geodesic ray g t starting from o does not leave Teich (S) eventually, then the direction of g t is uniquely ergodic.

One feature of the Teichmüller compactification is that the action of Mod(S) cannot be extended continuously to Teich(S) [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF]. However, uniquely ergodic measured foliations are nice points in terms of Mod(S)-action in the following sense:

Lemma 2.2.5. (Masur [START_REF]Two boundaries of Teichmüller space[END_REF]) The mapping class group acts continuously on Teich(S)

at uniquely ergodic points on the boundary.

The following Kerckhoff's formula concerning the calculation of Teichmüller distances will be frequently used.

Lemma 2.2.6 (Kerckhoff [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF]).

∀x, y ∈ Teich(S), d T (x, y) = 1 2 sup

[ξ]∈PMF (S) ln Ext x (ξ) Ext y (ξ) .

Hyperbolicity. It was first proved in [START_REF] Masur | Teichmüller space is not Gromov hyperbolic[END_REF] that the Teichmüller space (Teich(S), d T )

is not hyperbolic in the sense of Gromov. However some triangles in (Teich(S), d T )

are indeed thin. We now collect several related results in order to compare neighborhoods in PMF(S) defined by projections of balls in Teich(S) and the ones defined by intersection numbers.

The first result describes triangles with vertices in a thick part of Teich(S). Recall that, for > 0, the -thick part Teich (S) of the Teichmüller space Teich(S) is defined to be

Teich (S) = {y ∈ Teich(S) : ∀c ∈ C(S), Ext y (c) ≥ }.
Theorem 2.2.7 (Duchin [START_REF] Duchin | Curvature, stretchiness, and dynamics[END_REF]). Let > 0 and Teich (S) be the -part of Teich(S).

Then there exists M 0 = M 0 ( ) > 0 and k = k( ) > 0 such that, for every trian-

T hk % [I] . = |{a ≤ s ≤ b : I(s) ∈ Teich (S)}| b -a ≥ θ.
Theorem 2.2.8 (Dowdall-Duchin-Masur [START_REF] Dowdall | Statistical hyperbolicity in Teichmüller space[END_REF]). Given > 0 and 0 < θ ≤ 1, there exist constants D = D( , θ), L 0 = L 0 ( , θ) such that if I ⊂ [x, y] is a geodesic subinterval in Teich(S) of length at least L 0 and at least proportion θ of I is in Teich (S), then for every z ∈ Teich(S), we have

I ∩ N D ([x, z] ∪ [y, z]) = ∅.
The following result will also be used later. Recall that two parametrized geodesics segment δ(t) and δ (t) defined on [a, b] are said to P -fellow travel in a parametrized

fashion if, for every t ∈ [a, b], d T (δ(t), δ (t)) ≤ P .
Theorem 2.2.9 (Rafi [START_REF] Rafi | Hyperbolicity in Teichmüller space[END_REF]). Let > 0. Then there exists P = P ( ) > 0 such that whenever x 1 , x 2 , y 1 , y 2 are in Teich (S) with

d T (x 1 , x 2 ) ≤ 1, d T (y 1 , y 2 ) ≤ 1,
the geodesic segment [x 1 , y 1 ] and [x 2 , y 2 ] are P -fellow travelling.

Boundary representations of mapping class groups.

We are in a position to discuss a special type of quasi-regular unitary representations of mapping class groups. Fix o ∈ Teich(S), we first define a Radon measure ν o on PMF(S). Let ν T h be the Thurston measure on MF(S). For any open subset U ⊂ PMF(S), one defines ν o (U ) to be

ν o (U ) = ν T h ({ξ : [ξ] ∈ U, Ext o (ξ) ≤ 1}) .
One could verify that ∀γ ∈ Mod(S),

γ * ν o = ν γ.o and [ν x ] = [ν y ], ∀x, y ∈ Teich(S).
Therefore, one has

∀x, y ∈ Teich(S), [ξ] ∈ PMF(S), dν x dν y ([ξ]) = Ext y (ξ) Ext x (ξ) 6g-6 2
.

By the definition of extremal length, the function

[ξ] → Exty(ξ) Extx(ξ) 6g-6 2
is well-defined on PMF(S). We have, in particular,

∀γ ∈ Mod(S), [ξ] ∈ PMF(S), dγ * ν o dν o ([ξ]) = Ext o (ξ) Ext γ.o (ξ) 6g-6 2
.

Hence one has a quasi-regular unitary representation π νo of Mod(S) on the Hilbert space L 2 (PMF(S), ν o ). The quasi-regular representation π νo of Mod(S) is called the boundary representation of Mod(S) (with respect to o).

As intersection numbers will be the main tool, we embed PMF(S) into MF(S). For

L(g) = d T (o, g • o).
Inspired by [START_REF] Gekhtman | Stable type of the mapping class group[END_REF] and [START_REF] Dowdall | Statistical hyperbolicity in Teichmüller space[END_REF], we first describe our choice of E n that fits in Theorem 2.2.13. Let g q t be a Teichmüller geodesic ray starting from o in the direction of q ∈ QD 1 (X). For every m > 0, recall that

T hk % [o, g m ] . = |{0 ≤ s ≤ m : g s ∈ Teich (S)}| d T (o, g m ) .
Theorem 2.2.10 ([16] Proposition 5.5). For all 0 < θ < 1, there exists > 0 such that for all o = (X, φ) ∈ Teich(S) lim

R 0 →∞ ν o {q ∈ QD 1 (X) : T hk % [o, g q m ] ≥ θ, ∀m > R 0 } = 1.
We then fix any θ ∈ (0, 1) and take > 0 by the above theorem. We identify QD 1 (X)

with PMF(S) and g q t with V(q)(t). For each R > 0, we define

U (R, θ, ) = {ξ ∈ PMF(S) : T hk % [o, ξ m ] ≥ θ, ∀m > R}.
Then if R 2 ≥ R 1 > 0, we have

U (R 1 , θ, ) ⊂ U (R 2 , θ, ). Define U (θ, ) = ∪ R>0 U (R, θ, ),
then, by Theorem 2.2.10, one has Later, we will prove that E(θ, , n, o, ρ) has exponential growth. We first state one obvious property of the boundary representation. Lemma 2.2.12. Let π ν be the boundary representation of Mod(S). For every g ∈ Mod(S), π νo (g)1 PMF (S) L ∞ (PMF (S),νo) < ∞ Proof. The lemma is an easy consequence of Kerckhoff's formula, namely Lemma ≤ e d T (x,y) .

ν o (U (θ, ))) = 1.
As π νo (g)1 PMF (S) = Exto(ξ)

Extγ•o(ξ) 6g-6
4 , one has

π νo (g)1 PMF (S) L ∞ (PMF (S),νo) ≤ e 6g-6 2 d T (o,γ•o) < ∞.
The following theorem is a slight variant of Theorem 2.2.1 whose proof is the same as its original proof. (3) for every n ρ, there are two sequences of reals {h rn (n, ρ)} and {r n } such that lim n→∞ h rn (n, ρ) = lim n→∞ r n = 0 and such that

∀n ∈ N, ∀γ ∈ E n , π νo (γ)1 PMF (S) , 1 {x∈PMF (S):i(x,en(γ))≥rn} Φ(γ) ≤ h rn (n, ρ), (4)
sup n M 1 PMF (S) En 1 PMF (S) L ∞ (PMF (S),νo) < ∞.
Then the quasi-regular representation π νo is ergodic with respect to (E n , e n ) and any f ∈ H L ∞ (PMF (S),νo) , where

H =< 1 U : ν o (∂U ) = 0
= m(n) large so that U ∩ W n (r m ) = ∅. Now replace π ν (g)1 B , 1 W by π ν (g)1 B , 1
Wn and follow essential the same proof. The proof of [Theorem 2.2, [START_REF] Boyer | Ergodic boundary representations[END_REF]] is modified similarly by taking limit simultaneously with respect to n and r rather than taking limit first on n than on r as done in [START_REF] Boyer | Ergodic boundary representations[END_REF].

Our main result is the following theorem. Proof. We argue as Proposition 6.3 in [START_REF] Garncarek | Boundary representations of hyperbolic groups[END_REF]. By the main Theorem in [START_REF] Kuhn | Amenable actions and weak containment of certain representations of discrete groups[END_REF], we need to verify that the action of Mod(S) on PMF(S) is amenable. This is Proposition 8.1 in [START_REF]Geometry of the mapping class groups. I. Boundary amenability[END_REF] as a corollary of topological amenability of the action of Mod(S) on PMF(S). Proof. As Mod(S) is not amenable for hyperbolic surface S, the trivial representation is not contained in the regular representation. According to the above proposition and the fact that weak containment is transitive, we can conclude the proof.

Notations. We make some conventions for later use.

• S: a genus g ≥ 2, closed, oriented, connected surface;

• h = 6g -6;

• o: the base point in Teich(S) which is chosen to be generic in the sense that Stab o (Mod(S)) = id. Denote ν = ν o and the measure is normalized so that ν(PMF(S)) = 1;

• The projective measured foliation space PMF(S) is regarded as a subset of MF(S) by τ and an element [ξ] in PMF(S) is then written as ξ, so both [ξ] and ξ will be called directions when there are no confusions;

• Fix arbitrary ρ > 0 and assume n ρ;

• • : if A(t), B(t) are two functions, we use the notation A B to mean A(t) B(t) → 1 as t → ∞ and A <B to mean lim t→∞ A(t) B(t) ≤ 1. The notation A >B is defined similarly;

• A ∼ θ B: there is multiplicative constants C 1 > 0, C 2 > 0 depending on θ so that C 1 A ≤ B ≤ C 2 A.
A ≺ θ B: there is a multiplicative constant D = D(θ) > 0 so that A ≤ DB.

And A θ B is defined similarily;

• Denote U = U (θ, ) and E n = E(θ, , n, o, ρ) in the sequel which is described before Lemma 2. In this subsection, we will show that |E n | goes to infinity. In fact, we will show that |E n | grows exponentially. For any Borel subset W of PMF(S), denote by Sect W the union of geodesics starting from o and ending at W . We first recall the following theorem in [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF] in our setting. Let

C(n, ρ) = {γ ∈ Mod(S) : d T (γ • o, o) ∈ (n -ρ, n + ρ)} .
∈ C(n, ρ) : γ • o ∈ Sect W and γ -1 • o ∈ Sect V } Ke hn ν(W )ν(V ).
where K is a constant depending on g, ρ and o. In fact, using the notations in [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF], one has K = 2sinh(hρ) ν(PMF (S)) 2 hm(Mg)

, where m(M g ) is the push forward of the Masur

|E n | < Ke hn . We now show that |E n | > Ke hn . Recall that U (θ, ) = ∪ R>0 U (R, θ, )
with ν(U (θ, )) = 1 and U (S, θ, ) ⊂ U (T, θ, ) for T > S. Let δ 1 > 0 small enough and choose R 0 such that

1 -δ 1 ≤ ν(U (R, θ, )) ≤ 1, ν(∂U (R, θ, )) = 0.
By Theorem 2.3.1 again, for any δ 2 > 0 small enough, one could choose n large enough so that 1 3h ln ln n > R and so that

{γ ∈ C(n, ρ) : γ • o ∈ Sect U (R,θ, ) and γ -1 • o ∈ Sect U (R,θ, ) } ≥ Ke δ 2 e hn (ν(U (R, θ, ))) 2 ≥ Ke δ 2 (1 -δ 1 ) 2 e hn .
On the other hand, by the choice of n and the definition of U (R, θ, ), Recall that U has a full measure. We need a lemma that relates Busemann functions to extremal lengths. Recall that if (X, d X ) is a metric space and ξ is a geodesic ray starting from a point x 0 ∈ X, then the Busemann function associated to the geodesic ray ξ is the function b ξ on X defined by

{γ ∈ C(n, ρ) : γ • o ∈ Sect U (R,θ, ) and γ -1 • o ∈ Sect U (R,θ, ) } ⊂ E n . Therefore, we have |E n | ≥ Ke δ 2 (1 -δ 1 )
1 |E n | γ ∈ E n : P r(γ -1 ) ∈ W and P r(γ) ∈ V ≤ lim sup n→∞ 1 |E n | γ ∈ C(n, ρ) : P r(γ -1 ) ∈ W and P r(γ) ∈ V = lim sup n→∞ |C(n, ρ)| |E n | 1 |C(n, ρ)| γ ∈ C(n, ρ) : P r(γ -1 ) ∈ W and P r(γ) ∈ V ≤ ν(W )ν(V ). 2
b ξ : x → lim t→∞ (d X (x, ξ(t)) -t) .
For (X = Teich(S), d X = d) and ξ be a geodesic ray starting from o, one has, Lemma 2.3.4 (Walsh [START_REF] Walsh | The asymptotic geometry of the Teichmüller metric[END_REF]). If [ξ] is uniquely ergodic, then the Busemann function associated to the geodesic ray in the direction

[ξ] is ∀x ∈ Teich(S), b [ξ] (x) = 1 2 ln Ext x (ξ) Ext o (ξ) .
The following lemma is contained in [[23], Lemma 5.1] and will be used in the proof of uniform boundedness (Section 2.5.1), we include the proof for the sake of completeness.

Lemma 2.3.5. Let n ρ. Then elements in n E n are O-mapping classes, where C depends on R, θ, .

Proof. Let g ∈ E n and R > 0. As

ν(P r(B(g • o, R))) = {η∈P r(B(g•o,R)} Ext g•o (η) Ext o (η) h 2 dν g•o (η)
By Lemma 2.3.4 and the fact that uniquely ergodic measured foliations has a full measure [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF], so we have, for almost every η ∈ P r(B(g

• o, R), d(o, g • o) -2R ≤ - 1 2 ln Ext g•o (η) Ext o (η) ≤ d(o, g • o).
Notice that as such η is dense, the above inequality actually holds for all points in P r(B(g • o, R). By Lemma 2.2.5, Mod(S) acts on the Teichmüller compactification continuously at ν-almost every point, thus on one hand

ν(P r(B(g • o, R))) ≥ ν g•o (P r(B(g • o, R)))e -hd(o,g•o) = ν(P r g -1 •o (B(o, R)))e -hd(o,g•o) .
One the other hand, we have

ν(P r(B(g • o, R))) ≤ e 2hR-hd T (o,g•o) ν g•o (P r(B(g • o, R))) ≤ e 2hR-hd T (o,g•o) ν g•o (PMF(S)) = e 2hR-hd T (o,g•o) ν(PMF(S)) = C(R)e -hd(o,g•o)
.

We now claim that there exists a constant D > 0 independent of g ∈ ∪ n E n such that

ν(P r g -1 •o B(o, R)) ≥ D.
If not, then there is a sequence g

-1 k •o converges to a point [ζ] such that ν(P r g -1 k •o (B(o, R))) tends to 0. As {g k } ⊂ n E n , so by Masur's criterion, namely Lemma 2.2.4, [ζ] is uniquely ergodic, thus ν(P r [ζ] (B(o, R))) is zero, which is impossible since P r [ζ] (B(o, R))
contains a small open subset in PMF(S) and ν is not atomic. Then we have, by Kerckhoff's formula and Lemma 2.4.2,

i 2 (ξ γ , η) = Ext y (η) Ext y (ξ γ ) g,o,θ, Ext q (η) Ext q (ξ γ ) = Ext q (η)e -2d(o,q) ≥ Ext γ•o (η)e -2n ,
which means that, in this case, we always have i

2 (ξ γ , η) g,o, ,θ Ext γ•o (η)e -2n .
-A ln N + D 1 ≤ Ψ(γ) ≥N ≤ -B ln N + D 2 .
Proof. The proof is same as the one in [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF]. We repeat here for completeness. Namely,

Ψ(γ) ≥N = {η∈PMF (S):i(ξγ ,η)≥N } 1 i(ξ γ , η) h 2 dν(η) = R ν η ∈ PMF(S) : 1 i(ξ γ , η) h 2 ≥ t dt = 1 N h 2 1 ν η ∈ PMF(S) : i(ξ γ , η) ≤ 1 t 2 h dt = N 0 1 ν η ∈ PMF(S) : i(ξ γ , η) ≤ 1 t 2 h dt + 1 N h 2 N 0 ν η ∈ PMF(S) : i(ξ γ , η) ≤ 1 t 2 h
dt.

(2.4.2)

By the assumption and ν is a probability measure, one can easily have the conclusion.

A basic example.

Before continuing our discussions, we digress for the case of once-punctured torus Notice that all identifications here are Mod(S)-equivariant. Hence PMF(S) can be represented as {[x : y] : x 2 + y 2 = 0, x, y ∈ R}, or R ∪ {∞}. Teich(S) is then the usual compactification of H 2 . In this case, Mod(S) acts on Teich(S) via linear fractional transformations. For (x, y) ∈ R 2 , the extremal length at o is

Ext o ((x, y)) = x 2 + y 2 ,
hence the image of PMF(S) under τ is the circle. We will ignore the difference between R 2 and R 2 /{I, -I}. For two points (x, y), (p, q) ∈ MF(S), the intersection number is |qx -py|. Write the image of PMF(S) in the form of (sin(θ), cos(θ)), and fix any ξ = (sin(θ 0 ), cos(θ 0 )) ∈ PMF(S). Let M to be small enough, then

{η ∈ PMF(S) : i(ξ, η) ≤ M } = {θ ∈ [0, 2π] : | sin(θ) cos(θ 0 ) -cos(θ) sin(θ 0 )| ≤ M } = {θ ∈ [0, 2π] : | sin(θ -θ 0 )| ≤ M } = {θ ∈ [0, 2π] : -M ≤ sin(θ -θ 0 ) ≤ M }. (2.4.3)
As M is enough small, sin(θ) is almost the same as θ, so there exist constants A and B, so that

AM ≤ ν({η ∈ PMF(S) : i(ξ, η) ≤ M }) ≤ BM, Notice that, when S is S 1,1 , we have h = 6g -6 + n = 6 × 1 -6 + 2 × 1 = 2, hence h 2 = 1.

Approximation by pants curves.

Now we want to prove that the assumption in Lemma 2.4.6 holds, however, instead of proving it directly, we will approximate by pants curves using the map considered in [START_REF] Masur | Geometry of the complex of curves. I. Hyperbolicity[END_REF].

Recall that, thanks to Bers' theorem, there is a constant C 1 = C(g), depending only on the genus g, such that for every point x ∈ Teich(S), there exists a pants differential on o in the direction of [o, t y ], that is, under the above notations, q = q(y, α(y)). Let m i be the modulus of the cylinder C i determined by q, where C i has the same core curve with α i (y). According to the proof of Proposition 2 in [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF],

there is a metric σ in the conformal class of t y such that the core curve of α i (y) has σ-length 1 and area e T m i + A, where A is a constant depending on o. One also has We now summarize all discussions above. There are A > 0, B > 0, depending on , g, o, such that, for every γ ∈ E n , there is a sequence {ξ k (γ) ∈ PMF(S)} satisfying (a) ∀k, ξ k (γ) = [x k (γ) = 3g-3 i=1 α i (γ)] where {α i (γ)} 3g-3 i=1 is a pants decomposition of S;

1 e T m i ≥ Ext ty (α i (y)) ≥ 1 e T m i + A .
(b) For each i, there is t i such that Ae 2t i ≤ Ext o (α i (γ)) ≤ Be 2t i and lim i t i = ∞;

(c) The limit of {ξ k (γ) = Proof. In general, convergence in PMF(S), which is defined by weak convergence of measures, is hard to understand. However, in our case, the convergence ξ k (γ) → ξ γ can be understood as follows (see [START_REF] Hamenstädt | Train tracks and the Gromov boundary of the complex of curves, Spaces of Kleinian groups[END_REF] for backgrounds). Recall that the projective measured foliation space PMF(S) can be identified with the projective measured lamination space PML(S) on S. Regarding both ξ k (γ) and ξ γ as measured laminations, then according to [START_REF] Hamenstädt | Train tracks and the Gromov boundary of the complex of curves, Spaces of Kleinian groups[END_REF], ξ k (γ) converges to ξ γ in the sense of coarse Hausdorff topology. That is, as subsets, the limit ζ of ξ k (γ) contains ξ γ as a sublamination and the complement ζ -ξ γ consists of isolated leaves. For each > 0, take k large enough, then we could find a subsurface F of S, so that, on F , any η ∈ PML(S) with i(η, ξ γ ) is uniformly approximated by ξ k (γ) and outside F , i(η, ξ k γ) ≤ . Therefore such η has intersection number less than N + with ξ k (γ), we then have proved the upper bound for the measure. The left inequality can be proved similarly.

Regularity at pants curves.

We are now in a position to prove that the assumption in Lemma 2.4.10 holds. We first summarize all properties of ξ k (γ) that we really need.

More conventions:

From now on, we will use the hyperbolic length function Set-up 0: Let α = {α 1 , • • • , α 3g-3 } be a pants decomposition of S and consider it to be a measured foliation still denoted by α. Then [α] defines a unit holomorphic quadratic differential q on o, namely the unique q such that [V(q)] = [α]. Let ξ = α o(α) , then ξ is the image of [α] under τ . We denote g t the Teichmüller geodesic defined by q. We assume that for all i ∈ {1, • • • , 3g -3}, o (α i ) is bounded bleow and above, up to multiplicative constants depending only on g, o, , by e T for some T . Let also be the coordinate map in Theorem 2.4.12. Then, on the one hand, by hyperbolic geometry, there is δ 0 > 0 and M 0 > 0, depending on o, such that for all δ ≤ δ 0 and M ≤ M 0 , one has -1 ( (η 0 ) + W 0 (a, M )) ⊂ V.

Notice that one could choose a large enough such that -1 is a homeomorphism on (η 0 )+W 0 (a, M ). Therefore µ(V ) ≥ ν( -1 ( (η 0 )+W 0 (a, M ))). On the other hand, ν( -1 ( (η 0 ) + W 0 (a, M )))

= ν( -1 (W 0 (a, M ))).

The last measure is easy to see to be at least CM

Ergodicity of boundary representation

In this section, we will prove the main theorem: Theorem 2.2.14, namely, Take h(n, ρ) = c(g, o, ρ)

ln n-D a 1 n-c 1 ln ln n+b 1 , we complete the proof.

Uniform boundedness.

In this section, we complete our proof of the main theorem by proving the uniform boundedness. We start by some lemmas comparing of two types of neighborhoods. Now, since d(q, y) ≤ D, by Kerckhoff's formula, we have e -2D Ext q (ξ) ≤ Ext y (ξ), e -2D Ext q (η) ≤ Ext y (η).

Therefore, e -4D Ext q (ξ) Ext q (η) ≤ i 2 (ξ, η).

On the other hand, we have Ext q (ξ) = e -2d(o,q) , i(ξ, η) ≤ Ce -2n , which implies that e -4D Ext q (η)e -2d(o,q) ≤ C 2 e -4n .

That is, e -4D Ext q (η)e 2(n-d(o,q)) ≤ C We now deal with terms IV and II. Take H n = E 2 n ∪ F 2 n . These two terms can be put together to obtain: (2.5.4)

IV + II = 1 |E n | γ∈Hn 1 Ext γ.o (ζ)

  measures, see the examples in Chapter 1. The last question is based on the following observation: the proof of Theorem 1.4.1 uses that some subgroups of Mod(S) act on MF(S) essential freely. One then could ask, what happens for dynamics without this property? For instance, Question 0.3.4. Let S = S g and X = Rep(π 1 (S), SU (2)) be the SU (2)-character variety of the fundamental group π 1 (S). Let µ be the symplectic volume on X which is Mod(S)-invariant. Does the representation of Mod(S), given by the orthogonal complement of C1 X in L 2 (X, µ), have almost invariant vectors? Introduction en Français L'objectif principal de cette thèse est de comprendre certaines représentations unitaires de dimension infinie du groupe modulaire d'une surface. Pour les représentations unitaires (projectives) de dimension finie des groupes modulaires, on peut se référer, par exemple, à Roberts [55] ou Blanchet-Habegger-Masbaum-Vogel [10] et aux récents travaux associés.Soit S = S g,n une surface fermée, connexe, orientable de genre g avec n points marqués. Le groupe modulaire Mod(S) de S est le groupe des classes d'isotopie d'homéomorphismes préservant l'orientation de S. Ces groupes jouent un rôle important dans la compréhension des variétés de petite dimension. La solution récente de la conjecture virtuellement fibrée (voir Agol[START_REF] Agol | The virtual Haken conjecture[END_REF], Wise[START_REF] Daniel | Research announcement: the structure of groups with a quasiconvex hierarchy[END_REF] et le travail de Thurston[START_REF] Otal | Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3[END_REF]) dit que, les groupes modulaires nous permettent essentiellement de construire toutes les variétés hyperboliques fermées en dimension trois.Nous nous intéressons à deux espaces équipés d'une action de Mod(S). Premièrement, l'espace des feuillages mesurés MF(S) de S qui est l'ensemble des classes d'équivalence de feuillages mesurés sur S, et deuxièmement l'espace des feuilletages mesurés projecties PMF(S) qui est le quotient de MF(S) par les réels positifs R + .

Lemma 1 . 2 . 2 .

 122 Let (V, π) be a unitary representation of G and W be a G-invariant vector subspace of V such that the closure W = V . Then π does not have almost invariant vectors if and only if the representation π | W in W does not have almost invariant vectors.

Theorem 1 .

 1 3.2 is not true in general as shown by the following Example 1.3.4. Consider the genus 2 closed surface S, regarded as a quotient along boundaries of holed sphere with four disjoint open disks deleted. Let γ = 2γ 1 +3γ 2 , δ = γ 1 + γ 2 , where γ 1 and γ 2 are isotopy classes of two distinct images of boundaries.

[ 61 ]

 61 . We only discuss Thurston's construction for closed surfaces. For surfaces with boundaries, one can modify the construction without any difficulty. Let S = S g (g ≥ 2)and choose two essential simple closed curves α and β on S so that all connected components of S -α β are open topological disks. For each intersection point p of α and β, one can assign a rectangle to p so that S has a flat structure σ and, with respect to this flat structure, both Dehn twists T α and T β act as affine transformations (since we have flat structure, we can talk about affine transformations) with linear parts given by elements in P SL(2, R). An element in the subgroup of Mod(S) generated by T α and T β is pseudo-Anosov if the associated affine transformation has a hyperbolic linear part.We now mention some facts about the set L(S, σ) of linear measured foliations on S induced by the flat structure σ above. Note that unstable and stable foliations of pseudo-Anosov mapping classes obtained by Thurstion's construction are in L(S, σ) and L(S, σ) is a closed subset of MF(S). If we arrange all rectangles mentioned above on the plane such that α-sides are horizontal and label the rectangles from left to right by {2 1 , 2 2 , ..., 2 m }, then a linear measured foliation F ∈ L(S, σ) is given by parallel lines of the plane and a train track τ in S carrying F has the form that the restriction of τ in each rectangle 2 i is one of τ i and all such τ i appearing in τ are the same.

Theorem 1 . 4 . 1 .

 141 For a compact surface S = S g,n with 3g+n ≥ 4 and each generalized Thurston measure µ, the associated representation π µ of Mod(S) does not have almost invariant vectors. By using Theorem 1.2.1, Remark 1.3.3 and Remark 1.3.4, we have: Corollary 1.4.2. Let S = S g,n be a compact surface with 3g + n ≥ 4 and µ be a generalized Thurston measure, then H 1 (Mod(S), π µ ) = H 1 (Mod(S), π µ ), where π µ is the associated representation of Mod(S). Proof. By Theorem 1.2.1, we only need to show that the representation π µ has no nonzero invariant vectors. The corollary is thus concluded by using Remark 1.3.3 for discrete measures and Remark 1.3.4 for non-discrete measures.

7 . 1 . 4 . 2 . 1 . 5 . 1

 7142151 one can take an H-related cover of K f , then by the orthogonality similar to Fact 2 in Lemma 1.4.4 and follow the last few lines in the proof of Lemma 1.4.4, one can also choose the pair (K , ), where is a suitable multiple of , to complete the proof. Proof of Theorem 1.4.1. As any pseudo-Anosov subgroup acts freely on MF(S), by Lemma 1.4.4 and Proposition 1.3.7, the theorem is true for R = ∅. When R = S or R is of middle type, it is concluded by Lemma 1.4.5 and Proposition 1.3.Remark The same trick can be used to show that representations of mapping class groups in the space of L 2 -functions on the Teichmüller spaces with respect to Weil-Petersson volumes also have no almost invariant vectors. As one can show that such representations do not have non-trivial invariant vectors, we have the same conclusion about corresponding cohomology groups. 1.5 Classification of quasi-regular representations up to weak containment Irreducible decompositions. As pointed out in Section 1.3.1, for unitary representations of mapping class groups associated to discrete measures on the space of measured foliations, both reducible and irreducible ones exist. By examining Example 1.3.4 carefully, one sees that, reducible representations have an irreducible decomposition. For any multi-curve γ = k i=1 c i γ i on S, where c i > 0 for all i, we form γ = k i=1 γ i . Recall that {γ i } is a collection of pairwise disjoint isotopy classes of essential simple closed curves on S. As before, denote by G γ = M od(S, γ) and G γ = M od(S, γ) the corresponding subgroups of Mod(S). Hence G γ is a subgroup of G γ of finite index. Proposition 1.5.1. Let S = S g,n be a compact surface with 3g + n ≥ 4 and γ, γ as above.

2 .

 2 The space of measured foliations MF(S) is the set of equivalence classes of non-zero measured foliations on S. The mapping class group Mod(S) acts on MF(S) and preserves a Radon measure ν, called the Thurston measure on MF(S). Moreover, the space MF(S) is equipped with an R + -action that commutes with the Mod(S)-action. Therefore, Mod(S) acts on the quotient PMF(S), called the projective measured foliation space, of MF(S) by R + preserving a measure class [ν], called the Thurston measure (class) on PMF(S), defined by the Thurston measure on MF(S).

Theorem 2 . 1 . 1 ( 2 .

 2112 See Corollary 2.2.15). Let S = S g be a closed surface of genus g ≥ The quasi-regular unitary representation of the mapping class group Mod(S)

50 2. 2

 502 Quasi-regular unitary representations 2.2.1 Quasi-regular representations of discrete groups.

  gle (x, y, z) with vertices x, y, z in Teich (S) and side lengths a = d T (x, y), b = d T (y, z), c = d T (x, z) all at least M 0 and for every ρ > 0, one has d ≤ kaρ, whenever a + b -c ≤ aρ, where w ∈ T (S) be the unique point in the geodesic [x, z] such that d T (x, w) = d T (x, y) and d = d T (y, w). The next result, generalizing a theorem of Rafi [53], also describes when triangles are thin. We denote N D (A) for a subset A of Teich(S) by the D-neighborhood of A. Recall that a geodesic segment I : [a, b] → Teich(S) has at least proportion θ in Teich (S) if

1 .

 1 each [ξ], define τ (ξ) ∈ MF(S) to be the unique element in [ξ] such that Ext o (τ (ξ)) = Hence, the map τ : PMF(S) -→ MF(S) is a section of the projection π : MF(S) -→ PMF(S). When talking about intersection numbers for two points in PMF(S), we will always use the image of τ . Ergodic boundary representation. From now on, let S = S g (g ≥ 2) be a genus g closed, orientable surface and fix a point o = [(X, φ)] ∈ Teich(S). Normalize ν o to be a probability measure. Denote h = 6g -6 and let > 0 and θ > 0. Let also L be the length function on G induced by the Teichmüller distance d T , namely

Furthermore, after a

  suitable choice of θ, one has ν o (∂(U (θ, ))) = 0 and by Masur's criterion (Lemma 2.2.4), the set U ( , θ) consists of uniquely ergodic directions. We now fix the choice of and θ and for γ ∈ Mod(S), denote the direction determined by the oriented geodesic [o, γ •o] by ξ γ . Now we are in a position to describe E n . Fix ρ > 0 and let L 0 = L 0 (θ, ) be the constant as Theorem 2.2.8. For 1 3h ln ln n > max {L 0 , ρ}, define the set E(θ, , n, o, ρ) to be the set of all elements γ in Mod(S) satisfying:(a) d(γ • o, o) ∈ (n -ρ, n + ρ); (b) Both ξ γ and ξ γ -1 are in U (θ, ); (c) If g(t)is either the geodesic ray ξ γ (t) or ξ γ -1 (t), then the segment [o, g( 1 3h ln ln n)] has at least proportion θ in Teich (S).

Lemma 2 . 2 . 11 .

 2211 Let n large enough as before. Then for γ ∈ E(θ, , n, o, ρ), there exists a geodesic segment I γ of length 1 3h ln ln n in the geodesic [o, γ • o] that has at least proportion θ in Teich (S) and containing γ • o. Proof. Let γ ∈ E(θ, , n, o, ρ). Since the geodesic ray ξ γ -1 (t) satisfies (c) in the definition of E(θ, , n, o, ρ), the first segment I γ of [o, γ -1 • o] of length 1 3h ln ln n has at least proportion θ in Teich (S). As γ • [o, γ -1 • o] = [γ • o, o], therefore, the geodesic [o, γ • o] has a subinterval γ • I γ -1 of length 1 3h ln ln n that at least proportion θ in Teich (S) and starting at point γ • o.

2. 2 . 6 ,

 26 on Teichmüller distances. By Lemma 2.2.6, ∀x, y ∈ Teich(S), ∀[ξ] ∈ PMF(S), Ext x (ξ) Ext y (ξ) 1 2

Theorem 2 . 2 . 13 .( 1 )

 22131 Let π νo be the associated quasi-regular representation of Mod(S) on L 2 (PMF(S), ν o ). Let i be the intersection number function defined on PMF(S) induced by the section τ : PMF(S) -→ MF(S). Let n ρ and letE n = E n (ρ) ⊂ {g ∈ Mod(S) : d T (o, g • o) ∈ [n -ρ, n + ρ]} be symmetric Let e n = P r : E n -→PMF(S) be the radial projection from o. Assume that the following conditions hold:lim n→∞ |E n | = ∞,(2)for all Borel subsets W, V ⊂ PMF(S) such that ν o (∂W ) = ν o (∂V ) = 0, lim sup n→∞ 1 |E n | | γ ∈ E n : e n (γ -1 ) ∈ W and e n (γ) ∈ V | ≤ ν o (W )ν o (V ),

Corollary 2 . 2 . 18 .

 2218 The trivial representation 1 does not weakly contained in the boundary representation π νo . In other words, the boundary representation π νo does not have almost invariant vectors.

  P r y : Teich(S) -{y} -→ PMF(S): the radial projection from Teich(S) to PMF(S) that assigns every point z ∈ Teich(S) -{y} to the vertical measured foliation of the unit quadratic differential defined by the oriented geodesic [y, z]. For y = o, we simply denote P r o to be P r; • B(y, R): the closed ball in Teich(S) of radius R at y with respect to the Teichmüller distance d = d T . L: the length function on G induced by the Teichmüller distance d T through the orbit map: g → g • o;

2 . 11 ;

 211 • ξ γ ∈ PMF(S) (for γ ∈ Mod(S) -{id}): the direction of the oriented geodesic segment [o, γ • o].

2. 3

 3 Exponential growth and shadow lemma 2.3.1 Exponential growth.

Theorem 2 .

 2 3.1 ([4] Theorem 2.10). Let W and V be two Borel subsets of PMF(S) with measure zero boundaries. Then as R tends to ∞, {γ

Proof.

  First we remark that, since both η and ξ γ are uniquely ergodic, by [[34], Proposition 5.1], there is a geodesic whose horizontal and vertical measured foliations are in the projective classes ξ γ and η respectively. Hence we have a geodesic triangle (o, ξ γ , η). As γ ∈ E n , Lemma 2.2.11 implies that there is a geodesic segment I of length = 1 3h ln ln n in [o, γ • o] ending at γ • o that has at least proportion θ in Teich (S). By Theorem 2.2.8,I ∩ N D ([o, ξ γ ] ∩ [o, η]) = ∅,where D comes from Theorem 2.2.8. Choose q ∈ I ∩ N D ([o, ξ γ ] ∩ [o, η]). Then there are two possibilities: Case 1: d(q, y) ≤ D with y ∈ [ξ γ , η].

Case 2 :

 2 d(q, y) ≤ D with y ∈ [o, η]. Then we have then there exist A, B, D 1 , D 2 such that

S 1 , 1 .

 11 Some standard facts are taken from [[49], 7.2 Examples]. Let S = S 1,1 . Then Mod(S) = SL(2, Z) and Teich(S) = H 2 , the upper half plane.Take o to be i ∈ H 2 . The space MF(S) of measured foliations can be identified with the real plane module the inversion, namely {R 2 -(0, 0)}/{I, -I}. By the ergodicity of the Thurston measure ν T h , up to a constant multiple, the measure ν T h , which is defined by the weak limit of counting measures on MF(S), can be identified with the Lebesgue measure on R 2 . Rays in {R 2 -(0, 0)}/{I, -I}} are then identified with points in PMF(S). It implies that PMF(S) can be identified with RP 1 .

  Now consider the metric Σ = f * σ on o defined by the pullback of σ via the Teichmüller mapping f . As f preserves the area but shrink the vertical length, the core curve ofα i (y) has Σ-length e T . Thus Ext o (α i (y)) ≥ e 2T e T m i + A .As e T m i ≺ ,g 1,one then further hasExt o (α i (y)) g, ,o e 2T .

  x k (γ) √ Exto(x k (γ)) } in PMF(S) is ξ γ .Lemma 2.4.10. If there exists N 0 > 0 small enough such that, for every γ ∈ E n and ξ k (γ), one has∀N ≤ N 0 , aN h 2 ≤ ν({η ∈ PMF(S) : i(η, ξ k (γ)) ≤ N }) ≤ bN h 2 , then ∀N ≤ N 0 , aN h 2 ≤ ν({η ∈ PMF(S) : i(η, ξ γ ) ≤ N }) ≤ bN h 2 .

o

  (•). Since 2 o (•) ∼ o Ext o (•), we can use o (•) to replace Ext o (•) without affecting the result when we defining the measure ν o , the embedding τ : PMF(S) -→ MF(S) and ξ k (γ). For instance, for a measurable subset U ∈ PMF(S), we have ν 0 (U ) = µ({η : [η] ∈ U, o (η) ≤ 1}).

Theorem 2 . 4 . 11 . 2 .

 24112 Under the above Set-up 0, there exist M 0 > 0, C > 0 and D > 0, depending on g, o, such that when M < M 0 , we haveCM h 2 ≤ ν({η ∈ PMF(S) : i(η, ξ) ≤ M }) ≤ DM hThe main tool to prove the above theorem is the following Dehn-Thurston theorem. Let P = {α k } be a pants decomposition. For each α k , let m k : MF(S) -→ R ≥0 , ξ → i(α k , ξ) be the intersection function defined by α k and t k = tw k be the twist function associated to α k . Theorem 2.4.12 (The Dehn-Thurston theorem [52], Theorem 3.1.1). Let S = S g and α = {α 1 , • • • , α 3g-3 } be a pants decomposition of S. Then the map : MF(S) -→ R 6g-6 F → (m 1 (F), • • • m 3g-3 (F), t 1 (F), • • • , t 3g-3 (F)).

( 2 . 4 . 5 )

 245 gives a global coordinate for MF(S).

  Define a set of 6g -6-tuples byW 0 (a, M ) = {(x 1 , • • • , x 3g-3 , y 1 , • • • , y 3g-3 ) : ∀i, 0 ≤ ax i ≤ 1 3g -3 M, 0 ≤ (y i -x i ) o (α i ) ≤ δ}.

h 2 .

 2 Hence the proof is finished. Proof of Theorem 2.4.1. Ler γ ∈ E n and take M = e -2n in Corollary 2.4.3 and M = F ln ne -2n in Corollary 2.4.5 where F is the constant in Lemma 2.4.4. Then Theorem 2.4.11 implies the assumption in Lemma 2.4.10, hence Lemma 2.4.6 is true and hence Ψ(γ) ≥M ∼ g,o, n and Ψ(γ) ≥M ∼ g,o, a 1 n -c 1 ln n. Then by Corollary 2.4.3, Corollary 2.4.5 and Theorem 2.4.11 again, the proof is finished.

Theorem 2 . 5 . 1 .Proposition 2 . 5 . 2 .

 251252 Let S = S g (g ≥ 2) and π ν be the associated quasi-regular representation of the mapping class group Mod(S) on L 2 (PMF(S), ν). Let d be a metric on PMF(S) which is compatible with the topology of PMF(S). Let n ρ and E n = E(θ, , n, o, ρ). Let e n = P r : E n -→ PMF(S) be the radial projection which assigns g ∈ E n to the direction ξ g of the oriented geodesic [o, g • o].Then the quasi-regular representation π ν is ergodic with respect to (E n , e n ) and anyf ∈ H L ∞ (PMF (S),ν), whereH =< 1 U : ν(∂U ) = 0and U is a Borel subset of PMF(S) > . Proof. The proof consists of verifying all assumptions in Theorem 2.2.13 for E n . The first two will be verified by showing E n is of exponential growth (namely, Corollary 2.3.2, Corollary 2.3.3). The third one is by Proposition 2.5.2. The last one is Theorem 2.5.6. For every n ρ, there are two sequences of real numbers {h rn (n, ρ)} and {r n } such that lim n→∞ h rn (n, ρ) = lim n→∞ r n = 0 and such that ∀n ∈ N, ∀γ ∈ E n , π ν (γ)1 PMF (S) , 1 {x∈PMF (S):i(x,en(γ))≥rn} Φ(γ) ≤ h rn (n, ρ). Proof. Let n ρ and γ ∈ E n . Let x γ as before. Consider the intersection function on PMF(S) defined by ξ γ , namely, i(ξ γ , •). Let r n = 1 n . By Corollary 2.4.3, Lemma 2.4.6 and the proof of its assumption (Theorem 2.4.11), π ν (γ)1 PMF (S) , 1 {x∈PMF (S):i(x,ξγ )≥ 1 n } Φ(γ) ≤ c(g, o, ρ) ln n -D a 1 n -c 1 ln ln n + b 1 .

Case 1 :

 1 d(q, y) ≤ D with y ∈ [o, η]. Then d(q, o) -D ≤ d(o, y) ≤ d(q, o) + D. Since n --ρ ≤ d(q, o) ≤ n + ρ, we have 0 ≤ d(x, y) ≤ + D + ρ. Hence, d(x, γ • o) ≤ d(x, y) + d(y, q) + d(q, p) ≤ + D + D + + ρ ≤ 2( + D + ρ) ≤ 3 .

Case 2 :

 2 d(q, y) ≤ D with y ∈ [ξ, η]. Then by Lemma 2.4.2, one has i 2 (η, ξ) = Ext y (ξ) Ext y (η).

Corollary 2 . 5 . 5 .

 255 2 e -2n . By Kerckhoff's formula again, Ext p (η) ≤ C 2 e 2ρ+4D e -2n , or 1 2 ln Ext p (η) ≤ ln(Ce ρ+2D ) -n. Apply Lemma 2.3.4, one could choose z ∈ [o, η]∩Teich (S) so that, if denote d(o, p) = t, d(p, z) = a and d(z, o) = b, then a -b ≤ -n + ln(Ce 2ρ+4D ) + 1. Therefore, we have 0 ≤ t + a -b ≤ ln(Ce 2ρ+4D ) + 1 + ρ. By Theorem 2.2.7, we have d(p, y) ≤ k(ln(Ce 2ρ+4D ) + 1 + ρ) ≤ 3 . We complete the proof. Let η ∈ PMF(S) and suppose that η does not leave Teich eventually. Let x ∈ [o, η] such that d(x, o) = n. Let further C > 0 and n large enough. Then {γ ∈ E n : γ • o ∈ Sec I C (η,x) } ≺ g,o,ρ ln n. Proof. By Theorem 1.2 in [4] (note that Λ in the theorem is a constant function), when n is large enough, there exists a constant N 0 > 0, such that |B(x, R) ∩ Mod(S) • o| ≤ N 0 e hR . Apply Lemma 2.5.4, we have the conclusion. (γ • o, R)) O(γ•o,R) Note that there are bounded number intersections of open sets on the form O(γ • o, R) and the bound depends on R and ρ. Thus, since |E n | e hn (Corollary 2.3.2) and Φ(γ) g,o,ρ (a 1 n -c 1 ln ln n + b 1 )e -hn 2 (Harish-Chandra estimates), substitute all these together, one has, III ≺ g,o,ρ,R 1 a 1 n -c 1 ln ln n + b 1 {η∈PMF (S):i(η,ζ)>Ce -2n } 1 i(η, ζ) h 2 dν(η) ≺ g,o,ρ,R 1.

( 2 . 5 . 3 )

 253 The last inequality follows from the fact that ζ ∈ U ( , θ) and the proof of Harish-Chandra estimats.

2 (a 1 n 2 = 1 a 1 n

 21211 -c 1 ln ln n + b 1 )e -hn -c 1 ln ln n + b 1 |H n |.

  Definition 1.3.1. Let G be a group and (X, µ) be a Borel space. Suppose that G Example 1.3.6. Let H ≤ P SL(2, Z) be a Schottky group, then its limit set Λ(H) ⊆ S 1 , as a Cantor set, has zero Lebesgue measure, and thus it acts on {R 2 -(0, 0)}/{±1} almost properly discontinuously. If S = S 0,4 or S 1,1 , then, in both cases, MF(S) can be identified with {R 2 -(0, 0)}/{±1} and PMF(S) can be identified with S 1 . Moreover, there is a finite index subgroup of Mod(S) such that the action of this subgroup on PMF(S) is equivalent to the action of P SL(2, Z) on S 1 , see [[START_REF] Farb | A primer on mapping class groups[END_REF],Chapter 15] for the case of S 0,4 . By taking H to be any subgroup given in Example 1.3.6 and considering the set

	acts on X by measure-preserving homeomorphisms. We say that G acts on X almost
	properly discontinuously if there exists a G-invariant subset K with µ(K) = 0 such
	that G acts on X -K properly discontinuously.

Although the action of Mod(S) on MF(S) is ergodic with respect to generalized Thurston measures, the action of subgroups of Mod(S) on MF(S) is not always ergodic. The following proposition allows us to use properties of the "properly discontinuou" action. Proposition 1.3.7. For each complete pair (R, γ), there exists a rank 2 free pseudo-Anosov subgroup H of Mod(S) that acts on MF(S) almost properly discontinuously with respect to the generalized Thurston measure µ [(R,γ)] T h .

Any such free group will be called a p-rank 2 free subgroup.

The first case is when R = ∅ or each component of R is S 0,3 , then this proposition is obvious by taking H to be any free pseudo-Anosov subgroup generated by two pseudo-Anosov mapping classes (this works the same for non-integral multicurves as for integral multicurves). For other cases, we prove this proposition through two lemmas.

Lemma 1.3.8. There exists a p-rank 2 free subgroup H of Mod(S) that acts on MF(S) almost properly discontinuously with respect to the Thurston measure µ T h .

Proof.

  We will follow the idea of [[START_REF] Lindenstrauss | Ergodic theory of the space of measured laminations[END_REF], Lemma 3.1] to prove this lemma. Fix any hyperbolic structure X on S and consider the continuous function

	Lemma 1.3.9. For a complete pair (R, γ) of middle type, there exists a p-rank 2
	free subgroup H of Mod(S) that acts on MF(S) almost properly discontinuously with
	respect to the measure µ	[(R,γ)] T h	.
	Proof.

, Proposition 6.1] that Z(Λ(H)) -Λ(H) consists of no uniquely ergodic foliations and uniquely ergodic foliation has full µ T h -measure, we know that P r -1 (Z(Λ(H))) has null µ T h -measure. By [

[START_REF] Mccarthy | Dynamics on Thurston's sphere of projective measured foliations[END_REF]

,Theorem 7.17], H acts properly discontinuously on PMF(S) -Z(Λ(H)), thus properly discontinuously on MF(S) -P r -1 (Z(Λ(H))). Hence H acts almost properly discontinuously on

MF(S).

For R = S, a complete pair (R, γ) is called a middle type if R = ∅ and there is a connected component = S 0,3 . X : MF(S) → R + extending the geodesic length function. Thus

  Given a group H and a Borel space (X, µ). Suppose that H acts on X almost properly discontinuously and freely. Examples for such (H, X, µ) are given by Proposition 1.3.7. By definition of almost properly discontinuous action,

there is a null set Y such that H acts on X -Y properly discontinuously. For any compact subset K of X -Y , we will describe a "nice" cover of K. Since X -Y is the domain of discontinuity of H, for every p in K, there is an open neighbourhood U p of p in X -Y with finite µ-measure such that for all h ∈ H, one has h.U p U p = ∅. Thus there is an open cover of K. By compactness of K, choose a finite sub-cover of this cover. Label the sub-cover by U 1 , ..., U n and for each i ∈ 1, ..., n, consider A i = {h.U i |h ∈ H}. Starting from i = 1, form a family

  and U is a Borel subset of PMF(S) > . For the proof of Theorem 2.2.13, we modify the proof of Proposition 2.21 and thus Theorem 2.2 in[START_REF] Boyer | Ergodic boundary representations[END_REF] as follows. We use the same notations for convenience. In the proof of the first part of [Proposition 2.21,[START_REF] Boyer | Ergodic boundary representations[END_REF]], first approximate W by subsets W n consisting of uniquely ergodic measured foliations and such that ν o (W -W n ) ≤ k n with k n tends to 0. Then denote W n (r) = {η ∈ PMF(S) : i(η, W n ) ≤ r}, by Lemma 2.2.3, choose m

	Remark 2.2.3. As Theorem 2.2.13 is slight different from Theorem 2.2.1, it is worth
	making a few comments. One could easily find the only difference is the point (3) here
	since the original point (1) has been replaced automatically by Lemma 2.2.12. The
	assumption (3) in Theorem 2.2.13 is different from the assumption (4) in Theorem
	2.2.1.

  Theorem 2.2.14. There exists θ and such that, if E n = E(θ, , n, o, ρ), which is described before Lemma 2.2.11, then the boundary representation π νo is ergodic with respect to (E n , P r) and any f ∈ H The boundary representation π νo of Mod(S) is irreducible.We then mention a property of the boundary representation π νo . Recall that a unitary representation of a group G is called tempered if it is weakly contained in the regular representation L 2 (G). The boundary representation π νo of Mod(S) is tempered.

	Proposition 2.2.17.
	L ∞	as above. In other words, the pair (E n =
	E(θ, , n, o, ρ), P r) satisfies all conditions listed in Theorem 2.2.13.
	As ν o is a Radon measure, one has immediately the following two corollaries by
	Proposition 2.2.2 and Remark 2.2.1.
	Corollary 2.2.15. Corollary 2.2.16. The mapping class group Mod(S) acts ergodically on PMF(S)
	with respect to the measure class [ν o ].

  Let n 0 and K be the constant in Theorem 2.3.1. Then |E n | Ke hn . In particular, lim n→∞ |E n | = ∞. Proof. As E n ⊂ C(n, ρ) and, by Theorem 2.3.1, |C(n, ρ)| Ke hn , it is obvious that

	-Veech
	volume.
	Corollary 2.3.2.

  .3.2 Shadow lemma.

	Definition 2.3.1. (O-points) A point y in Teich(S) is called an O-point (with
	respect to o) if for every R > 0, there is a real number C ≥ 1 depending on R, such
	that
	1 C	exp (-hd(o, y)) ≤ ν(P r(B(y, R))) ≤ C exp (-hd(o, y))
	Remark 2.3.1. We call such points O-points because they are the points that satisfy
	the classic shadow lemma [58].

Definition 2.3.2. An element g ∈ Mod(S) is called an O-mapping class (with respect to o ∈ Teich(S)) if g • o is an O-point.

-function on Y γ defines an 2 -function on X γ , and such correspondence produces

Acknowledgements

I would like to thank my advisor Indira Chatterji for her constant help, encouragement and guidance over the past four years. For math, she not only encourages me to work on mapping class groups from the viewpoint of unitary representations, but also encourages and supports me to attend infinitely many conferences which broaden my limited knowledge of Geometric Group Theory. Without her constant guidance and patience, this thesis can never be completed. Especially, I would like to thank her for tolerating my very poor English and helping me to correct English errors whenever I need. For life, she is always kindness and helpful. She is a very great advisor and friend.

Com G (L) = L, then two unitary representations 2 (G/H) and 2 (G/L) of G are weakly equivalent if and only if L is conjugate to H. Theorem 1.5.4. Let S = S g,n be a compact surface with 3g + n ≥ 4. Let γ and δ be two 1-multi-curves on S with k, l geometric components, respectively.

(1) If at least one of k, l is not 3g -3 + n, then the associated unitary representations π γ and π δ are weakly equivalent if and only if γ and δ are of the same type.

(2) Suppose S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If the number of geometric components of γ is 3g -3 + n, then π γ is weakly equivalent to the regular representation λ S .

(3) Suppose S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 . If the number of geometric components of γ is not 3g -3 + n, then π γ is not weakly contained in λ S .

Proof. For any 1-multi-curve γ on S, Com Mod(S) (G γ ) = G γ (see [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF]). Given two 1-multi-curves γ and δ with k, l geometric components, respectively, such that at least one of k and l is not 3g -3 + n, then by Lemma 1.5.2, Lemma 1.5.3 and the fact that G γ is conjugate to G δ if and only if γ and δ are of the same type, we complete the proof for [START_REF] Agol | The virtual Haken conjecture[END_REF]. For [START_REF] Ellegaard | Cohomology of mapping class groups and the abelian moduli space[END_REF], by [START_REF] Bridson | Mapping class groups and outer automorphism groups of free groups are C * -simple[END_REF], if S is not S 0,4 , S 1,1 , S 1,2 , S 2,0 , the mapping class group Mod(S) is C*-simple. By the result of [START_REF] Breuillard | C * -simplicity and the unique trace property for discrete groups[END_REF] which states that a discrete group is C*-simple if and only if, for any amenable subgroup M of G, the quasi-regular representation 2 (G/M ) is weakly equivalent to the regular one. So combine with Lemma 1.5.2, we complete the proof of [START_REF] Ellegaard | Cohomology of mapping class groups and the abelian moduli space[END_REF]. The statement (3) is deduced from [START_REF] Ellegaard | Cohomology of mapping class groups and the abelian moduli space[END_REF] and the definition of C*-simplicity.

Remark 1.5.1. The "only if" part of (1) is a stronger version of Corollary 5.5 in [START_REF] Paris | Actions and irreducible representations of the mapping class group[END_REF].

Remark 1.5.2. If S is one of S 0,4 , S 1,1 , S 1,2 , S 2,0 , it is easy to show that, if the number of components of γ is 3g-3+n, then π γ is weakly contained in the regular representation to [START_REF] Farb | A primer on mapping class groups[END_REF], [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF] and [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF].

Mapping class groups and Teichmüller spaces.

Let S = S g be a genus g, closed, connected, orientable surface. We always assume that g ≥ 2. All arguments here work for hyperbolic surfaces with punctures as well. The mapping class group Mod(S) of S is the group of isotopy classes of orientation-preserving homeomorphisms of S. Namely, if the group of orientation-preserving homeomorphisms of S is denoted by Homeo + (S) and the group of homeomorphisms of S that isotopic to the identity is denoted by Homeo 0 (S), then Mod(S) = Homeo + (S)/Homeo 0 (S).

We remark here that mapping class groups of surfaces are finitely presented and considered to be discrete groups. The Teichmüller space Teich(S) of S is the space of homotopy classes of hyperbolic structures. The Teichmüller space Teich(S) is homeomorphic to R 6g-6 and the mapping class group Mod(S) acts on Teich(S) by changing markings. The quotient M(S) = Teich(S)/ Mod(S) is the moduli space of S. There are several distances on Teich(S) so that Mod(S) acts as isometries, the one that we will use is the Teichmüller distance d = d T . It is defined as follows: For

where f : X -→ Y is the Teichmüller mapping, locally in the form of x + iy → e t x + ie -t y, in the isotopy class of ψ • φ -1 , namely the quasi-conformal homeomorphism with minimal dilatation in the isotopy class of ψ • φ -1 and K f is the dilatation of f . It is obvious that Mod(S) ⊂ Isom(Teich(S), d). Neither the Teichmüller space Teich(S) nor M(S) is compact.

Measured foliations. The Teichmüller space can be compactified in several ways.

The compactification we will use in this paper is the Teichmüller compactification. Fix

Harish-Chandra estimates.

This section is devoted to prove the following Harish-Chandra estimates.

Theorem 2.4.1. Given n ρ. There exist

Recall that

Remark 2.4.1. 1. In [START_REF] Boyer | Equidistribution, ergodicity, and irreducibility in CAT(-1) spaces[END_REF], the left side is of the form (an + b)e -α 2 n . However, some other terms like ln ln n should be added for mapping class groups if we require

2. The following oberservation will be useful, namely Φ(γ) ne -hn 2 .

The proof is divided into several steps and will be given at the end of this section.

Reduction to intersection numbers.

By our convention, for every ξ ∈ PMF(S), one has Ext o (ξ) = 1, we then have

In order to estimate Φ(γ), we will relate it to the following integrations on intersection numbers:

The first step is to bound Φ (γ) from above. This can be easily done by Minsky's inequality. Namely, Lemma 2.4.2 (Minsky's inequality [START_REF] Yair | Teichmüller geodesics and ends of hyperbolic 3-manifolds[END_REF]). Let ξ and η be two measured foliations on S and x ∈ Teich(S), then

where the equilty holds if and only if there is a qudratic differential q so that the vertical measured foliation of q on X is ξ and the horizontal measured foliation is η.

Corollary 2.4.3. There exist constants

2 by e -2n , so one has

which gives the bound for the term II.

In order to bound Φ(γ) from below, we will use the fact that γ ∈ E n .

Lemma 2.4.4. There exists a constant F depending on g, o, , θ, ρ such that if i(ξ γ , η) ≥

Therefore, in this case we have a constant F 1 depending on g, o, , θ, ρ such that

Thus if we take F F 1 and require i(ξ γ , η) ≥ F ln ne -2n , it forces us in the Case 1 which implies the conclusion that

Corollary 2.4.5. For every γ ∈ E n , take M = F ln ne -2n where F is the constant in Lemma 2.4.4. Then Φ(γ) g,o, ,θ,ρ e -h 2 n Ψ(γ) ≥M .

Proof. Note that U ( , θ) has a full measure. Hence, by Lemma 2.4.4

Lemma 2.4.6. Assume that there exist N 0 > 0, a > 0 and b > 0 such that

decomposition, namely a collection of 3g -3 essential simple closed curves P =

If x is further assumed to be in Teich (S), we can choose a collection of 3g -3 essential

Denote α(x) ∈ M F (S) to be the measured foliation α(x) = 3g-3 i=1 α i (x) and [α(x)] be its projective class in PMF(S). By the Jenkins-Strebel theorem (see also [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF] Theorem 2.1), there is a unit holomorphic quadratic differential q = q(x, α(x)) on o ∈ Teich(S) whose projective class of the vertical measured foliation is [α(x)].

Given γ ∈ E n , by our construction of E n , the ending point ξ γ of the geodesic

Hence, there is a sequence of points y(k, γ) ∈ Teich (S) in g γ t tends to ξ γ in Teich(S). By the above discussion, there is a sequence of pants curves α(y(k, γ)) and a sequence of points [α(y(k, γ))] in PMF(S). By Minsky's inequality, [α(y(k, γ))] converges to [ξ γ ] in PMF(S), which means using the map τ ,

converges to ξ γ . We first estimate Ext o (α(y(k, γ))). Now let y ∈ Teich (S) and denote α(y) and q(y) = q(y, α(y)) as above. Let g t be the Teichmüller geodesic ray starting from o in the direction of q(y). Let t y be the unique point in g t such that t y has maximal distance with o and

Lemma 2.4.7. There is a constant C 2 depending on g and such that

The proof is based on the following theorem. We use the theorem in the form of Theorem 5.3 in [START_REF] Athreya | Lattice point asymptotics and volume growth on Teichmüller space[END_REF]. For the definition of twist numbers tw(α, β), the reader is referred to [START_REF] Penner | Combinatorics of train tracks[END_REF].

Theorem 2.4.8 (Minsky [START_REF]Extremal length estimates and product regions in Teichmüller space[END_REF]). Let x ∈ Teich(S) and P = {α 1 , • • • , α 3g-3 } be a pants decomposition produced by the Bers' theorem mentioned above. Then for any simple closed curve β,

Proof of Lemma 2.4.7. By Kerckhoff's formula, we only need to bound the ratio

Extt y (β) for any essential simple closed curve β on S. However, by the construction of y(x), the two hyperbolic surfaces t y and y have the same pants decomposition which satisfies the condition in Theorem 2.4.8, namely α(γ) = {α 1 (y), • • • , α 3g-3 (y)}.

As, for 1 ≤ i ≤ 3g -3, both extremal lengths Ext y (α i ) and Ext ty (α i ) are bounded below by the constant and above by a constant C 2 1 depending only on g, we can conclude the proof of the lemma by using Equation (2.4.4). Proof. We only need to show the second statement. Let y = y(k, γ) and t y = t y(k,γ) .

Let T = d(o, t y ) and f : o → t y be the Teichmüller mapping with dilatation e 2T between o and t y . We know that T ∼ g, d(o, y(k, γ)). We want to show that

On the one hand, by Kerckhoff's formula, one has

In order to bound Ext ( α i (y)) from below, we construct a metric and use the analytic definition of extremal length in [START_REF] Steven | The asymptotic geometry of Teichmüller space[END_REF]. Fix 1 ≤ i ≤ 3g -3. Let q be the unit quadratic n! ω n on MF(S), hence up to a multiplicative constant, the measure µ coincides with the Thurston measure ν T h on MF(S).

Proof. First we note that, different pants decompositions give rise to different train tracks charts which locally differ by an element in SL(k, Z), for some k, keeping the volume form invariant, hence µ is Mod(S)-invariant and independent of the choice of the pants decomposition P . As ν T h is ergodic and both µ and ν T h are in the Lebesgue measure class, the conclusion follows.

We now use µ to replace ν T h with α is fixed to the one given in Set-up 0. Note that h 2 = 3g -3. We now prove the theorem.

Proof of Theorem 2.4.11. By Lemma 2.4.2, for every two elements ξ and η in PMF(S), the intersection number i(η, ξ) ≤ 1 and 1 is achievable. So we take M 0 = 1 4 and let M ≤ M 0 . The proof is then divided into two parts. In the sequel, we will denote a = 1 i o(αi) and to be

Then by our assumption, there exists A 1 > 0 and B 1 > 0 depending on g, o, , such

82 By the definition of ν and i(η, ξ) = a 3g-3 k=1 i(η, α k ), we have,

where A 2 is a constant depending only on o. In fact, A 2 depends on the diameter of X. The last step comes from the fact that large twists will make the length to be large. Thus we further have

)

Lower bound:

In order to bound the measure from below, we will construct a subset contained in the set. We first fix, for each i, a positive orientation for α i . Let

Then η 0 = 1 3g-3 ξ is in V . Let a and M as above, and δ > 0 be a positive number. Proof. By Lemma 2.4.2,

Let α = α(γ) as before. Then γ) . On the other hand, by Lemma 2.4.7, up to a multiplicative constant, one could replace

Collect all discussions together, one can finish the proof.

Recall that M

1 PMF (S) En

.

By using the embedding map τ of PMF(S) into MF(S). One can rewrite the above formula to be M

.

We first introduce a type of open sets IN in PMF(S) defined by intersection numbers. For every η ∈ PMF(S), C > 0, t > 0, we define

Proof of Theorem 2.5.6. Let U ( , θ) the subset of PMF(S) of full measure. We shall give a bound independent on n ρ for M

(2.5.1) First we want to bound term I in Equation (2.5.1). The set E 1 n can be further decomposed as two sets:

We first deal with term III. First notice that

on the other hand, by Lemma 2.4.2, for ν-almost every ξ As U ( , θ) has full measure, in particular, it is dense in PMF(S), thus one can choose q ∈ B(γ • o, R) so that the direction ξ q of [o, q] is in U ( , θ) ∩ IN (ζ, n, C). By Theorem 2.2.9, there is a P = P ( ), so that the two geodesics [o, γ • o] and [o, q] are P -fellow travelling in a parametrized fashsion. Now consider the P -neighborhood N P of Teich (S), namely the union of points in Teich(S) that has distance at most P with a point in Teich (S). As Mod(S) acts as isometries on Teich(S) and Teich (S) is Mod(S)-invariant and cocompact, the neighborhood N P is Mod(S)-invariant and cocompact. By Mumford's compactness, there is a small so that N P ⊂ Teich (S).

Then as γ ∈ E n , the geodesic segment [o, q] has the property that it contains a segment I = [a, q] of length 1 3h ln ln n such that I has at least θ in Teich (S). Note that is fixed, hence C depends on g and o. Hence by Theorem 2.2.8, there are two constants D = D ( , θ) and L 0 = L 0 ( , θ) satisfy Theorem 2.2.8. Take n large enough and follow the proof Lemma 2.5.4 and Corollary 2.5.5, one has that |F 2 n | ≺ ln n.