Thesis advisor: Martin J. Bridgeman / Thesis advisor: Ian Biringer / The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSU(1,1). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the one-holed torus with boundary condition, and we determine regions where the action is ergodic. This uses a combination of techniques developed by Goldman, and Pickrell and Xia. The basic result is an analogue of the result of Goldman's at the level of moduli.

In this thesis we deal with combinatorial and geometric properties of arc complexes and triangulation graphs, and we will provide some applications to the study of the mapping class group and to the Teichmüller theory of a bordered surface. The thesis is divided into two parts. In the former we deal with the problem of combinatorial rigidity of arc complexes. In the latter we study some large-scale properties of the arc complex and the 1-skeleton of its dual, the so-called ideal triangulation graph.

In this thesis we build on the theory concerning the metric geometry of relatively hyperbolic and mapping class groups, especially with respect to the difficulty of embedding such groups into Banach spaces. In Chapter 3 (joint with Alessandro Sisto) we construct simple embeddings of closed graph manifold groups into a product of three metric trees, answering positively a conjecture of Smirnov concerning the Assouad-Nagata dimension of such spaces. Consequently, we obtain optimal embeddings of such spaces into certain Banach spaces. The ideas here have been extended to other closed three-manifolds and to higher dimensional analogues of graph manifolds. In Chapter 4 we give an explicit method of embedding relatively hyperbolic groups into certain Banach spaces, which yields optimal bounds on the compression exponent of such groups relative to their peripheral subgroups. From this we deduce that the fundamental group of every closed three-manifold has Hilbert compression exponent one. In Chapter 5 we prove that relatively hyperbolic spaces with a tree-graded quasi-isometry representative can be characterised by a relative version of Manning's bottleneck property. This applies to the Bestvina-Bromberg-Fujiwara quasi-trees of spaces, yielding an embedding of each mapping class group of a closed surface into a finite product of simplicial trees. From this we obtain explicit embeddings of mapping class groups into certain Banach spaces and deduce that these groups have finite Assouad-Nagata dimension. It also applies to relatively hyperbolic groups, proving that such groups have finite Assouad-Nagata dimension if and only if each peripheral subgroup does.

Nous montrons que les morphismes du groupe de tresses à n brins dans le mapping class group d'une surface de bord éventuellement non vide et de genre inférieur ou égal à n/2 sont soit cycliques (i.e. dont l'image est un groupe cyclique), soit des transvections de monodromie géométriques (i.e. à multiplication près par un élément du centralisateur de l'image, un générateur standard du groupe de tresses est envoyé sur un twist de Dehn, et deux générateurs standards consécutifs sont envoyés sur deux twists de Dehn le long de deux courbes s'intersectant en un point). En corollaire, nous déterminons les endomorphismes, les endomorphismes injectifs, les automorphismes et le groupe d'automorphisme des groupes suivants : le groupe de tresses à n brins lorsque n est supérieur ou égal à 6, le mapping class group de toute surface de genre supérieur ou égal à 2. Pour chacun des énoncés impliquant le mapping class group, nous étudions deux cas : lorsque le bord est fixé point par point ou seulement composante par composante. Nous décrivons également l'ensemble des morphismes entre différents groupes de tresses dont le nombre de brins diffèrent d'au plus un, et l'ensemble des morphismes entre mapping class groups de surfaces (de bord éventuellement non vide) dont les genres (supérieurs ou égal à 2) différent d'au plus un.

Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n &gt / 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group M of N.

Cette thèse porte sur l'étude de certaines familles de représentations projectives des groupes modulaires de surfaces issues de théories topologiques quantiques de champs. Les résultats principaux portent sur leur décomposition en facteurs irréductibles. / This thesis deals with some families of projective representations of the mapping class groups of surfaces arising in topological quantum field theories. The main results concern their decomposition into irreducible factors.

Dans cette thèse nous étudions certains propriétés combinatoires et géométriques des complexes d'arcs des surfaces de type fini. Nous démontrons que le groupe d'automorphisme du complexe d'arcs est le mapping class group de la surface. Nous étudions aussi le graphe des triangulations idéales et nous donnons certains applications au espaces de Teichmueller des surfaces avec bord . / In this thesis we deal with combinatorial and geometric properties of the arc complex of a surface of finite type. We prove that its automorphism group is isomorphic to the mapping class of the surface. Furthermore, we investigate the geometric properties of the ideal triangulation graph of a surface and provide some application to Teichmueller theory of a surface with boundary .

Dans le premier chapitre de ce travail, nous rappelons la théorie des représentants efficaces d'un élément pseudo-Anosov du mapping class group d'une surface S compacte orientée munie de n+1 points marqués. Ces objets ont été introduits par Bestvina-Handel et Los.<br /><br />Le deuxième chapitre contient l'exposé de la théorie des bons représentants et des représentants super efficaces d'un homéomorphisme pseudo-Anosov f fixant le point marqué x_0. Nous montrons ensuite un résultat de structure sur l'ensemble des représentants super efficaces : cet ensemble est une union d'un nombre fini de cycles qui sont parcourus en appliquant des opérations combinatoires. Nous en déduisons des algorithmes permettant de décider si l'homéomorphisme f - ou, ce qui est équivalent, sa classe d'isotopie - admet une racine fixant x_0, ou commute avec un élément d'ordre fini fixant x_0. Nous en déduisons également une nouvelle solution au problème de conjugaison parmi les éléments pseudo-Anosov du mapping class group qui fixent x_0.<br /><br />Dans le troisième chapitre, nous considérons un homéomorphisme f du disque et O une orbite de période n>=3 pour f. Nous donnons une minoration de l'entropie topologique des homéomorphismes isotopes à f relativement à O. Cette minoration est obtenue à l'aide de la théorie des représentants efficaces.<br /><br />Dans le quatrième chapitre, nous donnons des conditions nécessaires et suffisantes pour qu'une tresse beta à n brins admette une déstabilisation ou un mouvement d'échange. Ces conditions sont des propriétés sur l'élément du mapping class group induit par la tresse beta.

There are several reasons to be interested in conformal field theories in two dimensions. Apart from arising in various physical applications, ranging from statistical mechanics to string theory, conformal field theory is a class of quantum field theories that is interesting on its own. First of all there is a large amount of symmetries. In addition, many of the interesting theories satisfy a finiteness condition, that together with the symmetries allows for a fully non-perturbative treatment, and even for a complete solution in a mathematically rigorous manner. One of the crucial tools which make such a treatment possible is provided by category theory. This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group ofthe Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theories whose chiral data are described by the category of left-modules over H.