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 .

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.

Given a reduced cooperad C, we consider the 2-colored operad Cyl(C) which governs diagrams U: V -> W, where V, W are Cobar(C)-algebras, and U is an infinity-morphism. We then investigate the deformation complexes of Cyl(C) and Cobar(C). Our main result is that the restriction maps between between the deformation complexes Der'(Cyl(C)) and Der'(Cobar(C)) are homotopic quasi-isomorphisms of filtered Lie algebras. We show how this result may be applied to modifying diagrams of homotopy algebras by derived automorphism. We then recall that Tamarkin's construction gives us a map from the set of Drinfeld associators to the homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld and T. Willwacher, both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group GRT. We use our earlier results to prove that this map from the set of Drinfeld associators to the set of homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains is GRT-equivariant. / Mathematics