Groupes modulaires et groupes d'automorphismes de complexes de surfaces de type infini

Abstract : Let sigma g,n be an orientable surface of genus g with n punctures. The mapping class group of sigma g,n acts on several complexes, for instance the curve complex or the pants complex of the surface. It is proved that the automorphism group of each of these complexes are isomorphic to the mapping class group. This implies in particular that the group of outer automorphisms of a finite index subgroup is finite. The purpose of this thesis is to prove a similar result on some surfaces of infinite type and genus zero. For this, we define an asymptotic mapping class group of these surfaces, and then a locally infinite cellular complex where the mapping class group acts naturally. It brings up some properties of the automorphism group of each cellular complex by making automorphisms act on auxiliary graphs. The first studied asymptotic mapping class group is isomorphic to the Thompson group T. The second one is an extension of the universal mapping class group of genus zero.
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Maxime Nguyen. Groupes modulaires et groupes d'automorphismes de complexes de surfaces de type infini. Théorie des groupes [math.GR]. Université de Grenoble, 2012. Français. ⟨NNT : 2012GRENM102⟩. ⟨tel-01557515⟩

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