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Automorphismes et admissibilité dans les groupes de Coxeter et les monoïdes d'Artin-Tits

Abstract : This thesis is a contribution to the combinatorical study of Coxeter groups and Artin-Tits groups. In the first part, we complete the description of the automorphism group of a right-angled Coxeter group, by studying the second of the two subgroups that appear in the semi-direct product established by Tits (the first one is described by Mühlherr). We thus recover Radcliffe's result on rigidity of right-angled Coxeter groups. In the second part, we introduce and study the notion of submonoids of an Artin-Tits monoid induced by an admissible partition of the associated Coxeter graph, in the sense of Mühlherr. We show that such a submonoid is an Artin-Tits monoid, and that this notion generalizes and unifies the situation of submonoids of fixed elements of an Artin-Tits monoid under the action of graph automorphisms, and the notion of LCM-homomorphisms of Crisp and Godelle. We complete Mühlherr's classification of admissible partitions of spherical Coxeter graphs ; this leads us to the classification of Crisp's LCM-homomorphisms. In the third part, we study the Krammer-Paris representation of an Artin-Tits monoid of simply laced type without triangle. The submonoid of fixed elements of such a monoid under the action of a group of graph automorphisms stabilizes the subspace of fixed points of the space of the representation under the action of this group. We use notions developped by Hée to show that the representation obtained in this way is faithful. This generalizes, without any case-by-case enumeration, results established by Digne in the spherical case.
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Contributor : Anatole Castella <>
Submitted on : Friday, March 16, 2007 - 12:55:36 AM
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  • HAL Id : tel-00136943, version 1



Anatole Castella. Automorphismes et admissibilité dans les groupes de Coxeter et les monoïdes d'Artin-Tits. Mathématiques [math]. Université Paris Sud - Paris XI, 2006. Français. ⟨tel-00136943⟩



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