# Groups, Geometry, and Dynamics

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**Volume 11, Issue 3, 2017, pp. 819–877**

**DOI: 10.4171/GGD/416**

Published online: 2017-08-22

Coxeter group in Hilbert geometry

Ludovic Marquis^{[1]}(1) Université de Rennes I, France

A theorem of Tits and Vinberg allows to build an action of a Coxeter group $\Gamma$ on a properly convex open set $\Omega$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe a hypothesis that makes those conditions necessary.

Under this hypothesis, we describe the Zariski closure of $\Gamma$, nd the maximal $\Gamma$-invariant convex set, when there is a unique $\Gamma$-invariant convex set, when the convex set $\Omega$ is strictly convex, when we can find a $\Gamma$-invariant convex set $\Omega$' which is strictly convex.

*Keywords: *Coxeter group, Hilbert geometry, Discrete subgroup of Lie group, convex projective structure on manifold and orbifold, geometric group theory

Marquis Ludovic: Coxeter group in Hilbert geometry. *Groups Geom. Dyn.* 11 (2017), 819-877. doi: 10.4171/GGD/416