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Journal Articles Groups, Geometry, and Dynamics Year : 2017

Coxeter group in Hilbert geometry

Abstract

A theorem of Tits - 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 an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $\Gamma$, find the maximal $\Gamma$-invariant convex, when there is a unique $\Gamma$-invariant convex, when the convex $\Omega$ is strictly convex, when we can find a $\Gamma$-invariant convex $\Omega'$ which is strictly convex.
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Dates and versions

hal-01050772 , version 1 (25-07-2014)
hal-01050772 , version 2 (01-07-2015)

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Cite

Ludovic Marquis. Coxeter group in Hilbert geometry. Groups, Geometry, and Dynamics, 2017, 11 (3), pp.819-877. ⟨10.4171/GGD/416⟩. ⟨hal-01050772v2⟩
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