Coxeter group in Hilbert geometry

Abstract

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.