Coxeter group in Hilbert geometry

  • Ludovic Marquis

    Université de Rennes I, France
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Abstract

A theorem of Tits and Vinberg allows to build an action of a Coxeter group €Γ\Gamma on a properly convex open set Ω\Omega of the real projective space, thanks to the data PP 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 €Γ\Gamma, nd the maximal €Γ\Gamma-invariant convex set, when there is a unique €Γ\Gamma-invariant convex set, when the convex set Ω\Omega is strictly convex, when we can find a €Γ\Gamma-invariant convex set Ω\Omega' which is strictly convex.

Cite this article

Ludovic Marquis, Coxeter group in Hilbert geometry. Groups Geom. Dyn. 11 (2017), no. 3, pp. 819–877

DOI 10.4171/GGD/416