Coxeter group in Hilbert geometry

  • Ludovic Marquis

    Université de Rennes I, France


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