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. 819–877DOI 10.4171/GGD/416