Coxeter group in Hilbert geometry

TL;DR

This paper studies actions of Coxeter groups on convex sets in Hilbert geometry, providing conditions for various finiteness properties and describing the structure of invariant convex sets.

Contribution

It establishes sufficient and necessary conditions for Coxeter group actions to be of finite covolume, convex-cocompact, or geometrically finite, and analyzes invariant convex sets.

Findings

Conditions for finite covolume, convex-cocompact, and geometrically finite actions.

Descriptions of the Zariski closure of Coxeter groups.

Characterizations of invariant convex sets in various convexity conditions.

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.