Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

Abstract

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant ${\epsilon }_n>0$ such that, for any properly open convex set $Ø$ and any point $x\in Ø$, any discrete group generated by a finite number of automorphisms of $Ø$, which displace $x$ at a distance less than ${\epsilon }_n$, is virtually nilpotent.

arXiv commentin French. To appear in Annales Mathématiques Blaise Pascal