Paper detail

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

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ 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 $\varepsilon_n$, is virtually nilpotent.