Un lemme de Kazhdan-Margulis-Zassenhaus pour les g\'eom\'etries de Hilbert

TL;DR

This paper establishes a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries, showing that groups generated by automorphisms displacing a point less than a certain threshold are virtually nilpotent.

Contribution

It extends the Kazhdan-Margulis-Zassenhaus lemma to Hilbert geometries, providing a uniform displacement threshold in all dimensions.

Findings

Existence of a universal displacement constant _n in each dimension

Groups generated by automorphisms with small displacement are virtually nilpotent

The result applies to all properly open convex sets in Hilbert geometries

Abstract

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 $\O$ and any point $x \in \O$, any discrete group generated by a finite number of automorphisms of $\O$, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent.