A Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometry (Q395372)

The main result of this note is a Kazhdan-Margulis-Zassenhaus type lemma: given the dimension \(n\) there exists a constant \(\varepsilon _n>0\) such that for any Hilbert geometry \((\Omega \subset \mathbb{P}^n, d_{\Omega })\), any point \(x\in \Omega \) and any discrete subgroup \(\Gamma \in \text{Aut}(\Omega )\) the group \(\Gamma _{\varepsilon _n}\) generated by \(\{\gamma \in \Gamma ; d_{\Omega }(x, \gamma \cdot x)\leq \varepsilon _n\}\) is virtually nilpotent.