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

Crampon, Mickaël; Marquis, Ludovic

Annales mathématiques Blaise Pascal, Tome 20 (2013) no. 2, pp. 363-376.

On montre un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert. Plus précisément, en toute dimension $n$, il existe une constante ${\epsilon }_n>0$ telle que, pour tout ouvert proprement convexe $\Omega$, pour tout point $x\in \Omega$, tout groupe discret engendré par un nombre fini d’automorphismes de $\Omega$ qui déplacent le point $x$ de moins de ${\epsilon }_n$ est virtuellement nilpotent.

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 convex open set $\Omega$ and any point $x\in \Omega$, any discrete group generated by a finite number of automorphisms of $\Omega$, which displace $x$ at a distance less than ${\epsilon }_n$, is virtually nilpotent.

Classification : 22E40, 22F50, 57M99
Mot clés : Géométrie de Hilbert, lemme de Margulis, action géométriquement finie
Keywords: Hilbert’s geometry, lemma of Margulis, action geometrically finite

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