Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert
[A Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometry]
Mickaël Crampon; Ludovic Marquis
Annales Mathématiques Blaise Pascal, Volume 20 (2013) no. 2, p. 363-376

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

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

DOI : https://doi.org/10.5802/ambp.330
Classification:  22E40,  22F50,  57M99
Keywords: Hilbert’s geometry, lemma of Margulis, action geometrically finite
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     author = {Crampon, Micka\"el and Marquis, Ludovic},
     title = {Un lemme de Kazhdan-Margulis-Zassenhaus pour les g\'eom\'etries de Hilbert},
     journal = {Annales Math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {20},
     number = {2},
     year = {2013},
     pages = {363-376},
     doi = {10.5802/ambp.330},
     mrnumber = {3138033},
     zbl = {1282.22007},
     language = {fr},
     url = {https://ambp.centre-mersenne.org/item/AMBP_2013__20_2_363_0}
}
Crampon, Mickaël; Marquis, Ludovic. Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert. Annales Mathématiques Blaise Pascal, Volume 20 (2013) no. 2, pp. 363-376. doi : 10.5802/ambp.330. https://ambp.centre-mersenne.org/item/AMBP_2013__20_2_363_0/

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