Finitude géométrique en géométrie de Hilbert
Crampon, Mickaël ; marquis, Ludovic
Annales de l'Institut Fourier, Tome 64 (2014), p. 2299-2377 / Harvested from Numdam
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On étudie la notion de finitude géométrique pour certaines géométries de Hilbert définies par un ouvert strictement convexe à bord de classe 𝒞 1 .

La définition dans le cadre des espaces Gromov-hyperboliques fait intervenir l’action du groupe discret considéré sur le bord de l’espace. On montre, en construisant explicitement un contre-exemple, que cette définition doit être renforcée pour obtenir des définitions équivalentes en termes de la géométrie de l’orbifold quotient, similaires à celles obtenues par Brian Bowditch en géométrie hyperbolique.

We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with 𝒞 1 boundary.

In Gromov-hyperbolic spaces, geometrical finiteness is defined by a property of the group action on the boundary of the space. We show by constructing an explicit counter-example that this definition has to be strenghtened in order to get equivalent characterizations in terms of the geometry of the quotient orbifold, similar to those obtained by Brian Bowditch in hyperbolic geometry.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2914
Classification:  22E40,  20F67,  20F65,  53C60
Mots clés: géométrie de Hilbert, finitude géométrique, espace Gromov-hyperbolique, sous-groupes discrets des groupes de Lie, variété projective convexe 
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     title = {Finitude g\'eom\'etrique en g\'eom\'etrie de Hilbert},
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     volume = {64},
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     pages = {2299-2377},
     doi = {10.5802/aif.2914},
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Crampon, Mickaël; marquis, Ludovic. Finitude géométrique en géométrie de Hilbert. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2299-2377. doi : 10.5802/aif.2914. http://gdmltest.u-ga.fr/item/AIF_2014__64_6_2299_0/

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