Finitude géométrique en géométrie de Hilbert

Mickaël Crampon (1) , Ludovic Marquis (2)

1 USACH - Universidad de Santiago de Chile [Santiago]
2 IRMAR - Institut de Recherche Mathématique de Rennes

We study the notion of geometrical finiteness for those Hilbert geometries defined by strictly convex sets with $\mathcal{C}^1$ 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.

Mots clés: Hilbert geometry; Geometrical finiteness; geometric group theory

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]; Théorie des groupes [math.GR]

Mickaël Crampon, Ludovic Marquis. Finitude géométrique en géométrie de Hilbert. Annales de l'Institut Fourier, 2014, 64 (6), pp.2299-2377. ⟨10.5802/aif.2914⟩. ⟨hal-00673118v3⟩