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Convexes Hyperboliques et Quasiisométries

Hyperbolic Convexes and Quasiisometries

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Abstract

For any m ≥ 3, we construct properly convex open sets Ω in the real projective space \(\mathbb{P}^m\) whose Hilbert metric is Gromov hyperbolic but is not quasiisometric to the hyperbolic space \(\mathbb{H}^m\). We show that such examples cannot exist for m = 2.

Some of our examples are divisible, i.e. there exists a discrete group Г of projective transformations preserving Ω with a compact quotient Г\Ω. The open set Ω is strictly convex but the group Г is not isomorphic to any cocompact lattice in the isometry group of \(\mathbb{H}^m\).

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  1. Ecole Normale Supérieure-CNRS, 45 rue d’Ulm, 75230, Paris, France

    Yves Benoist

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  1. Yves Benoist
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Benoist, Y. Convexes Hyperboliques et Quasiisométries. Geom Dedicata 122, 109–134 (2006). https://doi.org/10.1007/s10711-006-9066-z

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