Rank-one Hilbert geometries

Islam, Mitul¹

¹Department of Mathematics, University of Michigan, Ann Arbor, MI, United States, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Geometry & topology, Tome 29 (2025) no. 3, pp. 1171-1235

Résumé

We develop a notion of rank-one properly convex domains (or Hilbert geometries) in real projective space. This is in the spirit of rank-one nonpositively curved Riemannian manifolds and CAT(0) spaces. We define rank-one isometries for Hilbert geometries and characterize them as being equivalent to contracting elements (in the sense of geometric group theory). We prove that if a discrete subgroup of automorphisms of a Hilbert geometry contains a rank-one isometry, then the subgroup is either virtually cyclic or acylindrically hyperbolic. This leads to several applications like infinite dimensionality of the space of quasimorphisms, counting results for conjugacy classes and genericity results for rank-one isometries.

Détail

DOI : 10.2140/gt.2025.29.1171

Keywords: rank one, Hilbert geometry, properly convex domain, divisible hilbert geometry, nonpositive curvature, contracting elements, acylindrically hyperbolic groups

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