Rank One Hilbert Geometries

Abstract

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively 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 quasi-morphisms, counting results for conjugacy classes and genericity results for rank one isometries.

arXiv commentMajor re-writing in Section 8 which also improves the results in that section. Results in all other sections remain unaffected. Other minor revisions based on referees' comments, notably in Sections 1, 4, 5, 12, and Appendix B. Final version to appear in Geometry & Topology. 72 pages. 1 figure. Comments welcome

References in corpus (10)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Rank rigidity for CAT(0) cube complexes
• Anosov representations and proper actions
• Convex projective structures on Gromov--Thurston manifolds
• Convex projective structures on non-hyperbolic three-manifolds
• Recognizing a relatively hyperbolic group by its Dehn fillings
• A flat torus theorem for convex co-compact actions of projective linear groups
• Convex co-compact actions of relatively hyperbolic groups
• A higher rank rigidity theorem for convex real projective manifolds
• A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries