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.
Cite
@article{arxiv.1912.13013,
title = {Rank One Hilbert Geometries},
author = {Mitul Islam},
journal= {arXiv preprint arXiv:1912.13013},
year = {2025}
}
Comments
Major 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