Rank-one Hilbert geometries

Islam, Mitul

doi:10.2140/gt.2025.29.1171

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Rank-one Hilbert geometries

Mitul Islam

Geometry & Topology 29 (2025) 1171–1235

DOI: 10.2140/gt.2025.29.1171

Abstract

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.

Keywords

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

Mathematical Subject Classification

Primary: 20F65, 20F67, 53A20

Secondary: 53C60, 57N16

References

Publication

Received: 2 July 2021

Revised: 26 November 2023

Accepted: 30 December 2023

Published: 31 May 2025

Proposed: Mladen Bestvina

Seconded: Benson Farb, David Fisher

Authors

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

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Volume 29, issue 3 (2025)