Work

Rank-one Hilbert geometries

Year: 2025

Type: article

Abstract: MITUL ISLAMWe 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.

Source: Geometry & Topology

DOI: 10.2140/gt.2025.29.1171

Author: Mitul Islam

Institutions: University of Michigan, Max Planck Institute for Mathematics in the Sciences

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funder: National Science Foundation

Award: Rigidity Phenomena in Geometry and Dynamics