2019/12/30 by Mitul Islam, Islam, Mitul · 1 citation
Mathematics · #20F65 #20F67 #53A20 #57N16 #58B20 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1912.13013
openalex publication_date 2019/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.