Rank-one Hilbert geometries

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.20F65, 20F67, 53A20; 53C60, 57N16 1. Introduction I. Preliminaries 2. Notation 3. Hilbert geometries 4. Dynamics of automorphisms II.Rank-one Hilbert geometries 5. Axis of isometries 6. Rank-one isometries: definition and properties 7. Rank-one axis and thin triangles 8. Rank-one Hilbert geometry: Zariski density and limit sets III.Contracting elements in Hilbert geometry 9. Contracting elements: definition and properties 10.Rank-one isometries are contracting 11.Contracting isometries are rank one 12.Acylindrical hyperbolicity: proof of Theorem 1.4 IV.Applications 13. Second bounded cohomology and quasimorphisms 14.Counting of conjugacy classes 15.Proofs of Propositions 1.10, 1.11 and 1.12 Appendix A. Rank-one Hilbert geometries: generalization, examples and nonexamples Appendix B.