On purely loxodromic actions

2015/04/30 by Ilya Kapovich · 1 citation
Mathematics · #Action (physics) #Biology #Combinatorics #Computer science #Embedding #Geometric and Algebraic Topology #Geometry #Graph #Isometric exercise #Mathematical Dynamics and Fractals #Mathematics #Orbit (dynamics) #Physics #Point processes and geometric inequalities #Pure mathematics #Quasiconvex function #Regular polygon #math.GR #math.GT #msc:20F65

paper · pdf · doi:10.1007/s00605-015-0795-7

published as Monatshefte für Mathematik \textbf{181} (2016), no. 1, 89--101 · several minor fixes and updates; 9 pages, no figures; to appear in Monatshefte für Mathematik

arxiv created 2015/07/07 · openalex publication_date 2015/07/20 · arxiv updated 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We construct an example of an isometric action of F(a,b) on a δ-hyperbolic graph Y, such that this action is acylindrical, purely loxodromic, has asymptotic translation lengths of nontrivial elements of F(a,b) separated away from 0, has quasiconvex orbits in Y, but such that the orbit map F(a,b)→ Y is not a quasi-isometric embedding.

Citations

• Small cancellation in acylindrically hyperbolic groups
• Word hyperbolic extensions of surface groups
• Geometry of the complex of curves I: Hyperbolicity
• Tight geodesics in the curve complex
• Laminations, trees, and irreducible automorphisms of free groups
• Train Tracks and Automorphisms of Free Groups
• Acylindrical accessibility for groups
• Metric Spaces of Non-Positive Curvature
• RELATIVELY HYPERBOLIC GROUPS
• Acylindrically hyperbolic groups

Cited by

• The co‐surface graph and the geometry of hyperbolic free group extensions