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Two Characterizations of Geometrically Infinite Actions on Gromov Hyperbolic Spaces

Chaodong Yang, Wenyuan Yang

Abstract

We provide two new characterizations of geometrically infinite actions on Gromov hyperbolic spaces: one in terms of the existence of escaping geodesics, and the other via the presence of uncountably many non-conical limit points. These results extend corresponding theorems of Bonahon, Bishop, and Kapovich-Liu from the settings of Kleinian groups and pinched negatively curved manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces.

Two Characterizations of Geometrically Infinite Actions on Gromov Hyperbolic Spaces

Abstract

We provide two new characterizations of geometrically infinite actions on Gromov hyperbolic spaces: one in terms of the existence of escaping geodesics, and the other via the presence of uncountably many non-conical limit points. These results extend corresponding theorems of Bonahon, Bishop, and Kapovich-Liu from the settings of Kleinian groups and pinched negatively curved manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces.
Paper Structure (14 sections, 29 theorems, 53 equations, 4 figures)

This paper contains 14 sections, 29 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence actions
  4. Geometrically finite actions
  5. Escaping Closed Geodesics
  6. The main proposition
  7. Identifying the cusp regions
  8. Completion of the proof of Theorem \ref{['geometricallyFiniteNearOrbit']}
  9. Necessity direction.
  10. Sufficiency direction.
  11. Uncountably Many Non-conical Points
  12. Existence of escaping hyperbolic elements in one direction
  13. Construction of escaping geodesic rays
  14. Proof of Theorem \ref{['geometricallyFiniteNonconicalCountable']}

Key Result

Theorem 1.1

Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.

Figures (4)

  • Figure 1: Illustration for the proof of Proposition \ref{['lineNearOrbit']}.
  • Figure 2: Ideal triangle and a nearby orbit point.
  • Figure 3:
  • Figure 4:

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3: bridson2013metric, Page 405
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 42 more