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On the approaching geodesics property

Leandro Arosio, Matteo Fiacchi

TL;DR

The paper surveys the approaching geodesics property in proper geodesic Gromov hyperbolic spaces and its implications for relating the Gromov and horofunction compactifications. It shows that when approaching geodesics hold, dynamical quantities such as the minimal displacement $\tau(f)$ and divergence rate $c(f)$ of non-expanding maps coincide, and hyperbolic isometries admit a unique axis, with consequences for boundary dynamics. It also collects results in complex analysis on Kobayashi geometry, horospheres, and domain types (e.g., strongly pseudoconvex and finite type) that guarantee approaching geodesics and equivalence of compactifications, while highlighting open questions in higher dimensions and for convex domains. Finally, it provides a concrete counterexample—a Gromov hyperbolic planar domain without approaching geodesics—to demonstrate the sharpness of the assumptions and to motivate unresolved questions about horofunction versus Gromov compactifications in this setting.

Abstract

We survey some recent results and open questions on the approaching geodesics property and its application to the study of the Gromov and horofunction compactifications of a proper geodesic Gromov metric space. We obtain results on the dynamics of isometries and we exhibit an example of a Gromov hyperbolic domain of $\mathbb{C}$ which does not satisfy the approaching geodesic property.

On the approaching geodesics property

TL;DR

The paper surveys the approaching geodesics property in proper geodesic Gromov hyperbolic spaces and its implications for relating the Gromov and horofunction compactifications. It shows that when approaching geodesics hold, dynamical quantities such as the minimal displacement and divergence rate of non-expanding maps coincide, and hyperbolic isometries admit a unique axis, with consequences for boundary dynamics. It also collects results in complex analysis on Kobayashi geometry, horospheres, and domain types (e.g., strongly pseudoconvex and finite type) that guarantee approaching geodesics and equivalence of compactifications, while highlighting open questions in higher dimensions and for convex domains. Finally, it provides a concrete counterexample—a Gromov hyperbolic planar domain without approaching geodesics—to demonstrate the sharpness of the assumptions and to motivate unresolved questions about horofunction versus Gromov compactifications in this setting.

Abstract

We survey some recent results and open questions on the approaching geodesics property and its application to the study of the Gromov and horofunction compactifications of a proper geodesic Gromov metric space. We obtain results on the dynamics of isometries and we exhibit an example of a Gromov hyperbolic domain of which does not satisfy the approaching geodesic property.
Paper Structure (5 sections, 16 theorems, 58 equations, 1 figure)

This paper contains 5 sections, 16 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gromov and horofunction compactifications
  3. The approaching geodesics property in complex analysis
  4. Dynamics of non-expanding self-maps
  5. A Gromov hyperbolic planar domain without approaching geodesics

Key Result

Theorem 2.11

AFGG Let $(X,d)$ be a proper geodesic Gromov hyperbolic metric space with approaching geodesics. Then the continuous map $\Psi\colon \overline{X}^G\to \overline{X}^H$ defined by is the inverse of $\Phi$, thus $\overline{X}^G$ and $\overline{X}^H$ are equivalent.

Figures (1)

  • Figure 1: The domain $D$ if $C=\mathbb{Z}$

Theorems & Definitions (40)

  • Definition 2.1: Compactification
  • Definition 2.2: Geodesic
  • Definition 2.3: Gromov hyperbolicity
  • Definition 2.4: Gromov compactification
  • Definition 2.5: Horofunction compactification
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9: Busemann function
  • Remark 2.10
  • ...and 30 more