On the approaching geodesics property

TL;DR

This paper surveys recent developments and open questions regarding the approaching geodesics property, exploring its implications for Gromov and horofunction compactifications, isometry dynamics, and providing a counterexample in hyperbolic domains.

Contribution

It compiles recent results, discusses open problems, and presents a new example illustrating limitations of the approaching geodesics property in hyperbolic domains.

Findings

The approaching geodesics property influences the structure of compactifications.

An example of a hyperbolic domain lacking the property is provided.

Results on isometry dynamics related to the property are discussed.

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.