Otal-Peign\'e's Theorem for Gromov-hyperbolic spaces

TL;DR

This paper generalizes Otal-Peigné's Theorem to certain Gromov-hyperbolic spaces, establishing a relationship between group action properties and dynamical invariants, with counterexamples illustrating the theorem's limitations.

Contribution

It extends the classical theorem to line-convex Gromov-hyperbolic spaces and identifies conditions under which the theorem holds or fails.

Findings

Critical exponent equals topological entropy in the specified setting

Counterexamples show the theorem does not hold universally

Provides new insights into group actions on hyperbolic spaces

Abstract

We extend the classical Otal-Peign\'e's Theorem to the class of proper, Gromov-hyperbolic spaces that are line-convex. Namely, we prove that when a group acts discretely and virtually freely by isometries on a metric space in this class then its critical exponent equals the topological entropy of the geodesic flow of the quotient metric space. We also show examples of proper, Gromov-hyperbolic spaces that are not line-convex and for which the statement fails.