Otal-Peigné's Theorem for Gromov-hyperbolic spaces

Abstract

We extend the classical Otal-Peigné'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.

This paper comes as an extension of a work that has been divided in two parts. The original first part corresponds to arXiv:2105.11774. The second part contained a mistake that has been corrected with a completely new strategy. The new technique allows to extend the main result to a much braoder class of metric spaces