Boundary rigidity of Gromov hyperbolic spaces

TL;DR

This paper establishes a connection between boundary rigidity and geometric properties of Gromov hyperbolic spaces, showing that boundary rigidity is characterized by uniform perfectness and related to nonamenability and geodesic richness.

Contribution

It introduces boundary rigidity for Gromov hyperbolic spaces and characterizes it via uniform perfectness, nonamenability, and geodesic richness, providing new insights into their geometric structure.

Findings

Boundary rigidity is equivalent to uniform perfectness of the Gromov boundary.

In Gromov hyperbolic manifolds and graphs, boundary rigidity correlates with positive Cheeger constant and nonamenability.

Hyperbolic fillings are boundary rigid if and only if the underlying metric spaces are uniformly perfect.

Abstract

We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are uniformly perfect. Also, boundary rigidity is shown to be equivalent to being geodesically rich, a concept introduced by Shchur (J. Funct. Anal., 2013).