Boundary rigidity of Gromov hyperbolic spaces

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 264(3):815–836, 2013).