  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).
