Boundary rigidity of Gromov hyperbolic spaces (Q6618805)

A Gromov hyperbolic metric space \((X,d)\) is boundary rigid if for any quasi-isometry \(f: X \rightarrow X\) whose induced map on \(\partial X\) is the identity map, we have finite displacement. The displacement of \(f\) is defined as \(\sup_{x\in X}d(x,f(x))\). Let \[QI(X)=\{f \mid f: X \rightarrow C \mbox{ is a a quasi-isometry} \} \big / \thicksim\] be the group of self quasi-isometries of \(X\), where \(f \thicksim g\) if and only if \(\sup_{x\in X}d(f(x),g(x)) < \infty\). Every self quasi-isometry \(f\) of \(X\) induces a self homeomorphism \(\partial f\) of \(\partial X\) and quasi-isometries uniformly close to each other induce the same homeomorphism of \(\partial X\). Therefore, there is a natural homomorphism \(\partial : QI(X) \rightarrow \mathrm{Homeo}(\partial X)\) which sends \(f\) to \(\partial f\). Boundary rigidity has the following equivalent definition: a Gromov hyperbolic metric space \(X\) is called boundary rigid if the natural homomorphism \(\partial: QI(X) \rightarrow \mathrm{Homeo}(\partial X)\) is injective.\N\NIn the paper under review, the authors 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, they 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 non-amenable. 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 \textit{V. Shchur} [J. Funct. Anal. 264, No. 3, 815--836 (2013; Zbl 1260.53078)] (corrected in [\textit{S. Gouëzel} and \textit{V. Shchur}, J. Funct. Anal. 277, No. 4, 1258--1268 (2019; Zbl 1419.53049)]).