2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68773We investigate the relationship between the metric boundary and the Gromov boundary of a hyperbolic metric space. We show that the Gromov boundary is a quotient topological space of the metric boundary, and that therefore a word-hyperbolic group has an amenable action on the metric boundary of its Cayley graph. This result has significance for the study of Lip-norms on group C*-algebras.9 pages. AMSLaTeXMetric GeometryGroup TheoryOperator Algebras20F65 (Primary) 46L87, 53C23 (Secondary)text