Denote by \(\Gamma\) a hyperbolic group in the sense of Gromov, fix a finite generating set of \(\Gamma\) and let \(\partial\Gamma\) denote the boundary of the Cayley graph. Let \(\mu\) denote any finite Borel measure on \(\partial\Gamma\) and assume that \(\mu\) is quasi-invariant under the action of \(\Gamma\) on \(\partial\Gamma\). Then the action of \(\Gamma\) on \((\partial\Gamma, \mu)\) is amenable.
The following application of this result is given. A group is almost simple if every normal subgroup is finite. Let \(G\) be a connected, almost simple Lie group with \(\mathbb{R}\text{-rank}(G)\geq 2\). Let \(M\) be a compact manifold and suppose there is a real analytic action of \(G\) on \(M\) preserving a real analytic connection and a finite measure. Then \(\pi_1 (M)\) is not isomorphic to a subgroup of a hyperbolic group.