Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space (Q784887)

Summary: For a non-elementary discrete isometry group \(G\) of divergence type acting on a proper geodesic \(delta\)-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of \(G\). As applications of this result, we have: (1) under a minor assumption, such a discrete group \(G\) admits no proper conjugation, that is, if the conjugate of \(G\) is contained in \(G\), then it coincides with \(G\); (2) the critical exponent of any non-elementary normal subgroup of \(G\) is strictly greater than the half of that for \(G\).