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

Abstract

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 .