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

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 half of that for G.