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For a non-elementary discrete isometry group of divergence type acting on a proper geodesic -hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of . As applications of this result, we have: (1) under a minor assumption, such a discrete group admits no proper conjugation, that is, if the conjugate of is contained in , then it coincides with ; (2) the critical exponent of any non-elementary normal subgroup of is strictly greater than the half of that for .
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Katsuhiko Matsuzaki, Yasuhiro Yabuki, Johannes Jaerisch, Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space. Groups Geom. Dyn. 14 (2020), no. 2, pp. 369–411DOI 10.4171/GGD/548