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

2015/11/09 by Katsuhiko Matsuzaki, Matsuzaki, Katsuhiko, Yasuhiro Yabuki +3 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1511.02664

Abstract

For a non-elementary discrete isometry group G of divergence type acting on a proper geodesic δ-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.

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