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

Katsuhiko Matsuzaki; Yasuhiro Yabuki; Johannes Jaerisch

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arXiv · 1511.02664

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

Katsuhiko Matsuzaki·Yasuhiro Yabuki·Johannes Jaerisch

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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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Katsuhiko Matsuzaki, Yasuhiro Yabuki, Johannes Jaerisch. 2019-11-08. Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space. https://arxiv.org/abs/1511.02664

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