JournalsggdVol. 14, No. 2pp. 369–411

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

  • Katsuhiko Matsuzaki

    Waseda University, Tokyo, Japan
  • Yasuhiro Yabuki

    Tokyo Metropolitan College of Industrial Technology, Tokyo, Japan
  • Johannes Jaerisch

    Nagoya University, Japan
Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space cover
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Abstract

For a non-elementary discrete isometry group GG of divergence type acting on a proper geodesic deltadelta-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of GG. As applications of this result, we have: (1) under a minor assumption, such a discrete group GG admits no proper conjugation, that is, if the conjugate of GG is contained in GG, then it coincides with GG; (2) the critical exponent of any non-elementary normal subgroup of GG is strictly greater than the half of that for GG.

Cite this article

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. 369–411

DOI 10.4171/GGD/548