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

Katsuhiko Matsuzaki, Yasuhiro Yabuki, Johannes Jaerisch

Research output: Contribution to journalArticle

Abstract

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

Original languageEnglish
Pages (from-to)369-411
Number of pages43
JournalGroups, Geometry, and Dynamics
Volume14
Issue number2
DOIs
Publication statusPublished - 2020

Keywords

  • Conical limit set
  • Discrete group
  • Divergence type
  • Ergodic action
  • Gromov hyperbolic space
  • Normal subgroup
  • Patterson measure
  • Poincaré series
  • Proper conjugation
  • Quasiconformal measure
  • Shadow lemma

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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