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

Katsuhiko Matsuzaki; Yasuhiro Yabuki; Johannes Jaerisch

doi:10.4171/GGD/548

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

Katsuhiko Matsuzaki, Yasuhiro Yabuki, Johannes Jaerisch

School of Education

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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 language	English
Pages (from-to)	369-411
Number of pages	43
Journal	Groups, Geometry, and Dynamics
Volume	14
Issue number	2
DOIs	https://doi.org/10.4171/GGD/548
Publication status	Published - 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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10.4171/GGD/548

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@article{6a6c05fa64494920ba5ced725eaa9cf6,

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

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.",

keywords = "Conical limit set, Discrete group, Divergence type, Ergodic action, Gromov hyperbolic space, Normal subgroup, Patterson measure, Poincar{\'e} series, Proper conjugation, Quasiconformal measure, Shadow lemma",

author = "Katsuhiko Matsuzaki and Yasuhiro Yabuki and Johannes Jaerisch",

note = "Funding Information: Acknowledgments. Theorems 1.2 and 1.3 were studied by the first two authors and announced in the conference “Rigidity School” held at University of Tokyo on March 19, 2012. Theorem 1.4 began as a different subject by the third author but was recently incorporated in the present study. This work was supported by JSPS KAKENHI 25287021 and 16K13767. Publisher Copyright: {\textcopyright} European Mathematical Society.",

year = "2020",

doi = "10.4171/GGD/548",

language = "English",

volume = "14",

pages = "369--411",

journal = "Groups, Geometry, and Dynamics",

issn = "1661-7207",

publisher = "European Mathematical Society Publishing House",

number = "2",

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TY - JOUR

T1 - Normalizer, divergence type, and Patterson measure for discrete groups of the Gromov hyperbolic space

AU - Matsuzaki, Katsuhiko

AU - Yabuki, Yasuhiro

AU - Jaerisch, Johannes

N1 - Funding Information: Acknowledgments. Theorems 1.2 and 1.3 were studied by the first two authors and announced in the conference “Rigidity School” held at University of Tokyo on March 19, 2012. Theorem 1.4 began as a different subject by the third author but was recently incorporated in the present study. This work was supported by JSPS KAKENHI 25287021 and 16K13767. Publisher Copyright: © European Mathematical Society.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Conical limit set

KW - Discrete group

KW - Divergence type

KW - Ergodic action

KW - Gromov hyperbolic space

KW - Normal subgroup

KW - Patterson measure

KW - Poincaré series

KW - Proper conjugation

KW - Quasiconformal measure

KW - Shadow lemma

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U2 - 10.4171/GGD/548

DO - 10.4171/GGD/548

M3 - Article

AN - SCOPUS:85091284333

SN - 1661-7207

VL - 14

SP - 369

EP - 411

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

IS - 2

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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