@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",
}