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Original Article

Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups

Beijing International Center for Mathematical Research and School of Mathematical Sciences, Peking University, Beijing, 100871, China
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Abstract

We prove that for a relatively hyperbolic group, there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of the group. Under natural assumptions, a similar result holds for the critical exponent of a cusp-uniform action of the group on a hyperbolic metric space. As a corollary, we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of proper quotient groups. This resolves a question of Dal’bo–Peigné–Picaud–Sambusetti. Our approach is based on the study of Patterson–Sullivan measures on Bowditch boundary of a relatively hyperbolic group and gives a series of results on growth functions of balls and cones.

Keywords

GrowthRelatively hyperbolic groupsPatterson–Sullivan measuresSmall cancellation

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Peking Mathematical Journal
Volume 5 Issue 1,
March 2022
Pages 153-212
DOI: 10.1007/s42543-020-00033-3
Cite this article:
Yang, WY. Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups. Peking Math J 5, 153-212 (2022). https://doi.org/10.1007/s42543-020-00033-3
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