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Central limit theorems for counting measures in coarse negative curvature

Part of: Limit theorems Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  03 November 2022

Ilya Gekhtman ,
Samuel J. Taylor  and
Giulio Tiozzo
Ilya Gekhtman
Affiliation:
Faculty of Mathematics, Technion-Israel Institute of Technology, Haifa 3200003, [email protected]
Samuel J. Taylor
Affiliation:
Department of Mathematics, Temple University, 1805 North Broad Street, Philadelphia, PA 19122, USA [email protected]
Giulio Tiozzo
Affiliation:
Department of Mathematics, University of Toronto, 40 St George St, Toronto, ON M5S 2E4, Canada [email protected]
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Abstract

We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

Keywords

central limit theoremhyperbolic spacesgroup actionsdisplacementtranslation lengthcounting measurerandom walk

MSC classification

Primary: 60F05: Central limit and other weak theorems 57M60: Group actions in low dimensions 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 10 , October 2022 , pp. 1980 - 2013
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

Gekhtman is partially supported by NSERC. Taylor is partially supported by NSF grants DMS-1744551, DMS-2102018 and the Sloan Foundation. Tiozzo is partially supported by NSERC, an Ontario Early Researcher Award and the Sloan Foundation.

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