Central limit theorems for counting measures in coarse negative curvature

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 seve...

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Bibliographic Details
Published in:Compositio mathematica 2022-10, Vol.158 (10), p.1980-2013
Main Authors: Gekhtman, Ilya, Taylor, Samuel J., Tiozzo, Giulio
Format: Article
Language:English
Subjects:
Central limit theorem
Geodesy
Hyperbolic coordinates
Manifolds (mathematics)
Smoothness
Theorems
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Summary: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.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X22007680