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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Statistical limit laws for hyperbolic groups


Author: Stephen Cantrell
Journal: Trans. Amer. Math. Soc. 374 (2021), 2687-2732
MSC (2020): Primary 37D40; Secondary 20F10, 20F67, 20F65
DOI: https://doi.org/10.1090/tran/8266
Published electronically: December 18, 2020
MathSciNet review: 4223030
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Abstract: Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text {CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.


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Additional Information

Stephen Cantrell
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: S.J.Cantrell@warwick.ac.uk

Received by editor(s): June 3, 2019
Received by editor(s) in revised form: April 20, 2020, April 27, 2020, June 24, 2020, and July 20, 2020
Published electronically: December 18, 2020
Article copyright: © Copyright 2020 American Mathematical Society