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Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups.
- Source :
-
Proceedings of the London Mathematical Society . Sep2023, Vol. 127 Issue 3, p589-652. 64p. - Publication Year :
- 2023
-
Abstract
- Given a Gromov‐hyperbolic group G$G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of G$G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00246115
- Volume :
- 127
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Proceedings of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 171348490
- Full Text :
- https://doi.org/10.1112/plms.12550