Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Stephen Cantrell; Cagri Sert

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arXiv · 2201.00737

Counting and boundary limit theorems for representations of Gromov-hyperbolic groups

Stephen Cantrell·Cagri Sert

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Abstract

Given a Gromov-hyperbolic group $G$ 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 $G$. 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 provide a positive answer to 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.

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Stephen Cantrell, Cagri Sert. 2022-01-03. Counting and boundary limit theorems for representations of Gromov-hyperbolic groups. https://arxiv.org/abs/2201.00737

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