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

TL;DR

This paper investigates statistical properties of Gromov-hyperbolic groups under linear representations, establishing limit theorems, large deviation estimates, and connections to Brownian motion and Patterson–Sullivan measures.

Contribution

It introduces new counting limit theorems and large deviation estimates for Gromov-hyperbolic groups, extending classical probabilistic results to this geometric setting.

Findings

Proves a weak law of large numbers for subadditive functions.

Establishes a counting central limit theorem with error rates.

Shows convergence of matrix norms to Brownian motion and a law of iterated logarithm.

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.