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

Given a Gromov-hyperbolic group G $G$ endowed with finite symmetric generating set, we study the statistics of counting measures on spheres associated Cayley graph under linear representations . More generally, obtain weak law large numbers for subadditive functions, echoing classical Fekete lemma. For strongly irreducible and proximal representations, prove central limit theorem Berry–Esseen type error rate exponential deviation estimates. Moreover, in same setting, show convergence interpolated normalized matrix norms along geodesic rays to Brownian motion functional iterated logarithm, paralleling analogous results theory random products. Our estimates address question Kaimanovich–Kapovich–Schupp. In most cases, our theorems will be obtained from stronger almost sure laws Patterson–Sullivan boundary group.