Central limit theorems for Green metrics on hyperbolic groups

Abstract

Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\hat{I}“$. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\hat{I}“$ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.

23 pages