Central limit theorems for Green metrics on hyperbolic groups

Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> . In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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.