Growth of conjugacy classes in Gromov hyperbolic groups

Abstract

Let \( \Gamma \) be a group acting properly and cocompactly by isometries on a proper geodesic \( \delta \)-hyperbolic metric space X whose boundary contains more than two points. Let P(t) denote the number of conjugacy classes of primitive elements \( \gamma \in \Gamma \) such that \( {\rm inf}_{x\in X}d(x,\gamma x) \le t \). We prove that there are positive constants A, B, h and t0 such that \( Ae^{ht}/t \le P(t) \le Be^{ht} \) for all \( t \ge t_0 \).