On the orbit function of discrete groups in negative curvature (Q5956594)

Let \(\Gamma\) be a non-elementary discrete group of isometries of a metric space \(X\) and \(O\) be a fixed point in \(X\). Denote by \(N_\Gamma(R)\) the number of points of the orbit point \(O\) with respect to \(\Gamma\), which belong to the ball with center \(O\) and radius \(R\). Set \(\delta(\Gamma)=\overline\lim_{R\to\infty}(\log N_\Gamma(R)/R)\). Theorem. \(\log N_\Gamma(R)/R\to\delta(\Gamma)\) as \(R\to\infty\).