A lattice point problem in hyperbolic space (Q792493)

Let G be a discrete group of isometries of the \((n+1)\)-dimensional hyperbolic space \({\mathbb{H}}^{n+1}\) of finite covolume. Let d denote the distance function in \({\mathbb{H}}^{n+1}\). Let \(S\subset {\mathbb{H}}^{n+1}\) and denote by \(N_ S(w_ 1,w_ 2;X)\) the number of \(g\in G\) such that \(gw_ 1\in S\) and \(d(gw_ 1,w_ 2)\leq X\). The author determines the asymptotic behaviour of \(N_ S(w_ 1,w_ 2;X)\) as \(X\to \infty\) when S is a hyperbolic cone with centre \(w_ 2\), and derives an interesting application to the case when S is a Stolz cone at a limit point of G (when G is cocompact). The proof is based on Hopf's ergodic theorem and subsidiary geometric considerations.