Ein Gitterpunktproblem in der hyperbolischen Ebene. (A lattice point problem in the hyperbolic plane) (Q583267)

Let \(H^ n\) be the n-dimensional hyperbolic space with its group of orientation preserving isometries \(G=SO^ o(n,1)\) acting on \(H^ n\) by Möbius transformations. Given a discrete subgroup \(\Gamma\leq G\), the hyperbolic lattice point problem asks for the asymptotics of the counting function \[ A_{\Gamma}(t,z,z')=card\{\gamma \in \Gamma | d(z,\gamma z')<t\} \] for \(t\to \infty\). Here d is the hyperbolic distance on \(H^ n\). The most general result in this direction is contained in the work by \textit{P. D. Lax} and \textit{R. S. Phillips} [J. Funct. Anal. 46, 280-350 (1982; Zbl 0497.30036)]. The author establishes a special case of this result: \(n=2\) and \(\Gamma\) not cocompact but of finite covolume. Although the result of this paper is covered by the work of Lax and Phillips the method might still be of interest. It is different from that of Lax and Phillips and uses the theory of a particular class of integral operators introduced by Selberg.