The hyperbolic lattice point problem in conjugacy classes

For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces $\Gamma\backslash \mathbb H$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound $O(X^{2/3})$, due to A. Good. For $\hbox{SL}(2,{\mathbb Z})$ we interpret our results in terms of indefinite quadratic forms.