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

The hyperbolic lattice point problem in conjugacy classes