The hyperbolic lattice point problem in conjugacy classes.

The hyperbolic lattice point problem deals with estimating the number of elements in a Fuchsian group , as measured by their action on the hyperbolic space. But what if, instead of the whole group, we look at subsets of the group ? In this talk, we concentrate on a conjugacy class of , and: (1) We explain the geometric significance in terms of geodesic segments. (2) We explain the main term in the counting problem and provide many theorems on the error term which are conjecturally optimal. (3) We interpret our results in terms of quadratic forms.

This is a joint work with Yiannis Petridis (UCL).