Homogeneous Asymptotic Limits of Uniform Averages on Fuchsian Groups

We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit under appropriate scaling. This generalizes some of the results of Maucourant to subgroups of infinite co-volume. The resulting disrtibution is expressed in terms of a measure considered by Mohammadi and Oh.