Homogeneous Asymptotic Limits of Uniform Averages on Fuchsian Groups

TL;DR

This paper establishes that uniform averages on geometrically finite Fuchsian groups, when embedded into matrix spaces, converge to a homogeneous limit under scaling, extending previous results to infinite co-volume subgroups.

Contribution

It generalizes Maucourant's results to infinite co-volume subgroups of Fuchsian groups using a measure from Mohammadi and Oh.

Findings

Averages on Fuchsian groups have a homogeneous asymptotic limit.

The results extend to geometrically finite groups with infinite co-volume.

Distribution described via a measure by Mohammadi and Oh.

Abstract

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.