Equidistribution of partial coverings defined from closed geodesics

TL;DR

This paper proves that partial coverings derived from equidistributing closed geodesics on hyperbolic surfaces also exhibit equidistribution, providing an alternative proof of a known result in the modular group case.

Contribution

It establishes the equidistribution of partial coverings from closed geodesics and offers a new proof for the modular group case.

Findings

Partial coverings from geodesics equidistribute with the geodesics themselves.

The result applies to hyperbolic surfaces of finite volume.

An alternative proof of a known equidistribution theorem for the modular group.

Abstract

Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in the unit tangent bundle then the associated partial coverings also equidistribute. In the case of the modular group, this yields an alternative proof of an equidistribution result due to Duke, Imamo\={g}lu, and T\'{o}th.