McShane's identity, using elliptic elements

TL;DR

This paper presents a novel approach to proving McShane's Identity by leveraging the action of elliptic elements of order two in the Fuchsian group, providing new insights into hyperbolic geometry.

Contribution

It introduces a new method for establishing McShane's Identity using elliptic elements, offering a different perspective from traditional proofs.

Findings

New proof of McShane's Identity using elliptic elements

Elliptic elements of order two exclude certain geodesic points

Enhanced understanding of hyperbolic surface geometry

Abstract

We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics.