Figure Eight Geodesics on 2-Orbifolds

TL;DR

This paper classifies and identifies the shortest figure eight geodesics on certain hyperbolic 2-orbifolds, revealing their geometric properties and minimal lengths.

Contribution

It provides a complete classification of figure eight geodesics on triangle group orbifolds and identifies the shortest such geodesic on specific hyperbolic 2-orbifolds.

Findings

Classified all figure eight geodesics on triangle group orbifolds

Identified the shortest figure eight geodesic on the (3,3,4)-triangle group orbifold

Showed the same geodesic is shortest on a hyperbolic 2-orbifold without orbifold points of order two

Abstract

It is known that the shortest non-simple closed geodesic on an orientable hyperbolic 2-orbifold passes through an orbifold point of the orbifold. This raises questions about minimal length non-simple closed geodesics disjoint from the orbifold points. Here we explore once self-intersecting closed geodesics disjoint from the orbifold points of the orbifold, called figure eight geodesics. Using fundamental domains and basic hyperbolic trigonometry we identify and classify all figure eight geodesics on triangle group orbifolds. This classification allows us to find the shortest figure eight geodesic on a triangle group orbifold, namely the unique one on the (3,3,4)-triangle group orbifold. We then show that this same curve is the shortest figure eight geodesic on a hyperbolic 2-orbifold without orbifold points of order two.