An upper bound for the volumes of complements of periodic geodesics

TL;DR

This paper establishes a linear upper bound on the hyperbolic volume of the complement of periodic geodesics' lifts in the unit tangent bundle, linking geometric length to topological complexity.

Contribution

It provides the first explicit linear upper bound for the volume of complements of periodic geodesics in hyperbolic surfaces.

Findings

Volume grows at most linearly with geodesic length

Provides a quantitative link between geodesic length and hyperbolic volume

Enhances understanding of the topology of geodesic complements

Abstract

A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.