Quantifying the sparseness of simple geodesics on hyperbolic surfaces

TL;DR

This paper provides explicit measures of how sparse simple closed geodesics are on hyperbolic surfaces, demonstrating that certain small disks are free of such geodesics, depending on the surface's topology.

Contribution

It introduces explicit quantifications of the non-density of simple closed geodesics on hyperbolic surfaces, linking geometric and topological properties.

Findings

Existence of a disk disjoint from simple closed geodesics within any embedded disk

Quantitative bounds depending on surface topology and initial disk size

Explicit measures of geodesic sparseness on hyperbolic surfaces

Abstract

The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded disk), which is disjoint from any simple closed geodesic.