Geometric filling curves on punctured surfaces

TL;DR

This paper establishes upper bounds on the lengths of the shortest dense closed geodesics and orthogeodesics on hyperbolic surfaces, contributing to understanding their geometric density properties.

Contribution

It provides new quantitative bounds on geodesic lengths that are dense on compact sets of hyperbolic surfaces, advancing geometric analysis in this area.

Findings

Upper bounds on shortest dense closed geodesics

Upper bounds on shortest dense orthogeodesics

Quantitative density measures on hyperbolic surfaces

Abstract

This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.