# Algorithm for filling curves on surfaces

## TL;DR

This paper introduces an efficient algorithm to determine if a geodesic curve on a hyperbolic surface is filling, and provides bounds relating combinatorial and hyperbolic lengths, aiding in identifying simple geodesics.

## Contribution

It presents a novel algorithm for deciding filling curves and establishes explicit bounds linking combinatorial and hyperbolic lengths on surfaces.

## Key findings

- Algorithm efficiently determines filling curves from word representations.
- Explicit bounds relate Dehn-Thurston coordinates to hyperbolic length.
- Method aids in identifying all simple geodesics below a certain length.

## Abstract

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $\pi_1(\Sigma)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed to contain all words corresponding to simple geodesics of bounded hyperbolic length.