Filling with separating curves

TL;DR

This paper characterizes the existence of separating filling pairs on surfaces, studies their symmetry actions, and constructs a Morse function to analyze their geometric properties.

Contribution

It provides a necessary and sufficient condition for separating filling pairs with two disks, explores the mapping class group action, and introduces a Morse function on moduli space.

Findings

Necessary and sufficient condition for existence of separating filling pairs

Analysis of mapping class group action on these pairs

Construction of a Morse function related to shortest filling pairs

Abstract

A pair $(\alpha, \beta)$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $\alpha$ is separating, then we call it as separating filling pair. In this article, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\M$ on the set of such filling pairs. Furthermore, we construct a Morse function $\mathcal{F}_g$ on the moduli space $\mathcal{M}_g$ which, for a given hyperbolic surface $X$, outputs the length of shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function $\mathcal{F}_g$ is the same as the number of $\M$-orbits of such filling pairs.