Filling with separating curves

Abstract

A pair [Formula: see text] of simple closed curves on a closed and orientable surface [Formula: see text] of genus [Formula: see text] is called a filling pair if the complement is a disjoint union of topological disks. If [Formula: see text] is separating, then we call it as separating filling pair. In this paper, we find a necessary and sufficient condition for the existence of a separating filling pair on [Formula: see text] with exactly two complementary disks. We study the combinatorics of the action of the mapping class group [Formula: see text] on the set of such filling pairs. Furthermore, we construct a Morse function [Formula: see text] on the moduli space [Formula: see text] which, for a given hyperbolic surface [Formula: see text], outputs the length of the shortest such filling pair with respect to the metric in [Formula: see text]. We show that the cardinality of the set of global minima of the function [Formula: see text] is the same as the number of [Formula: see text]-orbits of such filling pairs.