2023/01/14 by Bhola Nath Saha, Saha, Bhola Nath, Bidyut Sanki +1
Mathematics · #05C10 #57M15 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2301.05840
openalex publication_date 2023/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A pair (α, β) of simple closed curves on a closed and orientable surface Sg of genus g is called a filling pair if the complement is a disjoint union of topological disks. If α 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 Sg 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 Fg on the moduli space Mg 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 Fg is the same as the number of \M-orbits of such filling pairs.