Minimally intersecting filling pairs on surfaces

Abstract

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_g$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function ${\mathcal{ℱ}}_g$ over the moduli space of Riemann surfaces of genus $g$, which, given a hyperbolic metric $\sigma$, outputs the length of the shortest minimally intersecting filling pair for the metric $\sigma$. We completely characterize the global minima of ${\mathcal{ℱ}}_g$ and, using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grows at least exponentially in $g$.