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 grow at least exponentially in $g$.Comment: 26 pages, 11 figure

Minimally intersecting filling pairs on the punctured surface of genus two

“…For a fixed surface S g,p , we will let i g,p denote the minimal geometric intersection number of a filling pair on that surface. As discussed above, the values of i g,p were determined in almost all cases in the works of Aougab-Huang [1] and Aougab-Taylor [2]. However, in the case of S 2,p , p ≥ 3 odd, they showed only the bounds:…”

Small intersection numbers in the curve graph

“…For (1), [2] produce pairs of filling curves on a closed surface of genus g 3 that intersect 2g − 1 times by an explicit construction. For (2), first note that when p = 0 any pair of curves that fills and intersects 2g − 1 times must have a single disk as its complementary region.…”

“…Therefore, to finish the proof of (2) it suffices to exhibit a filling pair on S g,0 , g > 2, intersecting 2g times. Consider the polygonal decomposition of S 2,0 shown in Figure 2, originally constructed in [2].…”

Filling of closed surfaces

“…The combinatorial complexity of a filling X of F g (g ≥ 2), denoted by T k (X), is the number of simple closed curves which intersect the union of curves in X no more than k times. In [4] (Theorem 1.2), Aougab and Huang have proved that T 1 (α, β) ≤ 4g − 2 and equality holds if and only if (α, β) is a minimal filling pair of F g . Fillings of surfaces are well studied in topology and geometry.…”

“…It follows from the Euler's equation that if (α, β) is a minimal filling of F g , then i(α, β) = 2g − 1. In [4] (see Theorem 1.1), the authors have shown that for all g = 2, there exist minimal filling pairs of F g and for g = 2, if (α, β) is a filling pair of F 2 , then i(α, β) ≥ 4 (see Theorem 2.17 in [4]). Therefore, there is no minimal filling pair of F 2 .…”