Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces

Source abstract

Let $S_g$ be the closed oriented surface of genus $g \geq 2$, and let $Γ$ be a finite collection of closed curves on $S_g$ that fills, in the sense that its complement is a union of disks. The total number of double points of such a $Γ$ in minimal position is at least $2g-1$; we call $Γ$ minimally intersecting when equality holds. We prove that the number $N_g$ of mapping class group orbits of minimally intersecting filling curve systems satisfies $N_g \sim 16^g (2g)!/(64π\sqrt{2}\, g^3)$ as $g \to \infty$. We also study the single-curve subproblem, in which $Γ$ consists of a single component. Every such curve is in particular a filling system, so the corresponding count $N_g^{(1)}$ satisfies $N_g^{(1)} \leq N_g$, which improves the upper bound previously known for single filling curves. Finally, we compute $N_g$ and $N_g^{(1)}$ exactly for small genus and conjecture an asymptotic growth rate.