Curves intersecting exactly once and their dual cube complexes

Abstract

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the first author [1] and further answering a question of Malestein, Rivin, and Theran [10]. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev [12]. In particular, we show that for any even $k$ between $\lfloor g/2\rfloor$ and $g$, there exists such collections whose dual cube complexes have dimension $k$, and we prove a simplifying structural theorem for any cube complex dual to a collection of curves on a surface pairwise intersecting at most once.