Curves intersecting exactly once and their dual cube complexes

TL;DR

This paper constructs numerous collections of simple closed curves on surfaces that intersect exactly once, analyzes their dual cube complexes, and classifies these complexes based on their dimensions, advancing understanding of surface curve configurations.

Contribution

It introduces a method to distinguish collections of curves via dual cube complexes and constructs many such collections with specified intersection properties, extending prior results.

Findings

Existence of collections with dual cube complex dimension k for various k

Classification of dual cube complexes for curves intersecting at most once

Construction of exponentially many mapping class group orbits of such collections

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 and further answering a question of Malestein-Rivin-Theran. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev. 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.