Let denote the closed orientable surface of genus . We construct exponentially many mapping class group orbits of collections of simple closed curves on which pairwise intersect exactly once, extending a result of the first author  and further answering a question of Malestein, Rivin, and 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 between and , there exists such collections whose dual cube complexes have dimension , 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.
Cite this article
Tarik Aougab, Jonah Gaster, Curves intersecting exactly once and their dual cube complexes. Groups Geom. Dyn. 11 (2017), no. 3, pp. 1061–1101DOI 10.4171/GGD/422