Groups, Geometry, and Dynamics

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Volume 11, Issue 3, 2017, pp. 1061–1101
DOI: 10.4171/GGD/422

Published online: 2017-08-22

Curves intersecting exactly once and their dual cube complexes

Tarik Aougab[1] and Jonah Gaster[2]

(1) Brown University, Providence, USA
(2) Boston College, Chestnut Hill, USA

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.

Keywords: Curves on surfaces, curve systems

Aougab Tarik, Gaster Jonah: Curves intersecting exactly once and their dual cube complexes. Groups Geom. Dyn. 11 (2017), 1061-1101. doi: 10.4171/GGD/422