Origami edge-paths in the curve graph

Abstract

An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are "origami pairs of curves" -- filling pairs of simple closed curves, $ (\alpha,\beta)$, in $S_g$ such that their minimal intersection is equal to their algebraic intersection -- they are "coherent". An origami pair of curves is naturally associated with an origami on $S_g$. Our main result establishes that for any origami pair of curves there exists an "origami edge-path", a sequence of curves, $\alpha =\alpha_0, \alpha_1, \alpha_2, \cdots, \alpha_n = \beta $, such that: $\alpha_i$ intersects $\alpha_{i+1}$ at exactly once; any pair $(\alpha_i, \alpha_j)$ is coherent; and thus, any filling pair, $(\alpha_i, \alpha_j)$, is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.