Small intersection numbers in the curve graph

Abstract

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p\ge 0$ punctures, and let $\omega (g,p)=3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_0,v_1,v_2,...\right\}$ in the curve graph satisfying the following optimal intersection property: for any natural number $k$, the endpoints $v_i,v_{i+k}$ of any length $k$ subsegment intersect $O({\omega }^{k-2})$ times. By combining this with work of the first author, we answer a question of Dan Margalit.

13 pages, 6 figures

References in corpus (2)

• Minimally intersecting filling pairs on surfaces
• A short proof of the bounded geodesic image theorem

Cited by in corpus (4)

• Efficient geodesics and an effective algorithm for distance in the complex of curves
• Minimally intersecting filling pairs on the punctured surface of genus two
• Distance $4$ curves on closed surfaces of arbitrary genus
• MICC: A tool for computing short distances in the curve complex