Distance $4$ curves on closed surfaces of arbitrary genus

TL;DR

This paper investigates the behavior of Dehn twists on the curve complex of closed surfaces of genus g, demonstrating how they can produce vertices at distance 4 and establishing bounds on intersection numbers.

Contribution

It provides new insights into the action of Dehn twists on the curve complex, specifically constructing distance 4 vertices and bounding their intersection numbers.

Findings

Dehn twists can produce vertices at distance 4 in the curve complex.

Vertices at distance 4 have intersection numbers at most (2g-1)^2.

Examples of distance 4 vertices are explicitly constructed.

Abstract

Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ be the associated curve complex. The mapping class group of $S_g$, $Mod(S_g)$ acts on $\mathcal{C}(S_g)$ by isometries. Since Dehn twists about certain curves generate $Mod(S_g)$, one can ask how Dehn twists move specific vertices in $\mathcal{C}(S_g)$ away from themselves. We show that if two curves represent vertices at a distance $3$ in $\mathcal{C}(S_g)$ then the Dehn twist of one curve about another yields two vertices at distance $4$. This produces many tractable examples of distance $4$ vertices in $\mathcal{C}(S_g)$. We also show that the minimum intersection number of any two curves at a distance $4$ on $S_g$ is at most $(2g-1)^2$.