The curves not carried

TL;DR

This paper proves that the set of simple closed curves not carried by a train track on a surface forms a quasi-convex subset in the curve complex, complementing previous results about carried curves.

Contribution

It establishes the quasi-convexity of the complement of curves carried by a train track in the curve complex, extending prior work on carried curves.

Findings

The complement of $C( au)$ in the curve complex is quasi-convex.

This complements Masur and Minsky's result on $C( au)$ being quasi-convex.

Provides new geometric insight into the structure of the curve complex.

Abstract

Suppose $\tau$ is a train track on a surface $S$. Let $C(\tau)$ be the set of isotopy classes of simple closed curves carried by $\tau$. Masur and Minsky [2004] prove $C(\tau)$ is quasi-convex inside the curve complex $C(S)$. We prove the complement, $C(S) - C(\tau)$, is quasi-convex.