The curves not carried

Abstract

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