Subsurface 1-Distance of the Handlebody

Abstract

For a handlebody $H$ with $∂H = S$, let $F ⊂ S$ be an essential connected subsurface of $S$. Let $\mathcal{C}(S)$ be the curve complex of $S$, $\mathcal{AC}(F)$ be the arc and curve complex of $F$, $\mathcal{D}(H) ⊂ \mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ⊂ \mathcal{AC}(F)$ be the image of $\mathcal{D}(H)$ in $\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.