Prescribed Arc Graphs

Abstract

Given a compact surface $\Sigma$ with boundary and a relation $\Gamma$ on ${\pi }_0(\partial \Sigma )$, we define the prescribed arc graph $\mathcal{A}(\Sigma ,\Gamma )$ to be the full subgraph of the arc graph $\mathcal{A}(\Sigma )$ containing only classes of arcs between boundary components in $\Gamma$. We prove that $(\Sigma ,\Gamma )$ is connected and infinite-diameter (if $\Sigma$ is not the sphere with three boundary components), and classify when it is Gromov hyperbolic: in particular, $(\Sigma ,\Gamma )$ is Gromov hyperbolic if and only if $\Gamma$ is not bipartite, except in some sporadic cases.

arXiv comment21 pages, 7 figures