The asymptotic dimension of the grand arc graph is infinite

Abstract

Let $\Sigma$ be a compact, orientable surface of genus $g$, and let $\Gamma$ be a relation on ${\pi }_0(\partial \Sigma )$ such that the prescribed arc graph $\mathcal{A}(\Sigma ,\Gamma )$ is Gromov-hyperbolic and non-trivial. We show that $\mathrm{asdim}\mathcal{A}(\Sigma ,\Gamma )\ge -\chi (\Sigma )-1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $\Sigma$ is a graph of simple arc and curves on $\Sigma$, on which $\mathrm{PMap}(\Sigma )$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\mathrm{asdim}\mathcal{M}\ge g-\lceil \frac{1}{2}\chi (\Sigma )\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.

19 pages. Asymptotic dimension bounds now apply to connected graphs whose vertices are finite collections of (possibly intersecting) simple arcs and curves. We show any such graph on a compact surface S admitting a cocompact action of PMap(S) is equivariantly quasi-isometric to a graph of markings likewise admitting a cocompact action, which suffices to generalize the techniques of the paper