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 $\operatorname{ asdim} \mathcal{ A}(\Sigma,\Gamma) \geq -\chi(\Sigma) - 1$, 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 $\operatorname{ PMap}(\Sigma)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{ M}$ has $\operatorname{ asdim} \mathcal{ M} \geq g - \lceil\frac{ 1}{ 2} \chi(\Sigma)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension.
Categories: math.GT