Right-angled Artin groups and finite subgraphs of curve graphs

Abstract

We show that for a sufficiently simple surface $S$, if a right-angled Artin group $A(\Gamma)$ embeds into $\mathrm{Mod}(S)$ then $\Gamma$ embeds into the curve graph $\mathcal{C}(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(\Gamma) \to \mathrm{Mod}(S)$ such that $\Gamma$ is not contained in $\mathcal{C}(S)$ as an induced subgraph.