Right-angled Artin groups and finite subgraphs of curve graphs

We show that for a sufficiently simple surface $S$, a right-angled Artin group $A(Γ)$ embeds into $\Mod(S)$ if and only if $Γ$ embeds into the curve graph $\mC(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(Γ)\to\Mod(S)$ for some $Γ$ not contained in $\mC(S)$.

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