An Obstruction to Embedding Right-Angled Artin Groups in Mapping Class Groups

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension 2 which does not embed into the associated mapping class group. For a right-angled Artin group on a graph Γ to embed into the mapping class group of a surface S, we show that the chromatic number of Γ cannot exceed the chromatic number of the clique graph of the curve graph $\mathcal C(\mathcal S)$. Thus, the chromatic number of Γ is a global obstruction to embedding the right-angled Artin group A(Γ) into the mapping class group Mod(S).