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

Abstract

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 I" to embed into the mapping class group of a surface S, we show that the chromatic number of I" cannot exceed the chromatic number of the clique graph of the curve graph C(S). Thus, the chromatic number of I" is a global obstruction to embedding the right-angled Artin group A(I") into the mapping class group Mod(S).