Abstract commensurators of right-angled artin groups and mapping class groups

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various subgroups of the mapping class group - the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration - and additionally to the outer automorphism group of a free group and to certain linear groups.