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

TL;DR

This paper demonstrates that most mapping class groups of surfaces are not abstractly commensurable with right-angled Artin groups, highlighting fundamental differences in their algebraic structures.

Contribution

It establishes the non-commensurability between mapping class groups and right-angled Artin groups, extending the result to various subgroups and related groups.

Findings

Mapping class groups are not abstractly commensurable with right-angled Artin groups.

The result applies to subgroups generated by powers of Dehn twists and Johnson filtration.

Outer automorphism groups of free groups and certain linear groups also differ in this regard.

Abstract

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.