Mathematical Research Letters

Volume 21 (2014)

Number 3

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

Pages: 461 – 467



Matt Clay (Department of Mathematics, University of Arkansas, Fayetteville, Ark., U.S.A.)

Christopher J. Leininger (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.)

Dan Margalit (School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)


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.


right-angled Artin group, mapping class group, abstract commensurator

2010 Mathematics Subject Classification

Primary 20E36. Secondary 57M07.

Full Text (PDF format)