Abstract Commensurators of Right-angled Artin Groups and Mapping Class Groups

COMMENSURATORS OF RIGHT-ANGLED ARTIN GROUPS AND MAPPING CLASS GROUPS MATT CLAY, CHRISTOPHER J. LEININGER, AND DAN MARGALIT Abstract. We prove that, aside from the obvious exceptions, the mapping class 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. There are many analogies and interconnections between the theories of rightangled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest [12], the work of Koberda [26], and the work of the first two authors with Mangahas [11], there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved [29] that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups. We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index subgroups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph Γ, we can associate a right-angled Artin group: this is the group with one generator for each vertex of Γ, and one defining relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the marked points. As discussed in Koberda’s paper [26, Theorem 1.5], no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g, n) 6= (2, 0); see also [23]. In particular, such mapping class groups are not abstractly commensurable 2000 Mathematics Subject Classification. Primary: 20E36; Secondary: 57M07.