Commensurators of abelian subgroups and the virtually abelian dimension of mapping class groups

Let Mod(S) be the mapping class group of a compact connected orientable surface S, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup H of Mod(S), there is a subgroup H' commensurable with H such that the commensurator of H equals the normalizer of H As a consequence we give, for each n >= 2, an upper bound for the geometric dimension of Mod(S) for the family of abelian subgroups of rank bounded by n. These results generalize work by JuanPineda-Trujillo-Negrete and Nucinkis-Petrosyan for the virtually cyclic case.(c) 2023 Elsevier B.V. All rights reserved.