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

Let $\mathrm{M}\mathrm{o}\mathrm{d}(S)$ 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 $\mathrm{M}\mathrm{o}\mathrm{d}(S)$, there is a subgroup $H^{\mathrm{\prime }}$ commensurable with $H$ such that the commensurator of $H$ equals the normalizer of $H^{\mathrm{\prime }}$. As a consequence we give, for each $n\ge 2$, an upper bound for the geometric dimension of $\mathrm{M}\mathrm{o}\mathrm{d}(S)$ for the family of abelian subgroups of rank bounded by $n$. These results generalize work by Juan-Pineda–Trujillo-Negrete and Nucinkis–Petrosyan for the virtually cyclic case.