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

Abstract

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 Juan-Pineda–Trujillo-Negrete and Nucinkis–Petrosyan for the virtually cyclic case.