Abelian subgroups of the Torelli group

Abstract

Let $S$ be a closed oriented surface of genus $g\ge 2$, and let $\mathcal{T}$ denote its Torelli group. First, given a set $\mathfrak{E}$ of homotopically nontrivial, pairwise disjoint, pairwise nonisotopic simple closed curves on $S$, we determine precisely when a multitwist on $\mathfrak{E}$ is an element of $\mathcal{T}$ by defining an equivalence relation on $\mathfrak{E}$ and then applying graph theory. Second, we prove that an arbitrary Abelian subgroup of $\mathcal{T}$ has rank $\le 2g-3$.