#### Volume 2, issue 1 (2002)

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Abelian subgroups of the Torelli group

### William R Vautaw

Algebraic & Geometric Topology 2 (2002) 157–170
 arXiv: math.GT/0203131
##### Abstract

Let $S$ be a closed oriented surface of genus $g\ge 2$, and let $\mathsc{T}$ 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 $\mathsc{T}$ by defining an equivalence relation on $\mathfrak{E}$ and then applying graph theory. Second, we prove that an arbitrary Abelian subgroup of $\mathsc{T}$ has rank $\le 2g-3$.

##### Keywords
mapping class group, Torelli group, multitwist
Primary: 57M60
Secondary: 20F38
##### Publication
Received: 12 December 2001
Revised: 24 February 2002
Accepted: 28 February 2002
Published: 6 March 2002
##### Authors
 William R Vautaw Department of Mathematics Michigan State University East Lansing MI 48824 USA