#### Algebraic and Geometric Topology 2 (2002),
paper no. 8, pages 157-170.

## Abelian Subgroups of the Torelli Group

### William R. Vautaw

**Abstract**.
Let S be a closed oriented surface of genus g > 1, and let T denote
its Torelli group. First, given a set E of homotopically nontrivial,
pairwise disjoint, pairwise nonisotopic simple closed curves on S, we
determine precisely when a multitwist on E is an element of T by
defining an equivalence relation on E and then applying graph
theory. Second, we prove that an arbitrary Abelian subgroup of T has
rank < 2g-4.
**Keywords**.
Mapping class group,Torelli group, multitwist

**AMS subject classification**.
Primary: 57M60.
Secondary: 20F38.

**DOI:** 10.2140/agt.2002.2.157

**E-print:** `arXiv:math.GT/0203131`

Submitted: 12 December 2001.
(Revised: 24 February 2002.)
Accepted: 28 February 2002.
Published: 6 March 2002.

Notes on file formats
William R. Vautaw

Department of Mathematics, Michigan State University

East Lansing, MI 48824, USA

Email: vautawwi@pilot.msu.edu

AGT home page

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