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.

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William R. Vautaw
Department of Mathematics, Michigan State University
East Lansing, MI 48824, USA

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