Abelian subgroups of the Torelli group

Let $S$ be a closed oriented surface of genus $g \geq 2$ , 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 $\leq 2 g - 3$ .

Keywords

mapping class group, Torelli group, multitwist

Mathematical Subject Classification 2000

Primary: 57M60

Secondary: 20F38

References

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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

Volume 2, issue 1 (2002)

William R Vautaw