Algebraic & Geometric Topology

Abelian subgroups of the Torelli group

William R Vautaw

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Let S be a closed oriented surface of genus g2, 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 2g3.

Article information

Algebr. Geom. Topol., Volume 2, Number 1 (2002), 157-170.

Received: 12 December 2001
Revised: 24 February 2002
Accepted: 28 February 2002
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M60: Group actions in low dimensions
Secondary: 20F38: Other groups related to topology or analysis

mapping class group Torelli group multitwist


Vautaw, William R. Abelian subgroups of the Torelli group. Algebr. Geom. Topol. 2 (2002), no. 1, 157--170. doi:10.2140/agt.2002.2.157.

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