Algebraic & Geometric Topology

Abelian subgroups of the Torelli group

William R Vautaw

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882688

Digital Object Identifier
doi:10.2140/agt.2002.2.157

Mathematical Reviews number (MathSciNet)
MR1917048

Zentralblatt MATH identifier
0997.57035

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

Keywords
mapping class group Torelli group multitwist

Citation

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. https://projecteuclid.org/euclid.agt/1513882688


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References

  • Joan S. Birman, Alex Lubotzky, and John McCarthy: Abelian and Solvable Subgroups of the Mapping Class Group, Duke Mathematical Journal, Volume 50, Number 4, pp. 1107 – 1120; December 1983.
  • J. A. Bondy and U. S. R. Murty: Graph Theory with Applications, North-Holland, New York; 1976.
  • Nikolai V. Ivanov: Subgroups of Teichmüller Modular Groups, Translations of Mathematical Monographs, Volume 115, American Mathematical Society, 1992.
  • John D. McCarthy: Normalizers and Centralizers of Pseudo-Anosov Mapping Classes, (preprint), June 8, 1994.
  • Jerome Powell: Two Theorems on the Mapping Class Group of a Surface, Proceedings of the American Mathematical Society, Volume 68, Number 3, pp. 347 – 350; March 1978.