Hiroshima Mathematical Journal

The mapping class group of a punctured surface is generated by three elements

Naoyuki Monden

Full-text: Open access

Abstract

Let $\rm Mod(\Sigma_{\textit{g,p}})$ be the mapping class group of a closed oriented surface $\Sigma_{g,p}$ of genus $g\geq 1$ with $p$ punctures. Wajnryb proved that $\rm Mod({\Sigma_{\textit{g},0}})$ is generated by two elements. Korkmaz proved that one of these generators may be taken to be a Dehn twist. Korkmaz also proved the same result in the case of $\rm Mod(\Sigma_{\textit{g},1})$. For $p\geq 2$, we prove that $\rm Mod(\Sigma_{\textit{g,p}})$ is generated by three elements.

Article information

Source
Hiroshima Math. J., Volume 41, Number 1 (2011), 1-9.

Dates
First available in Project Euclid: 31 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1301586286

Digital Object Identifier
doi:10.32917/hmj/1301586286

Mathematical Reviews number (MathSciNet)
MR2809044

Zentralblatt MATH identifier
1229.57001

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M07: Topological methods in group theory

Keywords
Mapping class group punctured surface

Citation

Monden, Naoyuki. The mapping class group of a punctured surface is generated by three elements. Hiroshima Math. J. 41 (2011), no. 1, 1--9. doi:10.32917/hmj/1301586286. https://projecteuclid.org/euclid.hmj/1301586286


Export citation

References

  • T. E. Brendle and B. Farb. Every mapping class group is generated by 3 torsion elements and by 6 involutions. J. Algebra 278 (2004), $187\mbox{--}198$.
  • M. Dehn. Papers on group theory and topology. Springer-Verlag, New York, 1987 (Die Gruppe der Abbildungsklassen, Acta Math. Vol. 69 (1938), $135\mbox{--}206$)
  • S. Gervais. A finite presentation of the mapping class group of a punctured surface. Topology 40 (2001), No. 4, $703\mbox{--}725$
  • S. P. Humphries. Generators for the mapping class group. Topology of $low\mbox{-}dimensional$ manifolds Proc. Second Sussex Conf. Chelwood Gate 1977, Lecture Notes in Math. 722 (1979), Springer, $44\mbox{--}47$.
  • D. Johnson. The structure of Torelli group I: A finite set of generators for ${\cal I}$. Ann. of Math. 118 (1983), $423\mbox{--}442$.
  • M. Kassabov. Generating Mapping Class Groups by Involutions. arXiv:math.GT/0311455 v1 25 Nov 2003.
  • M. Korkmaz. Generating the surface mapping class group by two elements. Trans. Amer. Math. Soc. 357 (2005), $3299\mbox{--}3310$.
  • W. B. R. Lickorish. A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769–778.
  • N. Lu. On the mapping class groups of the closed orientable surfaces. Topology Proc. 13 (1988), 293–324.
  • B. Wajnryb. Mapping class group of a surface is generated by two elements. Topology. 35 (1996), 377–383.