Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T02:28:38.453Z Has data issue: false hasContentIssue false
  • English
  • Français

Generating the mapping class group by torsion elements of small order

Published online by Cambridge University Press:  03 October 2012

NAOYUKI MONDEN
NAOYUKI MONDEN*
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan. e-mail: n-monden@cr.math.sci.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated by 3 elements of order 3. Moreover, we show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated and by 4 elements of order 4.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 41 - 62
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Brendle, T. E. and Farb, B.Every mapping class group is generated by 3 torsion elements and by 6 involutions. J. Algebra 278 (2004), 187–198.CrossRefGoogle Scholar
[2]Dehn, M. Papers on group theory and topology (Springer-Verlag, New York, 1987) (Die Gruppe der Abbildungsklassen, Acta Math. Vol. 69 (1938), 135–206).CrossRefGoogle Scholar
[3]Farb, B. Some problems on mapping class groups and moduli space. Problems on mapping class groups and related topics, 11–55. Proc. Symp. Pure Math., 74. (Amer. Math. Soc., Providence, 2006).CrossRefGoogle Scholar
[4]Farb, B. and Margalit, D.A Primer on Mapping Class Groups. Princeton Math. Ser., (Princeton University Press, 2012), 623–658.Google Scholar
[5]Hirose, S.Presentations of periodic maps on oriented closed surfaces of genera up to 4. Osaka Journal of Mathematics, vol. 47, No. 2 (2010), 385–421.Google Scholar
[6]Humphries, S. P. Generators for the mapping class group. Topology of low-dimensional manifolds. Proc. Second Sussex Conf. Chelwood Gate 1977 Lecture Notes in Math. 722 (Springer, 1979), 44–47.CrossRefGoogle Scholar
[7]Johnson, D.The structure of Torelli group I: A finite set of generators for . Ann. of Math. 118 (1983), 423–442.CrossRefGoogle Scholar
[8]Kassabov, M. Generating Mapping Class Groups by Involutions. arXiv:math.GT/0311455 v1 25 Nov 2003.Google Scholar
[9]Korkmaz, M.Generating the surface mapping class group by two elements. Trans. Amer. Math. Soc. 357 (2005), 3299–3310.CrossRefGoogle Scholar
[10]Lickorish, W. B. R.A finite set of generators for the homeotopy group of a 2-manifold. Proc. Camb. Phils. Soc. 60 (1964), 769–778.CrossRefGoogle Scholar
[11]Luo, F. Torsion Elements in the Mapping Class Group of a Surface. arXiv:math.GT/0004048 v1 8 Apr 2000.Google Scholar
[12]Maclachlan, C.Modulus space is simply-connected. Proc. Amer. Math. Soc. 29 (1971), 85–86.CrossRefGoogle Scholar
[13]McCarthy, J. and Papadopoulos, A.Involutions in surface mapping class groups. Enseign. Math. 33 (1987), 275–290.Google Scholar
[14]Powell, J.Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350.CrossRefGoogle Scholar