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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
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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

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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