Generating the surface mapping class group by two elements

Author:
Mustafa Korkmaz

Journal:
Trans. Amer. Math. Soc. **357** (2005), 3299-3310

MSC (2000):
Primary 57M07, 20F05; Secondary 57M60, 20F38

Published electronically:
December 28, 2004

MathSciNet review:
2135748

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

**1.**Tara E. Brendle and Benson Farb,*Every mapping class group is generated by 6 involutions*, J. Algebra**278**(2004), no. 1, 187–198. MR**2068073**, 10.1016/j.jalgebra.2004.02.019**2.**M. Dehn,*Die gruppe der abdildungsklassen*, Acta Math.**69**(1938), 135-206.**3.**Stephen P. Humphries,*Generators for the mapping class group*, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 44–47. MR**547453****4.**Dennis Johnson,*The structure of the Torelli group. I. A finite set of generators for \cal𝐼*, Ann. of Math. (2)**118**(1983), no. 3, 423–442. MR**727699**, 10.2307/2006977**5.**W. B. R. Lickorish,*A representation of orientable combinatorial 3-manifolds*, Ann. of Math. (2)**76**(1962), 531–540. MR**0151948****6.**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. MR**0171269****7.**Bronislaw Wajnryb,*A simple presentation for the mapping class group of an orientable surface*, Israel J. Math.**45**(1983), no. 2-3, 157–174. MR**719117**, 10.1007/BF02774014**8.**Bronislaw Wajnryb,*Mapping class group of a surface is generated by two elements*, Topology**35**(1996), no. 2, 377–383. MR**1380505**, 10.1016/0040-9383(95)00037-2**9.**Bronislaw Wajnryb,*An elementary approach to the mapping class group of a surface*, Geom. Topol.**3**(1999), 405–466 (electronic). MR**1726532**, 10.2140/gt.1999.3.405

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57M07,
20F05,
57M60,
20F38

Retrieve articles in all journals with MSC (2000): 57M07, 20F05, 57M60, 20F38

Additional Information

**Mustafa Korkmaz**

Affiliation:
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Email:
korkmaz@arf.math.metu.edu.tr

DOI:
https://doi.org/10.1090/S0002-9947-04-03605-0

Received by editor(s):
November 8, 2003

Received by editor(s) in revised form:
January 8, 2004

Published electronically:
December 28, 2004

Additional Notes:
The author was supported in part by the Turkish Academy of Sciences under the Young Scientists Award Program (MK/TÜBA-GEBİP 2003-10)

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.