Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T02:00:18.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal closures of powers of Dehn twists in mapping class groups

Published online by Cambridge University Press:  18 May 2009

Stephen P. Humphries
Stephen P. Humphries
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah, 84602, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let F = F(g, n) be an oriented surface of genus g≥1 with n<2 boundary components and let M(F) be its mapping class group. Then M(F) is generated by Dehn twists about a finite number of non-bounding simple closed curves in F([6, 5]). See [1] for the definition of a Dehn twist. Let e be a non-bounding simple closed curve in F and let E denote the isotopy class of the Dehn twist about e. Let Nbe the normal closure of E2in M(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:

Theorem 1. The subgroup N is of finite index in M(F).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 3 , September 1992 , pp. 314 - 317
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Birman, J., Braids, links and mapping class groups, (Princeton University Press, Annals of Math Studies #82, 1975).CrossRefGoogle Scholar
2.Jones, V., Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987) 335–388.CrossRefGoogle Scholar
3.Johnson, D., An abelian quotient of the mapping class group Ig, Math. Ann. 249 (1980) 225–242.CrossRefGoogle Scholar
4.Johnson, D., The structure of the Torelli Group II. Topology 24 (1985) 113–126.CrossRefGoogle Scholar
5.Johnson, D., The structure of the Torelli Group I. Ann. Math. 118 (1983) 423–442.CrossRefGoogle Scholar
6.Lickorish, W. B. R., A finite set of generators for the homeotopy group of a 2-manifold, Proc. Camb. Phil. Soc.1 60 (1964) 769–778.CrossRefGoogle Scholar
7.Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, (Dover, 1976).Google Scholar
8.Mennicke, J., Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 157 (1965) 115–129.CrossRefGoogle Scholar
9.Newman, M., Integral matrices, (Academic Press, 1972).Google Scholar
10.Powell, J., Two theorems on the mapping class group of surfaces, Proc. AMS 68 (1978) 347–350.CrossRefGoogle Scholar