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Free products in mapping class groups generated by Dehn twists

Published online by Cambridge University Press:  18 May 2009

Stephen P. Humphries
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
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Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Birman, , Braids, links and mapping class groups, Ann. of Math. Stud. 82 (Princeton University Press, 1975).Google Scholar
2.Fathi, A., Laudenbach, F., Poenaru, V., Travaux de Thurston sur les surfaces, Astërisque 6667 (1979).Google Scholar
3.Humphries, S. P., Free subgroups of SL(n, Z), n>2, generated by transvections, J. Algebra 116 (1988), 155162.CrossRefGoogle Scholar
4.Humphries, S. P., Subgroups of SL(3, Z) generated by transvections and involutions I and II, preprints, 1987.Google Scholar
5.Husein-Zade, S. M., The monodromy groups of isolated singularities of hypersurfaces, Russian Math. Surveys 32 (1977), 2365.CrossRefGoogle Scholar
6.Lyndon, R., Schupp, P., Combinatorial group theory (Springer Verlag, 1976).Google Scholar
7.Magnus, D., Karass, A. and Solitar, D., Combinatorial group theory. (Dover, 1976).Google Scholar
8.Powell, J., Two theorems on the mapping class group of surfaces, Proc. Amer. Math. Soc. 68 (1978), 347350.CrossRefGoogle Scholar
9.Tits, J., Free groups in linear groups, J. Algebra 20 (1972), 250270.CrossRefGoogle Scholar
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