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Groups whose three-generator subgroups are free

Published online by Cambridge University Press:  17 April 2009

Gilbert Baumslag  and
Peter B. Shalen
Gilbert Baumslag
Affiliation:
Department of Mathematics, City College (CUNY), New York, NY 10031, United States of America
Peter B. Shalen
Affiliation:
Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, Chicago, IL 60680United States of America
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Abstract

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We define a certain class of groups, Ck, which we show to contain the class of all k-free groups. Our main theorem shows that certain amalgamated free products of groups in C3, are again in C3. In the appendix we show that many 3-manifold groups belong to Ck for suitable k.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 2 , October 1989 , pp. 163 - 174
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Baumslag, B., ‘Generalized free products whose two-generator subgroups are free’, J. London Math. Soc. 43 (1968), 601–606.CrossRefGoogle Scholar
[2]Fatou, P., ‘Fonctions automorphes’, Volume 2 of Théorie des Fonctions algébriques, Appell, R.E. and Goursat, E. (Gauthiers-Villars, Paris). 158–160.Google Scholar
[3]Hempel, J., 3-manifolds: Ann. of Math. Studies 86 (Princeton University Press, 1076).Google Scholar
[4]Jaco, W., Lectures on 3-manifold topology (Amer. Math. Soc. for CBMS, Providence, c., 1980).CrossRefGoogle Scholar
[5]Jaco, W. and Shalen, P.B., Seifert fibered spaces in 3-manifolds (Mem. Amer. Math. Soc. 21, no. 220, 1979).Google Scholar
[6]Karrass, A. and Solitar, P., ‘The subgroups of a free product of two groups with an amalgamated subgroup’, Trans. Amer. Moth. Soc. 150 (1970), 227–255.CrossRefGoogle Scholar
[7]Magnus, W., ‘Über freie Faktorgruppen und freie Untergruppen gegebener Gruppen’, Monatsh. Math. u. Phys. 47 (1939), 307–313.CrossRefGoogle Scholar
[8]Milnor, J., ‘A unique decomposition theorem for 3-manifolds’, Amer. J. Math. 84 (1962), 1–7.CrossRefGoogle Scholar
[9]Neumann, B.H., ‘An essay on free product of groups with amalgamations’, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554.Google Scholar
[10]Scott, G.P., ‘Compact submanifolds of 3-manifolds’, J. London Math. Soc. 7 (1973), 246–250.CrossRefGoogle Scholar
[11]Serre, J.-P, Arbres, amalgames, SL2, Astérisque 46: Translated as Trees (Springer-Verlag, Berlin, Heidelberg, New York, 1980).Google Scholar
[12]Shalen, P.B. and Wagreich, P., ‘Growth, volumes and free subgroups of 3-manifold groups’, (In preparation).Google Scholar
[13]Tucker, T., ‘On Kleinian groups and 3-manifolds of Euler characteristic zero’, (Unpublished).Google Scholar