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An extension of the Freiheitssatz

Published online by Cambridge University Press:  24 October 2008

B. Baumslag  and
S. J. Pride
B. Baumslag
Affiliation:
Imperial College, London, and University of Glasgow
S. J. Pride
Affiliation:
Imperial College, London, and University of Glasgow
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Let I be a set and let H(i) (i ∈ I) be non-trivial groups. If J is a subset of I, we denote the free product of the H(j) (j∈J) by H(J). We denote H(I) simply by H. Let R be a cyclically reduced element of Hof length at least two, and let

Let μ: H → G be the natural homomorphism. If J is a subset of I such that R ∉ H(J), we call H(J) a Magnus subgroup, or occasionally the J-Magnus subgroup (of H with respect to R). We will say that the Freiheitssatz holds if μ| M is an injection for each Magnus subgroup M. Magnus (4) showed that the Freiheitssatz holds if the H(i) are free, and this was extended by Pride (5) to the case where the H(i) are locally fully residually free. In this paper we prove Theorem 1. The Freiheitssatz holds if the H(i) are locally residually free.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 1 , January 1981 , pp. 35 - 41
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

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