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A Generalization of Lyndon's Theorem on the Cohomology of One-Relator Groups

Published online by Cambridge University Press:  20 November 2018

D. Gildenhuys
D. Gildenhuys*
Affiliation:
McGill University, Montreal, Quebec
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In this paper we generalize a theorem of Lyndon's [7], which states that a one-relator group G = F/(r) (F is free and r Ç F) has cohomological dimension cd (F/(r)) ≧ 2 if and only if the relator r is not a proper power in F. His proof relies on the Identity Theorem and recently he has shown [8] how a generalized version of this theorem and a generalized version of the Freiheitsatz can be simultaneously obtained by the methods of combinatorial geometry. These generalizations refer to a situation where the free group F is replaced by a free product of subgroups of the additive group of real numbers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 3 , 01 June 1976 , pp. 473 - 480
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Barr, M. and Beck, J., Homology and standard constructions, Seminar on Triples and Categorical Homology, Lecture Notes in Mathematics No. 80 (Springer-Verlag, New York, N.Y.).Google Scholar
2. Baumslag, G., Positive one-relator groups, Trans. Amer. Math. Soc. 156 (1971), 165–183.Google Scholar
3. Gildenhuys, D., The cohomology of groups acting on trees, J. Pure and Applied Algebra 6 (1975), 265–274.Google Scholar
4. Gruenberg, K. W., Cohomological topics in group theory, Lecture Notes in Mathematics No. 143 (Springer-Verlag, New York, N.Y.).Google Scholar
5. Hilton, P. and Stammbach, U., A course in homological algebra (Springer-Verlag, New York, N.Y.).Google Scholar
6. Karrass, A. and Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255.Google Scholar
7. Lyndon, R., Cohomology theory of groups with a single defining relation, Ann. Math. 52 (1950), 650–665.Google Scholar
8. Lyndon, R., On the Freiheitsatz, J. London Math. Soc. (2) 5 (1972), 95–101.Google Scholar
9. Magnus, W., Ueber diskontinuerliche Gruppen mit einer definierenden Relation (Der Freiheitsatz), J. Reine Angew. Math. 103 (1930), 141–165.Google Scholar
10. Magnus, W., Karass, A. and Solitar, D., Combinatorial group theory (Interscience Pub., N.Y.).Google Scholar
11. Serre, J.-P., Groupes discrets, Lecture Notes from Collège de France. (1968-1969).Google Scholar