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A note on the cohomology of metabelian groups

Published online by Cambridge University Press:  24 October 2008

P. H. Kropholler
P. H. Kropholler
Affiliation:
St. John's College, Cambridge
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Extract

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 437 - 445
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Bieri, R.. Homological Dimension of Discrete Groups. Queen Mary College Mathematics Notes (Q.M.C. London 1976).Google Scholar
[2]Bieri, R. and Groves, J. R. J.. Metabelian groups of type (FP)∞ are virtually of type (FP). Proc. London Math. Soc. (3) 45 (1982), 365–384.CrossRefGoogle Scholar
[3]Bieri, R. and Strebei, R.. Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41 (1980), 439–464.Google Scholar
[4]Groves, J. R. J.. Metabelian groups with finitely generated integral homology. Quart. J. Math. Oxford (2) 33 (1982), 405–420.CrossRefGoogle Scholar
[5]Hilton, P. J. and Stammbach, U.. A Course in Homological Algebra. Graduate Texts in Mathematics 4 (Springer-Verlag, 1970).Google Scholar
[6]Kropholler, P. H.. On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. (3) 49 (1984), 155–169.CrossRefGoogle Scholar
[7]Lennox, J. C.. On quasinormal subgroups of certain finitely generated groups. Proc. Edin. Math. Soc. 26 (1983), 25–28.CrossRefGoogle Scholar
[8]Musson, I.. On the structure of certain injective modules over group algebras of soluble groups of finite rank. J. Algebra 85 (1983), 51–75.CrossRefGoogle Scholar
[9]Robinson, D. J. S.. Finiteness Conditions and Generalized Soluble Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 62/63 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[10]Robinson, D. J. S.. On the cohomology of soluble groups of finite rank. J. Pure Appl. Alg. 6 (1975), 155–164.CrossRefGoogle Scholar
[11]Robinson, D. J. S.. Applications of cohomology to the theory of groups. In Groups - St Andrews 1981 (ed. Campbell, C. M. and Robertson, E. F.), London Math. Soc. Lecture Notes Series 71 (Cambridge University Press, 1982), 46–80.Google Scholar
[12]Robinson, D. J. S. and Wilson, J. S.. Soluble groups with many polycyclic quotients. Proc. London Math. Soc. (3) 48 (1984), 193–229.CrossRefGoogle Scholar
[13]Wehrfritz, B. A. F.. Infinite Linear Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 76 (Springer-Verlag, 1973).CrossRefGoogle Scholar