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Strongly torsion generated groups

Published online by Cambridge University Press:  24 October 2008

A. J. Berrick  and
C. F. Miller III
A. J. Berrick
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511
C. F. Miller III
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
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Extract

It has long been known that the integral homology of a non-trivial finite group must be non-zero in infinitely many dimensions 17. Recent work on the Sullivan conjecture in homotopy theory has made it possible to extend this result to locally finite groups. For more general groups with torsion it becomes more difficult to make such a strong statement. Nevertheless we prove that when a non-perfect group is generated by torsion elements its integral homology must also be non-zero in infinitely many dimensions. Remarkably, this result is best possible, in that for perfect torsion generated groups all (finite or infinite) sequences of abelian groups are shown below to be attainable as higher homology groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 219 - 229
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

1Atiyah, M. F.. Characters and cohomology of finite groups. Inst. Hautes tudes Sci. Publ. Math. 9 (1961), 2364.CrossRefGoogle Scholar
2Baumslag, G., Dyer, E. and Heller, A.. The topology of discrete groups. J. Pure Appl. Algebra 16 (1980), 147.CrossRefGoogle Scholar
3Baumslag, G., Dyer, E. and Miller, C. F. III. On the integral homology of finitely presented groups. Topology 22 (1983), 2746.CrossRefGoogle Scholar
4Baumslag, G. and Miller, C. F. III. Some odd finitely presented groups. Bull. London Math. Soc. 20 (1988), 239244.CrossRefGoogle Scholar
5Berrick, A. J.. An Approach to Algebraic K-Theory. Research Notes in Math. no. 56 (Pitman, 1982).Google Scholar
6Berrick, A. J.. Group extensions and their trivialisation. Enseign. Math. (2) 31 (1985), 151172.Google Scholar
7Berrick, A. J.. Remarks on the structure of acyclic groups. Bull. London Math. Soc. 22 (1990), 227232.CrossRefGoogle Scholar
8Berrick, A. J.. Torsion generators for all abelian groups. J. Algebra 139 (1991), 190194.CrossRefGoogle Scholar
9Berrick, A. J. and Casacuberta, C.. Groups and spaces with all localizations trivial. In Groups and Classifying Spaces, Lecture Notes in Math. (Springer-Verlag), to appear.Google Scholar
10Berrick, A. J. and Kropholler, P. H.. In preparation.Google Scholar
11Henn, H.-W.. A note on the homology of locally finite groups. Unpublished manuscript.Google Scholar
12Lyndon, R. C. and Schupp, P. E.. Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
13Malcev, A. I.. On isomorphie matrix representations of infinite groups. Mat. Sb. 8 (1940), 405422.Google Scholar
14Miller, C. F. III. Decision problems for groups: survey and reflections. In Proceedings of the Workshop on Algorithms, Word Problems and Classification in Combinatorial Group Theory (Springer-Verlag, 1990).Google Scholar
15Miller, H.. The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120 (1984), 3987.CrossRefGoogle Scholar
16Milnor, J.. Introduction to Algebraic K-Theory. Ann. of Math. Studies, no. 72 (Princeton University Press, 1971).Google Scholar
17Swan, R. G.. The nontriviality of the restriction map in the cohomology of groups. Proc. Amer. Math. Soc. 11 (1960), 885887.Google Scholar