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Minimal Cockcroft subgroups

Published online by Cambridge University Press:  18 May 2009

Jens Harlander
Jens Harlander
Affiliation:
Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 6000 Frankfurt/Main 11
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Consider any group G. A [G, 2]-complex is a connected 2-dimensional CW-complex with fundamental group G. If X is a [G, 2]-complex and L is a subgroup of G, let XL denote the covering complex of X corresponding to the subgroup L. We say that a [G, 2]-complex is L-Cockcroft if the Hurewicz map hL:π2(X)→;H2(XL) is trivial. In case L = G we call X Cockcroft. There are interesting classes of 2-complexes that have the Cockcroft property. A [G, 2]-complex X is aspherical if π2(X) = 0. It was observed in [4] that a subcomplex of an aspherical 2-complex is Cockcroft. The Cockcroft property is of interest to group theorists as well. Let X be a [G, 2]-complex modelled on a presentation (〈S; R〉 of the group G. If it can be shown that X is Cockcroft, then it follows from Hopf's theorem (see [2, p. 31]) that H2(G) is isomorphic to H2(X). In particular H2(G) is free abelian. For a survey on the Cockcroft property see Dyer [5]. A collection {Gα: α ∈ Ώ} of subgroups of a group G that is totally ordered by inclusion is called a chain of subgroups of G. Denning β ≤ α if and only if Gα ≤ Gβ makes Ώ into a totally ordered set. The main result of this paper is the following theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 87 - 90
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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