### Minimal Cockcroft subgroups

Jens Harlander
1994 Glasgow Mathematical Journal
Statement of results. 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 X L 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 h L :ji 2 (X)-*H 2 (X L ) 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
more » ... plex X is aspherical if n 2 {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 Hopfs theorem (see [2, p. 31]) that H 2 (G) is isomorphic to H 2 (X). In particular H 2 {G) is free abelian. For a survey on the Cockcroft property see Dyer [5] . A collection {G a :aeQ.} of subgroups of a group G that is totally ordered by inclusion is called a chain of subgroups of G. Denning /3 < a if and only if G a ^ Gp makes Q into a totally ordered set. The main result of this paper is the following theorem. THEOREM 1. Let {G a :a e Q} be a chain of subgroups of a group G. A [G, 2]-complex X that is G a -Cockcroft for all a e Q is also I p | G a )-Cockcroft. Theorem 1 together with Zorn's lemma give the next result. COROLLARY 1. Let X be a Cockcroft [G,2]-complex. Then G contains a minimal subgroup L such that X is L-Cockcroft.