Minimal Cockcroft subgroups

Abstract

Consider any groupG. A [G, 2]-complexis a connected 2-dimensional CW-complex with fundamental groupG. IfXis a [G, 2]-complex andLis a subgroup ofG, letXLdenote the covering complex ofXcorresponding to the subgroupL. We say that a [G, 2]-complex isL-Cockcroftif the Hurewicz maphL:π2(X)→;H2(XL)is trivial. In caseL=Gwe callX Cockcroft. There are interesting classes of 2-complexes that have the Cockcroft property. A [G, 2]-complexXisasphericalif 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. LetXbe a [G, 2]-complex modelled on a presentation (〈S; R〉 of the groupG. If it can be shown thatXis Cockcroft, then it follows from Hopf's theorem (see [2, p. 31]) thatH2(G)is isomorphic toH2(X). In particularH2(G)is free abelian. For a survey on the Cockcroft property see Dyer [5]. A collection {Gα: α ∈ Ώ} of subgroups of a groupGthat is totally ordered by inclusion is called achain of subgroups of G. Denning β ≤ α if and only ifGα≤Gβmakes Ώ into a totally ordered set. The main result of this paper is the following theorem.