The homology of Peiffer products of groups
The Peiffer product of groups \(G\bowtie H\) first arose in a topological setting in the work of \textit{J.~H.~C.~Whitehead} [Ann. Math., II. Ser. 42, 409-428 (1941; Zbl 0027.26404)] on the structure of relative homotopy groups, posing the following question about asphericity: Are subgroups of aspherical two-complexes themselves aspherical? which is closely related with the embedding question: when are the natural maps \(G\to G\bowtie H\leftarrow H\) injective? In this paper the authors prove that the Peiffer product of perfect groups is perfect, describe the second integral homology \(H_2(G\bowtie H)\) of a Peiffer product \(G\bowtie H\) in terms of information about the factors \(G\) and \(H\), obtaining that a Peiffer product of superperfect (\(H_1=H_2=0\)) groups is superperfect. They also introduce a double wreath product of cyclic groups as a Peiffer product and determine when a Peiffer product of cyclic groups has trivial second homology.
scientific article; zbMATH DE number 1409628