Group pairs with periodic cohomology (Q1813760)

A group pair \((G,{\mathbf S})\) with \({\mathbf S}\) a family of subgroups of \(G\) is said to be periodic if the (relative) cohomology is periodic, i.e. if there is an integer \(q\) such that for all \(k\geq 2\) and for all \(G\)-modules \(A\) one has a natural isomorphism \(H^ k(G,{\mathbf S};A)\equiv H^{k+q}(G,{\mathbf S};A)\). In this paper the authors give a classification of all periodic pairs \((G,{\mathbf S})\) with \(G\) finite and \({\mathbf S}\) a finite family. Also they prove a result which reduces the classification of arbitrary periodic pairs with \(G\) finitely generated and accessible and \({\mathbf S}\) a finite family to the case of the fundamental group of a graph of groups with finite edge groups and whose infinite vertex groups are conjugate to members of \({\mathbf S}\).