The \(p\)-period of an infinite group
A group \(\Gamma\) has virtual cohomological dimension (vcd) \(n\) if \(\Gamma\) has a subgroup of finite index which has cohomological dimension \(n\). For such groups the Farrell cohomology \(\widehat{H}^ i(\Gamma,-)\) is defined which has formal properties analogous to those of the Tate cohomology for finite groups (in which case \(n=0\)) and \(\widehat{H}^ i(\Gamma,-)\cong H^ i(\Gamma,-)\) for \(i > n\) [e.g. \textit{K. S. Brown}, ``Cohomology of groups'' (Graduate Texts Math. 87, 1982; Zbl 0584.20036)]. For \(\Gamma\) a group with vcd \(\Gamma=n < \infty\) and \(p\) a prime the \(p\)-period of \(\Gamma\) is defined to be the least positive integer \(d\) such that \(\widehat{H}^ i(\Gamma,M)\) and \(\widehat{H}^{i + d}(\Gamma,M)\) have naturally isomorphic \(p\)-primary components for all \(i \in \mathbb{Z}\) and \(\mathbb{Z} \Gamma\)-modules \(M\). Here the author generalizes a result of \textit{R. G. Swan} [Ill. J. Math. 4, 341-346 (1960; Zbl 0095.020)] on the \(p\)-period of a finite \(p\)-periodic group to a \(p\)-periodic group \(\Gamma\) with vcd \(\Gamma=n < \infty\) and proves: Theorem. If \(\Gamma\) has a normal subgroup of cohomological dimension \(n\) so that the associated quotient is a finite group whose \(p\)-Sylow subgroup is elementary abelian or cyclic, then the \(p\)-period of \(\Gamma\) is twice the least common multiple of \(\{N_ \Gamma(\langle x\rangle)/C_ \Gamma(\langle x\rangle)\}\), where \(\langle x\rangle\) ranges over all conjugacy classes of \(\mathbb{Z}/p\) subgroups of \(\Gamma\). Moreover, an example is given of a group \(\Gamma\) of finite vcd whose only finite subgroup is a \(\mathbb{Z}/2\) and yet the 2-period is greater than \(2 | N_ \Gamma(\mathbb{Z}/2)/C_ \Gamma(\mathbb{Z}/2)|\). More interesting results are obtained concerning \(p\)-periodic mapping class groups.
scientific article; zbMATH DE number 94220