Pro-\(p\) completions of Poincaré duality groups. (Q2017150)

The goal of the paper under review is to study conditions that ensure that the pro-\(p\) completion of a \(\text{PD}_n\)-group is virtually a pro-\(p\) \(\text{PD}_m\)-group, for some \(m<n-1\). Here \(p\) is a prime, and a group is said to be \(\text{PD}_n\) if it is an orientable Poincaré duality group of dimension \(n\). In particular, it is shown that if (i) \(G\) is an abstract orientable \(\text{PD}_n\) group, for some \(n\geq 3\), (ii) the pro-\(p\) completion \(\widehat G_p\) of \(G\) is infinite and not a virtual orientable pro-\(p\) \(\text{PD}_n\) group, and (iii) for \(2\leq i<n\) the inverse limit (over a directed set of subgroups \(U\) of \(p\)-power index in \(G\) which induces the pro-\(p\) topology of \(G\)) of the \(i\)-th homology group of \(U\) with coefficients in the field \(F\) with \(p\) elements has finite dimension over \(F\), then \(\widehat G_p\) is virtually a pro-\(p\) \(\text{PD}_m\)-group for some \(m<n-1\).