Centralizers and normalizers of subgroups of \(\text{PD}_3\)-groups and open \(\text{PD}_3\)-groups. (Q2576923)

A \(\text{PD}_3\)-group consists of a group \(G\) and a finite set of conjugacy classes of monomorphisms of various groups into \(G\) satisfying Poincaré duality in dimension three. Examples are fundamental groups of \(3\)-manifolds and the author introduces open \(\text{PD}_3\)-groups as an analogue of fundamental groups of open (aspherical) \(3\)-manifolds. In this paper the author obtains many interesting results about centralizers and normalizers of subgroups of \(\text{PD}_3\)-groups and open \(\text{PD}_3\)-groups. In particular he shows (a) Every strictly increasing sequence of centralizers in a \(\text{PD}_3\)-group or open \(\text{PD}_3\)-group has length at most \(4\). (b) Ascendant \(\text{FP}_2\)-subgroups of \(\text{PD}_3\)-groups and open \(\text{PD}_3\)-groups are usually normalized by a subgroup of finite index. (c) If \((G,H)\) is a \(\text{PD}_3\)-group pair such that \(H\) differs from its normalizer in \(G\), then \(H\) has index \(2\) and \((G,H)\) is the group system of a twisted \(I\)-bundle with connected boundary.