Centralizers and normalizers of subgroups of PD3PD3-groups and open PD3PD3-groups

We give algebraic proofs of some PD3PD3-analogues of theorems on centralizers and normalizers of finitely generated subgroups of 3-manifold groups. In particular, we introduce the notion of open PD3PD3-group, as an analogue of the fundamental group of an aspherical open 3-manifold, and we show that every strictly increasing sequence of centralizers in a PD3PD3-group or open PD3PD3-group has length at most 4. We show also that ascendant FP2FP2 subgroups of PD3PD3-groups and open PD3PD3-groups are usually normalized by a subgroup of finite index.