Pro-$p$ completions of $PD_n$-groups

TL;DR

This paper refines the understanding of pro-$p$ completions of orientable $PD_3$-groups, providing examples and optimal results across four cases, and extends analysis to some $PD_n$ groups for $n \,\leq\, 5$.

Contribution

It offers a detailed classification of pro-$p$ completions for $PD_3$-groups and explores extensions to higher dimensions, including explicit examples and optimality results.

Findings

Four distinct cases for pro-$p$ completions of $PD_3$-groups identified.

Examples of aspherical 3-manifolds for each case provided.

Results in three cases are shown to be optimal.

Abstract

We sharpen earlier work on the pro-$p$ completions of orientable $PD_3$-groups. There are four cases, and we give examples of aspherical 3-manifolds representing each case. In three of the four cases the new results are best possible. We also consider the pro-$p$ completion of some orientable $PD_n$ groups for $n \leq 5$, including surface-by-surface groups.