Subnormality in $PD_3$-groups and $L^2$-Betti numbers

TL;DR

This paper explores the algebraic properties of subnormal subgroups within $PD_3$-groups and group pairs, extending prior work on 3-manifold groups and highlighting the role of $L^2$-Betti numbers, which remain unproven in general.

Contribution

It extends algebraic analysis of subnormal subgroups from 3-manifold groups to $PD_3$-groups and group pairs, under an unverified $L^2$-Betti number hypothesis.

Findings

Extension of algebraic arguments to $PD_3$-groups

Identification of the $L^2$-Betti number hypothesis as crucial

Highlighting open problems in the general case

Abstract

We reconsider work of Elkalla on subnormal subgroups of 3-manifold groups, giving essentially algebraic arguments that extend to the case of $PD_3$-groups and group pairs. However the argument relies on an $L^2$-Betti number hypothesis which has not yet been shown to hold in general.