Dimension drop in residual chains

TL;DR

This paper characterizes the Linnell division ring for certain residually poly-$b Z$ virtually nilpotent groups, linking cohomology vanishing, group quotients, and $ ext{l}^2$-Betti numbers to structural properties like virtual freeness and nilpotency.

Contribution

It introduces a novel description of the Linnell division ring for RPVN groups and establishes new connections between cohomology, $ ext{l}^2$-Betti numbers, and group structure.

Findings

Vanishing top-degree cohomology implies existence of a specific normal subgroup.

Vanishing second $ ext{l}^2$-Betti number characterizes certain group quotients.

Finitely generated RPVN groups of dimension 2 are virtually free-by-nilpotent.

Abstract

We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existence of a normal subgroup $N \leqslant G$ such that $\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)$ and $G/N$ is poly-$\mathbb Z$ virtually nilpotent. As a consequence, we show that if $G$ is an RPVN group of finite type, then its top-degree $\ell^2$-Betti number vanishes if and only if there is a poly-$\mathbb Z$ virtually nilpotent quotient $G/N$ such that $\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)$. In particular, finitely generated RPVN groups of cohomological dimension $2$ are virtually free-by-nilpotent if and only if their second $\ell^2$-Betti number vanishes, and therefore $2$-dimensional RPVN groups with vanishing second $\ell^2$-Betti number are coherent. As another application, we show that if $G$ is a finitely generated parafree group with $\mathrm{cd}(G) = 2$, then $G$ satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.