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Dimension drop in residual chains

Sam P Fisher, Kevin Klinge

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.

Dimension drop in residual chains

Abstract

We give a description of the Linnell division ring of a countable residually (poly- virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group with coefficients in this Novikov ring implies the existence of a normal subgroup such that and is poly- virtually nilpotent. As a consequence, we show that if is an RPVN group of finite type, then its top-degree -Betti number vanishes if and only if there is a poly- virtually nilpotent quotient such that . In particular, finitely generated RPVN groups of cohomological dimension are virtually free-by-nilpotent if and only if their second -Betti number vanishes, and therefore -dimensional RPVN groups with vanishing second -Betti number are coherent. As another application, we show that if is a finitely generated parafree group with , then 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.

Paper Structure

This paper contains 20 sections, 35 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results
  3. Parafree groups
  4. The Linnell division ring of an RPVN group
  5. Acknowledgments
  6. Preliminaries
  7. Orders on nilpotent groups
  8. Cohomological dimension and finiteness properties
  9. The category of R-division rings and specialisations
  10. Novikov rings and division rings
  11. Novikov rings
  12. Division rings
  13. Summary
  14. DkG-Betti numbers
  15. Iterated fractions
  16. ...and 5 more sections

Key Result

Theorem A

Let $G$ be an RPVN group of type $\mathrm{FP}(\mathbb{Q})$ and of $\mathop{\mathrm{cd}}\nolimits_\mathbb{Q}(G) = d$, and let $G = N_0 \geqslant N_1 \geqslant \dots$ be a residual chain of normal subgroups such that every quotient $G/N_i$ is poly-$\mathbb{Z}$ and virtually nilpotent. Then $b_{d}^{(2)

Figures (1)

  • Figure 1: A commutative diagram summarising the various inclusions between the rings and modules introduced above.

Theorems & Definitions (88)

  • Theorem A
  • Remark 1.1
  • Corollary B
  • Remark 1.2
  • Corollary C
  • Corollary D
  • Remark 1.3
  • Theorem E
  • Remark 1.4
  • Theorem F
  • ...and 78 more