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BNSR invariants and $\ell^2$-homology

Sam Hughes, Dawid Kielak

TL;DR

The paper develops a ring-theoretic bridge between nonvanishing $\ell^2$-Betti numbers and BNSR invariants, showing that nonzero $n$th $\ell^2$-Betti numbers for a group $G$ of type $\textsf{FP}_n(\mathbb{Q})$ force $\Sigma^n(G;\mathbb{Q})$ to be empty. Central to the method is transferring vanishing information from Novikov homology to the algebra of affiliated operators via division/rational closures and a new ring of weakly rational elements, built through an approximate Ore framework. This transfer yields broad consequences for manifolds and groups, including cases of the Singer conjecture and computations for poly-free and poly-surface groups, and provides new algebraic proofs that avoid analytic techniques. The results extend to agrarian settings and positive characteristic under the existence of suitable skew fields, enabling characteristic-$p$ analogues for locally indicable and related groups, and yield concrete vanishing results for explicit families such as poly-elementarily-free groups and hyperbolic lattice groups.

Abstract

We prove that if the $n$th $\ell^2$-Betti number of a group is non-zero then its $n$th BNSR invariant over $\mathbb{Q}$ is empty, under suitable finiteness conditions. We apply this to answer questions of Friedl--Vidussi and Llosa Isenrich--Py about aspherical Kähler manifolds, to verify some cases of the Singer Conjecture, and to compute certain BNSR invariants of poly-free and poly-surface groups.

BNSR invariants and $\ell^2$-homology

TL;DR

The paper develops a ring-theoretic bridge between nonvanishing -Betti numbers and BNSR invariants, showing that nonzero th -Betti numbers for a group of type force to be empty. Central to the method is transferring vanishing information from Novikov homology to the algebra of affiliated operators via division/rational closures and a new ring of weakly rational elements, built through an approximate Ore framework. This transfer yields broad consequences for manifolds and groups, including cases of the Singer conjecture and computations for poly-free and poly-surface groups, and provides new algebraic proofs that avoid analytic techniques. The results extend to agrarian settings and positive characteristic under the existence of suitable skew fields, enabling characteristic- analogues for locally indicable and related groups, and yield concrete vanishing results for explicit families such as poly-elementarily-free groups and hyperbolic lattice groups.

Abstract

We prove that if the th -Betti number of a group is non-zero then its th BNSR invariant over is empty, under suitable finiteness conditions. We apply this to answer questions of Friedl--Vidussi and Llosa Isenrich--Py about aspherical Kähler manifolds, to verify some cases of the Singer Conjecture, and to compute certain BNSR invariants of poly-free and poly-surface groups.
Paper Structure (20 sections, 33 theorems, 102 equations)

This paper contains 20 sections, 33 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group von Neumann algebras
  4. l²-homology and Betti numbers
  5. Ore localisation
  6. The algebra of affiliated operators
  7. Division and rational closures
  8. Twisted polynomial rings
  9. BNSR invariants
  10. Novikov--Sikorav homology
  11. Passing vanishing of Novikov homology to a division-closed subring
  12. Approximate Ore condition
  13. Approximate Ore condition
  14. Asymptotic agreement
  15. Partial inverse
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $G$ be a group of type $\mathsf{F}_n$, that is admitting a classifying space with finite $n$-skeleton, and let $\varphi\colon G\twoheadrightarrow \mathbb{Z}$ be an epimorphism. If $b_n^{(2)}(G)\neq 0$, then $\ker\varphi$ is not of type $\mathsf{F}_n$.

Theorems & Definitions (104)

  • Theorem 1.1: Lück
  • Theorem B
  • Remark 1.3
  • Definition 2.1: Group von Neumann algebra
  • Definition 2.2: Polar decomposition
  • Theorem 2.3: Linnell
  • Definition 2.4: $\ell^2$-homology and Betti numbers
  • Definition 2.5
  • Definition 2.6: Right Ore condition
  • Definition 2.7: Right Ore localisation
  • ...and 94 more