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On vanishing criteria of $L^2$-Betti numbers of groups

Pablo Sánchez-Peralta

TL;DR

The paper advances the theory of $L^2$-Betti numbers of groups by proving higher-degree vanishing criteria that transfer information from infinite-index subgroups, via purely algebraic and groupoid methods. It constructs and analyzes the crossed product $S=\,\\mathcal{U}(N)*G/N$ to translate finite-dimensional behavior of $S$-modules into vanishing of $L^2$-invariants for $G$, and it extends Sauer–Thom and Gaboriau-type results to a broader algebraic setting using translation groupoids and groupoid algebras. A central achievement is a fully algebraic proof of Gaboriau’s first Betti number vanishing, together with a higher-degree analogue, and the paper provides new vanishing criteria that connect subnormal and Hillman-type questions to groupoid techniques. Collectively, these results furnish powerful tools to control $L^2$-Betti numbers from substructures, with potential applications to 3-manifold groups and related areas in geometric group theory. The work highlights the utility of groupoid algebras and dimension theory (via SMRF/BSMRF and von Neumann dimensions) as a robust alternative to measured equivalence relations in studying $L^2$-invariants.

Abstract

Let $G$ be a countable group and $k$ a positive integer, we show that the $L^2$-Betti numbers of the group $G$ vanish up to degree $k$ provided that there is some infinite index subgroup $H$ with finite $k$th $L^2$-Betti number containing a normal subgroup of $G$ whose $L^2$-Betti numbers are all zero below degree $k$. This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first $L^2$-Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for $k = 1$ and $H$ containing a subnormal subgroup instead.

On vanishing criteria of $L^2$-Betti numbers of groups

TL;DR

The paper advances the theory of -Betti numbers of groups by proving higher-degree vanishing criteria that transfer information from infinite-index subgroups, via purely algebraic and groupoid methods. It constructs and analyzes the crossed product to translate finite-dimensional behavior of -modules into vanishing of -invariants for , and it extends Sauer–Thom and Gaboriau-type results to a broader algebraic setting using translation groupoids and groupoid algebras. A central achievement is a fully algebraic proof of Gaboriau’s first Betti number vanishing, together with a higher-degree analogue, and the paper provides new vanishing criteria that connect subnormal and Hillman-type questions to groupoid techniques. Collectively, these results furnish powerful tools to control -Betti numbers from substructures, with potential applications to 3-manifold groups and related areas in geometric group theory. The work highlights the utility of groupoid algebras and dimension theory (via SMRF/BSMRF and von Neumann dimensions) as a robust alternative to measured equivalence relations in studying -invariants.

Abstract

Let be a countable group and a positive integer, we show that the -Betti numbers of the group vanish up to degree provided that there is some infinite index subgroup with finite th -Betti number containing a normal subgroup of whose -Betti numbers are all zero below degree . This generalizes previous criteria of both Sauer and Thom, and Peterson and Thom. In addition, we exhibit a purely algebraic proof of a well-known theorem of Gaboriau concerning the first -Betti number which was requested by Bourdon, Martin and Valette. Finally, we provide evidence of a positive answer for a question posted by Hillman that wonders whether the above statement holds for and containing a subnormal subgroup instead.
Paper Structure (12 sections, 40 theorems, 155 equations)

This paper contains 12 sections, 40 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ore localization
  4. Homological algebra
  5. Left ideals in group algebras
  6. Sylvester rank functions
  7. Von Neumann dimension
  8. Algebraic structure of discrete measured groupoids
  9. Homological Algebra and Dimension Theory
  10. Homological features of the translation groupoid
  11. Proof of Theorem \ref{['thm: module_beta_k']}
  12. New Vanishing Criteria

Key Result

Theorem 1.1

Let $G$ be a countable group, $N$ an infinite index normal subgroup in $G$, $M$ a left $\mathbb{C}[G]$-module with $N$ acting trivially and $k$ a positive integer. Suppose that $\beta_i^{(2)}(N)$ vanishes for all $0\leq i\leq k-1$ and $\dim_{\hbox{$\mathcal{U}(N)$}}(\mathop{\mathrm{Tor}}\nolimits_k^

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 90 more