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L^2-Betti numbers in prime characteristic and a conjecture of Wise

Grigori Avarmidi, Wolfgang Lueck

TL;DR

This paper develops a unified framework for $L^2$-Betti numbers in both zero and prime characteristic via division rings, focusing on $ ext{RALI}$-type groups and their weak variants. It extends fundamental properties such as multiplicativity, Künneth, Euler–Poincaré, and Poincaré duality to mod $p$ settings and establishes comparison results between characteristic zero and prime characteristics, including approximation results. A central theme links vanishing of $b_2^{(2)}$ to Wise's conjecture on non-positive towers of $2$-complexes, proving the equivalence in several cases (one-relator complexes, spines of aspherical $3$-manifolds with boundary, certain Davis-complex quotients) and providing mod $p$ variants for residually $p$-finite towers. The work also develops approximation by finite quotients, maximal residually coverings, and applications to $3$-manifolds and extensions, culminating in coherence-type questions for groups of cohomological dimension $2$ with vanishing top $L^2$-Betti numbers, and linking these to broader conjectures such as the Farrell–Jones framework.

Abstract

We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.

L^2-Betti numbers in prime characteristic and a conjecture of Wise

TL;DR

This paper develops a unified framework for -Betti numbers in both zero and prime characteristic via division rings, focusing on -type groups and their weak variants. It extends fundamental properties such as multiplicativity, Künneth, Euler–Poincaré, and Poincaré duality to mod settings and establishes comparison results between characteristic zero and prime characteristics, including approximation results. A central theme links vanishing of to Wise's conjecture on non-positive towers of -complexes, proving the equivalence in several cases (one-relator complexes, spines of aspherical -manifolds with boundary, certain Davis-complex quotients) and providing mod variants for residually -finite towers. The work also develops approximation by finite quotients, maximal residually coverings, and applications to -manifolds and extensions, culminating in coherence-type questions for groups of cohomological dimension with vanishing top -Betti numbers, and linking these to broader conjectures such as the Farrell–Jones framework.

Abstract

We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.
Paper Structure (38 sections, 58 theorems, 151 equations)

This paper contains 38 sections, 58 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. $L^2$-Betti numbers in prime characteristic
  3. $L^2$-Betti numbers and towers of $2$-complexes
  4. Acknowledgments
  5. $L^2$-Betti numbers defined over the group von Neumann algebra
  6. $L^2$-Betti numbers in arbitrary characteristic
  7. Basics on division rings
  8. (Weak) Linnell and (weak) Hughes groups
  9. The definition of $L^2$-Betti numbers over fields
  10. Restriction and induction
  11. The $0$th $L^2$-Betti number
  12. Künneth formula
  13. The Euler-Poincaré formula
  14. Poincaré duality
  15. On the top $L^2$-Betti number
  16. ...and 23 more sections

Key Result

Theorem 1.2

Suppose that $X$ and $Y$ are finite, connected $2$-complexes with RALI fundamental groups. Let $X\rightarrow Y$ be a $\mathbb Z$-tower, or more generally a RALI-tower. Suppose that $b_2^{(2)}(\widetilde{Y};\mathcal{N}(\pi_1(Y)))=0$. Then either $\chi(X)\leq 0$, or $X$ is contractible.

Theorems & Definitions (145)

  • Conjecture 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 3.1: (Weak) Linnell group
  • Conjecture 3.2: Torsionfree groups are Linnell groups
  • Definition 3.3: (Weak) Hughes group
  • Remark 3.4: Functoriality of $\mathcal{D}_{FG}$
  • Lemma 3.5
  • proof
  • Definition 3.6: $L^2$-Betti numbers over fields
  • ...and 135 more