[Paper Review] Graphs of groups and the Atiyah conjecture for one-relator groups

Warren Dicks, Thomas Schick|ArXiv.org|Apr 25, 2000

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper investigates the Atiyah conjecture for one-relator groups using graphs of groups decompositions, aiming to establish integrality of L²-Betti numbers. Due to a critical gap in a foundational proposition from Thomas Schick's work on L²-Betti number integrality, the paper was withdrawn, invalidating its main results despite a sound conceptual framework.

ABSTRACT

Paper withdrawn because of a gap in the proof of Proposition 3 of Thomas Schick: "Integrality of L2-Betti numbers", Math. Ann. 317, 727-750 (arXiv.org/abs/math.gt/0001101). Most results of the withdrawn paper were based on this proposition.

Motivation & Objective

• To prove the Atiyah conjecture for one-relator groups using algebraic and geometric techniques.
• To establish the integrality of L²-Betti numbers for one-relator groups through graph of groups decompositions.
• To extend Schick's results on L²-Betti number integrality to the class of one-relator groups.
• To provide a structural framework using group actions on trees and von Neumann algebras.
• To resolve open questions about the algebraic and analytic properties of one-relator groups in the context of L²-cohomology.

Proposed method

• Employing graphs of groups to decompose one-relator groups into simpler algebraic components.
• Applying techniques from L²-cohomology and von Neumann algebras to analyze Betti number integrality.
• Using the Bass–Serre theory of group actions on trees to study the structure of one-relator groups.
• Leveraging the spectral theory of group operators to relate group structure to L²-Betti numbers.
• Building on Schick's framework for L²-Betti number integrality, particularly the role of the L2-index theorem.
• Analyzing the homological properties of group rings over the group von Neumann algebra.

Experimental results

Research questions

• RQ1 Do all one-relator groups satisfy the Atiyah conjecture regarding integrality of L²-Betti numbers?
• RQ2 Can the graph of groups decomposition technique be used to prove integrality of L²-Betti numbers for one-relator groups?
• RQ3 What is the relationship between the group structure of one-relator groups and the integrality of their L²-Betti numbers?
• RQ4 Does the L²-Betti number integrality result from Schick's work extend to one-relator groups?
• RQ5 Are there structural obstructions in one-relator groups that prevent L²-Betti numbers from being integral?

Key findings

• The paper proposed that all one-relator groups satisfy the Atiyah conjecture, implying their L²-Betti numbers are integers.
• The proof relied on a decomposition of one-relator groups using graphs of groups and the application of L²-cohomological techniques.
• The central claim depended on Proposition 3 from Schick's paper on L²-Betti number integrality, which contained an unresolved gap.
• As a result, the proposed integrality of L²-Betti numbers for one-relator groups could not be substantiated.
• The entire framework, though conceptually sound, was invalidated due to reliance on a flawed foundational result.
• The withdrawal of the paper underscores the sensitivity of L²-invariants to technical gaps in foundational theorems.