Atiyah conjecture

The Atiyah conjecture is a family of statements restricting possible $L^2$-Betti numbers of groups and spaces. The scan found no complete proof, counterexample resolving the general question, or AI-generated solution.

Known results

• The strong form is known for several classes, including Linnell's class, Schick's class, and locally indicable groups.
• Without bounded torsion, the strong Atiyah conjecture is false in general, by work of Grigorchuk, Linnell, Schick, and Żuk.

September 2024 inheritance theorem

A preprint proves that, for a countable graph of groups with finite edge groups and finite $\operatorname{lcm}(G)$, the strong conjecture passes from the vertex groups to the whole group; free products are a special case. It also gives related algebraic, center-valued, and localization results, but does not claim a general solution.

Current status (as of August 2026): Partial progress is established through inheritance and group-class results, but the general Atiyah conjecture remains open, and its strong form is false without bounded torsion.