Math Conjectures

Strong Atiyah Conjecture

OPENMajorConjectureProposed 1976 · Standard version

Canonical statement

Let GG be a discrete group for which the orders of finite subgroups are bounded, and put L=lcm⁡{∣H∣:H≤G finite}L=\operatorname{lcm}\{|H|:H\le G\text{ finite}\}. For all m,n≥1m,n\ge1 and every matrix A∈Mm,n(CG)A\in M_{m,n}(\mathbb C G), let rA:ℓ2(G)m→ℓ2(G)nr_A:\ell^2(G)^m\to\ell^2(G)^n be the bounded GG-equivariant operator given by right convolution by AA. Then
L dim⁡N(G)ker⁡rA∈Z, L\,\dim_{\mathcal N(G)}\ker r_A\in\mathbb Z,
where N(G)\mathcal N(G) is the group von Neumann algebra and dim⁡N(G)\dim_{\mathcal N(G)} is its von Neumann dimension.
View source LaTeX
Let \(G\) be a discrete group for which the orders of finite subgroups are bounded, and put \(L=\operatorname{lcm}\{|H|:H\le G\text{ finite}\}\). For all \(m,n\ge1\) and every matrix \(A\in M_{m,n}(\mathbb C G)\), let \(r_A:\ell^2(G)^m\to\ell^2(G)^n\) be the bounded \(G\)-equivariant operator given by right convolution by \(A\). Then
\[
  L\,\dim_{\mathcal N(G)}\ker r_A\in\mathbb Z,
\] where \(\mathcal N(G)\) is the group von Neumann algebra and \(\dim_{\mathcal N(G)}\) is its von Neumann dimension.
ALG-L2-017:[Atiyah1976Elliptic][Lueck2002L2][SanchezPeralta2025Atiyah]
Algebra and representation theoryGeometry and topology

Notes

The strong Atiyah conjecture predicts integrality constraints on von Neumann dimensions. For a discrete group GG whose finite subgroups have bounded order, set L=lcm⁡{∣H∣:H≤G finite}L=\operatorname{lcm}\{|H| : H\le G \text{ finite}\}; the conjecture asserts that for every matrix AA over the group ring CG\mathbb C G, the kernel of the induced convolution operator rAr_A between finite direct sums of copies of ℓ2(G)\ell^2(G) satisfies L dim⁡N(G)ker⁡rA∈ZL\,\dim_{\mathcal N(G)}\ker r_A\in\mathbb Z. The question descends from Atiyah's 1976 work introducing L2L^2-Betti numbers via elliptic operators, discrete groups, and von Neumann algebras, which asked about the possible values of such dimensions [Atiyah1976Elliptic].

For torsion-free groups the statement forces kernels to have integer dimension, a condition closely tied to the zero-divisor problem for group rings. The conjecture is established for extensive classes of groups and enjoys strong stability under group-theoretic constructions, including recent results for graphs of groups obtained via universal localizations [SanchezPeralta2025Atiyah]; a systematic account of the surrounding L2L^2-theory is given by Lück [Lueck2002L2].

The bounded-torsion hypothesis is essential, since unrestricted rationality statements without it admit counterexamples. For arbitrary groups with bounded torsion the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.