Atiyah's conjecture for torsion-free groups

About 22 years old · traced to 2004

Let $G$ be a torsion-free group, let $A$ be a finite $n\times m$ matrix over $\mathbb ZG$ , and consider the right-multiplication map

\bigoplus_n \ell^2(G)\longrightarrow\bigoplus_m \ell^2(G),\qquad x\longmapsto xA.

Atiyah conjecture. The von Neumann dimension of the kernel of this map is an integer.

The conjecture concerns the possible von Neumann dimensions arising from matrices over the integral group ring and would imply strong integrality for $L^2$ -homological calculations. It is open; the source records cases proved for extensions of free groups by elementary amenable groups and for residually torsion-free elementary amenable groups, and notes stability under subgroups and free products.

References

Primary source

Dawid Kielak, “Virtual fibring of manifolds and groups”, arXiv:2510.01805 (2025).

Additional references

3 papers in this index state this conjecture (2004–2025). The statement above is taken from the most recent of them; the others are arXiv:1605.09067, arXiv:math/0408400.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The conjecture remains open in general, but several broad families of torsion-free groups are now known to satisfy it.

The conjecture asks whether every finite integral group-ring matrix over a torsion-free group has an integer-valued von Neumann kernel dimension. No general proof or torsion-free counterexample has been reported.

Known results

Linnell: the conjecture holds for classes containing free groups and closed under directed unions and extensions with elementary amenable quotients.
Schick: it holds over $\mathbb{Q}$ for a class containing residually torsion-free elementary amenable groups and closed under subgroups, limits, direct products, and free products.
Known examples include elementary amenable groups, braid groups, locally indicable groups, virtually compact special groups, and $3$ -manifold groups.

2025 graph-of-groups closure result

Pablo Sánchez-Peralta proved that Strong Atiyah Conjecture validity passes to fundamental groups of graphs of groups with finite edge groups under stated hypotheses, explicitly settling closure under free products. This is substantial inheritance progress, not a solution for all torsion-free groups.

Current status (as of August 2026): The conjecture remains open for arbitrary torsion-free groups; the recorded 2025 closure theorem establishes additional cases but not the general statement.

Sources

No solutions have been posted yet.