The Strong Atiyah Conjecture predicts that for a countable group G whose finite subgroups have bounded orders with least common multiple lcm(G), the von Neumann dimensions of kernels of finite matrices over the group ring K[G], for a subfield K of the complex numbers, always lie in the cyclic subgroup (1/lcm(G))Z; for torsion-free G this forces all such dimensions to be integers and implies Kaplansky's Zero Divisor Conjecture. The source paper, Kielak–Linton's 'Group rings of three-manifold groups', flagged as an acknowledged gap that the conjecture was not known to be preserved either under free products or under finite extensions of groups. The free-product half has since been resolved affirmatively: Sánchez-Peralta (2025) proved that the conjecture passes to fundamental groups of graphs of groups with finite edge groups, free products in particular. The surviving open core is the finite-extension half: if a countable group G contains a finite-index subgroup satisfying the Strong Atiyah Conjecture over K, must G satisfy it? As of June 2026 this is explicitly open (Fisher–Ng), with GL_m(Z) — whose finite-index subgroups satisfy the conjecture while the group itself is unresolved — as a concrete instance, and with substantial conditional and class-specific partial results (Linnell–Schick; Schreve) delimiting what a solution must overcome.

Background

The Strong Atiyah Conjecture concerns the possible values of L2L^2-Betti numbers, invariants of groups introduced through Atiyah's 1976 question about square-summable cohomology on covering spaces. For a countable group GG, let ℓ2(G)\ell^2(G) be the Hilbert space of square-summable complex-valued functions on GG. The group ring CG\mathbb{C}G, consisting of finite formal C\mathbb{C}-linear combinations of elements of GG, acts on ℓ2(G)\ell^2(G) by left convolution; the group von Neumann algebra N(G)\mathcal{N}(G) is the closure of these convolution operators in the weak operator topology. Its canonical finite trace assigns to every closed N(G)\mathcal{N}(G)-invariant subspace of ℓ2(G)n\ell^2(G)^n a von Neumann dimension dim⁡N(G)\dim_{\mathcal{N}(G)}, a real number in [0,n][0,n] generalizing ordinary Hilbert-space dimension; the L2L^2-Betti numbers of a GG-space are von Neumann dimensions of its L2L^2-cohomology. The Strong Atiyah Conjecture (SAC) over a subfield K⊆CK\subseteq\mathbb{C} predicts that if the orders of the finite subgroups of GG are bounded, with least common multiple lcm(G)\mathrm{lcm}(G), then for every finite matrix AA over the group ring K[G]K[G], viewed as an operator on ℓ2(G)n\ell^2(G)^n, one has dim⁡N(G)ker⁡(A)∈1lcm(G)Z\dim_{\mathcal{N}(G)}\ker(A)\in\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}; for torsion-free GG this says all such dimensions are integers. The boundedness hypothesis is necessary: Grigorchuk–Linnell–Schick–Żuk constructed groups with unbounded torsion admitting kernels of irrational dimension, so the conjecture is false without it.

The conjecture is known for large classes of groups: Linnell's class C\mathcal{C}, the smallest class containing all free groups and closed under directed unions and under extensions by elementary amenable groups (groups built from finite and abelian groups by iterated extensions and directed unions), by Linnell's 1993 theorem for groups in C\mathcal{C} with a uniform bound on torsion; locally indicable groups, in which every nontrivial finitely generated subgroup admits a homomorphism onto Z\mathbb{Z}, by Jaikin-Zapirain–López-Álvarez; virtually cocompact special groups, meaning finite extensions of fundamental groups of compact special cube complexes, by Schreve; and fundamental groups of three-manifolds, by Friedl–Lück and by Kielak–Linton. For torsion-free groups it implies Kaplansky's Zero Divisor Conjecture in characteristic zero, namely that the group ring has no zero divisors.

Closure properties determine how far these classes can be propagated. The conjecture passes trivially to subgroups. Kielak–Linton, proving that all three-manifold groups lie in Linnell's class C\mathcal{C}, highlighted exactly two missing closure properties: it was, in their words, 'not known in general' whether the SAC is preserved under free products or under finite extensions of groups; either property would have extended the Friedl–Lück theorem from admissible three-manifold groups (connected, orientable, irreducible, with empty or toral boundary and infinite fundamental group) to all finitely generated three-manifold groups. The free-product half was settled in 2025 by Sánchez-Peralta, who proved that the SAC passes to fundamental groups of graphs of groups with finite edge groups whenever all vertex groups satisfy it and lcm\mathrm{lcm} stays finite, free products being the case of trivial edge groups. The finite-extension half remains open and is the subject of this problem. Linnell–Schick established it only under strong additional hypotheses — essentially that the mod-pp cohomology of the group agrees with that of its pp-adic completion for every prime pp, plus a condition such as torsion-free quotients of the lower central or derived series — hypotheses that hold for free, surface, pure braid, certain link, and one-relator groups, but not in general.

Problem Statement

Let K⊆CK\subseteq\mathbb{C} be a subfield closed under complex conjugation (for example K=QK=\mathbb{Q} or K=CK=\mathbb{C}), let GG be a countable group, and let H≤GH\leq G be a subgroup of finite index. Suppose that HH satisfies the Strong Atiyah Conjecture over KK: the orders of the finite subgroups of HH are bounded, with least common multiple lcm(H)\mathrm{lcm}(H), and for every finite square matrix AA over the group ring K[H]K[H], viewed as a bounded operator on ℓ2(H)n\ell^2(H)^n by left convolution,

dim⁡N(H)ker⁡(A)∈1lcm(H)Z.\dim_{\mathcal{N}(H)}\ker(A)\in\frac{1}{\mathrm{lcm}(H)}\mathbb{Z}.

Note that lcm(G)\mathrm{lcm}(G) is then automatically finite, since every finite subgroup of GG has order dividing lcm(H)⋅[G:H]\mathrm{lcm}(H)\cdot[G:H]. Must GG satisfy the Strong Atiyah Conjecture over KK, that is, must

dim⁡N(G)ker⁡(B)∈1lcm(G)Z\dim_{\mathcal{N}(G)}\ker(B)\in\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}

hold for every finite matrix BB over K[G]K[G]? Since the conjecture passes to subgroups, the finite-index overgroup direction asked here is the open direction; and because a non-normal finite-index subgroup can be replaced by its normal core, a finite-index normal subgroup of GG contained in HH that inherits the conjecture from HH, the statement for arbitrary finite-index subgroups is equivalent to the statement for normal ones. A concrete instance noted by Fisher–Ng: it is not known whether GLm(Z)\mathrm{GL}_m(\mathbb{Z}) satisfies the conjecture, although it has finite-index subgroups that do. This is the surviving half of the two closure questions posed by the source; the companion free-product question was resolved affirmatively by Sánchez-Peralta in 2025.

References

  1. Dawid Kielak and Marco Linton, Group rings of three-manifold groups, Proc. Amer. Math. Soc. 152 (2024), no. 5, 1939–1946, DOI 10.1090/proc/16716; arXiv:2303.15907, https://arxiv.org/abs/2303.15907
  2. Pablo Sánchez-Peralta, Universal localizations, Atiyah conjectures and graphs of groups, Geom. Funct. Anal. 35 (2025), no. 3, 842–876, DOI 10.1007/s00039-025-00710-4; arXiv:2409.12268, https://arxiv.org/abs/2409.12268
  3. Peter Linnell and Thomas Schick, Finite group extensions and the Atiyah conjecture, J. Amer. Math. Soc. 20 (2007), no. 4, 1003–1051, DOI 10.1090/S0894-0347-07-00561-9; arXiv:math/0403229, https://arxiv.org/abs/math/0403229
  4. Sam P. Fisher and Andrew Ng, Outer automorphism groups and the Atiyah Conjecture, arXiv:2606.19606 (2026), https://arxiv.org/abs/2606.19606
  5. Kevin Schreve, The strong Atiyah conjecture for virtually cocompact special groups, Math. Ann. 359 (2014), no. 3-4, 629–636, DOI 10.1007/s00208-014-1007-9; arXiv:1305.1071, https://arxiv.org/abs/1305.1071
  6. Peter A. Linnell, Division rings and group von Neumann algebras, Forum Math. 5 (1993), 561–576, DOI 10.1515/form.1993.5.561, https://doi.org/10.1515/form.1993.5.561
  7. Rostislav I. Grigorchuk, Peter Linnell, Thomas Schick, and Andrzej Żuk, On a question of Atiyah, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 663–668, DOI 10.1016/S0764-4442(00)01702-X, https://doi.org/10.1016/S0764-4442(00)01702-X
  8. Stefan Friedl and Wolfgang Lück, L²-Euler characteristics and the Thurston norm, Proc. Lond. Math. Soc. (3) 118 (2019), no. 4, 857–900, DOI 10.1112/plms.12202, https://doi.org/10.1112/plms.12202
  9. Andrei Jaikin-Zapirain and Diego López-Álvarez, The strong Atiyah and Lück approximation conjectures for one-relator groups, Math. Ann. 376 (2020), no. 3-4, 1741–1793, DOI 10.1007/s00208-019-01926-0; arXiv:1810.12135, https://arxiv.org/abs/1810.12135
  10. Sam P. Fisher and Pablo Sánchez-Peralta, Division rings for group algebras of virtually compact special groups and 3-manifold groups, J. Comb. Algebra 10 (2024), no. 1, 153–193, DOI 10.4171/jca/89; arXiv:2303.08165, https://arxiv.org/abs/2303.08165
  11. Sam Hughes, Philip Möller, and Olga Varghese, Profinite properties of Coxeter groups, arXiv:2505.08701 (2025), https://arxiv.org/abs/2505.08701 (with an appendix by Sam Fisher and Sam Hughes proving the Strong Atiyah Conjecture for Coxeter groups)