For a finitely generated group Γ\Gamma with a finite generating set SS, the rate of relations τ(ΓS)\tau(\Gamma_S) is the von Neumann dimension of the closure of the space of 1-cycles of the Cayley graph of the marked group ΓS\Gamma_S; Pichot (2006) proved that the first ℓ2\ell^2-Betti number satisfies β1(2)(Γ)=∣S∣−1−τ(ΓS)\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S) for every infinite marked group. The question, recalled in that paper as a still open question due to Atiyah, asks: is the rate of relations τ(Γ)\tau(\Gamma) of a torsion-free finitely presented group Γ\Gamma always an integer — equivalently, is the first ℓ2\ell^2-Betti number of every torsion-free finitely presented group an integer? This is precisely the finitely presented, first-Betti-number instance of the Atiyah conjecture for torsion-free groups. Counterexamples to integrality are known only for groups whose finite subgroups have unbounded order (lamplighters and the constructions of Grigorchuk–Linnell–Schick–Żuk, Austin, Grabowski, and Pichot–Schick–Żuk), while integrality is proven for large classes such as free groups, elementary amenable groups of bounded torsion, locally indicable groups, and one-relator groups. As of 2026 the torsion-free case remains open.

Background

A countable group Γ\Gamma acts by left translations on the Hilbert space ℓ2(Γ)\ell^2(\Gamma) of square-summable complex functions on Γ\Gamma. The theory of Murray and von Neumann assigns to every closed Γ\Gamma-invariant subspace H⊆ℓ2(Γ)nH \subseteq \ell^2(\Gamma)^n a von Neumann dimension dim⁡ΓH∈[0,n]\dim_\Gamma H \in [0,n], a normalized, additive dimension function on equivariant subspaces; for a finite group it agrees with the ordinary dimension divided by the group order. If Γ\Gamma acts freely and cocompactly on a CW complex YY, the ℓ2\ell^2-Betti numbers βk(2)(Y,Γ)\beta_k^{(2)}(Y, \Gamma) are the von Neumann dimensions of the reduced ℓ2\ell^2-homology spaces of YY, and for a finitely presented group Γ\Gamma the numbers βk(2)(Γ)\beta_k^{(2)}(\Gamma) are defined this way from the universal cover of a presentation complex (they are independent of the chosen finite presentation). These invariants were introduced by Atiyah (1976) in index theory on non-compact covering manifolds; notably, every value he computed was rational, which prompted his famous question on the possible values of ℓ2\ell^2-Betti numbers of closed manifolds. The theory is developed in Lück's treatise (2002), and ℓ2\ell^2-Betti numbers are a standard tool in geometric group theory (for example, they vanish for infinite amenable groups by Cheeger–Gromov, and they are invariant under measure equivalence — the measure-theoretic analogue of quasi-isometry — by Gaboriau).

A marked group is a finitely generated group Γ\Gamma together with a finite generating set SS; the marking turns Γ\Gamma into a labeled graph, the Cayley graph, whose vertices are the elements of Γ\Gamma and in which each vertex gg is joined to the vertices gsgs and gs−1gs^{-1} for every s∈Ss \in S. Let YY be this graph and Z1(Y)=ker⁡∂1Z_1(Y) = \ker \partial_1 the space of finitely supported 1-cycles (here ∂1\partial_1 is the cellular boundary map from 1-cells to 0-cells). Pichot (2006) defined the rate of relations of the marked group by τ(ΓS)=dim⁡ΓZ1(Y)‾\tau(\Gamma_S) = \dim_\Gamma \overline{Z_1(Y)}, the closure being taken in the space of ℓ2\ell^2 1-chains, and the rate of relations of Γ\Gamma as τ(Γ)=inf⁡Sτ(ΓS)\tau(\Gamma) = \inf_S \tau(\Gamma_S) over finite generating sets. For instance, a finite group of order NN has τ(ΓS)=∣S∣−1+1/N\tau(\Gamma_S) = |S| - 1 + 1/N, which is already non-integer — illustrating that torsion genuinely produces denominators. Pichot proved (his Proposition 2) that for every infinite marked group

β1(2)(Γ)=∣S∣−1−τ(ΓS),\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S),

so the rate of relations is a generating-set-independent measure of 'how many relations' Γ\Gamma has, complementary to the first ℓ2\ell^2-Betti number; he used it to prove that β1(2)\beta_1^{(2)} is upper semi-continuous on the space of marked groups. He also observed that τ(ΓS)−τ(Γ)\tau(\Gamma_S) - \tau(\Gamma) is always a non-negative integer, so integrality of τ\tau does not depend on the marking.

The integrality question sits inside the well-known Atiyah conjecture landscape. The strong Atiyah conjecture predicts that if the orders of the finite subgroups of a countable group GG are bounded with least common multiple lcm(G)\mathrm{lcm}(G), then the von Neumann dimensions of kernels of matrices over Q[G]\mathbb{Q}[G] — equivalently the ℓ2\ell^2-Betti numbers of free cocompact GG-complexes — lie in 1lcm(G)Z\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}; for torsion-free groups it predicts integer values. It is known for Linnell's class C\mathcal{C} of groups (a class containing all free groups and, by closure under extensions and directed unions, the elementary amenable groups — the groups obtained from finite and abelian groups by iterated subgroups, quotients, extensions, and directed unions; Linnell 1993), for locally indicable groups (groups in which every non-trivial finitely generated subgroup admits a surjective homomorphism onto Z\mathbb{Z}), and hence for all one-relator groups (Jaikin-Zapirain–López-Álvarez 2020), and for further classes (fundamental groups of graphs of groups with finite edge groups by Sánchez-Peralta 2024; torsion-free finite-index subgroups of outer automorphism groups of right-angled Artin groups and of mapping class groups by Fisher–Ng 2026). On the negative side, Grigorchuk–Linnell–Schick–Żuk (2000) produced a closed 7-manifold with β3(2)=1/3\beta_3^{(2)} = 1/3, Austin (2013) produced finitely generated groups with irrational kernel dimensions, Grabowski (2014, 2016) showed that irrational and even transcendental values occur (already for the lamplighter groups Zp≀Z\mathbb{Z}_p \wr \mathbb{Z}, the wreath products (⨁i∈ZZp)⋊Z(\bigoplus_{i \in \mathbb{Z}} \mathbb{Z}_p) \rtimes \mathbb{Z}), and Pichot–Schick–Żuk (2015) realized many computable real numbers for finitely presented groups and closed manifolds. Crucially, every known counterexample lives in a group whose finite subgroups have unbounded order (the lamplighter Zp≀Z\mathbb{Z}_p \wr \mathbb{Z} has torsion elements of order at most pp, but its finite subgroups (Zp)n(\mathbb{Z}_p)^n have unbounded order), so none of them touches the torsion-free case. The torsion-free case is also significant algebraically: Linnell proved that a torsion-free group satisfies the strong Atiyah conjecture over a subfield of C\mathbb{C} if and only if the division closure of its group algebra — the smallest subring of the algebra of affiliated operators (the ring of closed, densely defined operators affiliated to the group von Neumann algebra, which plays the role of a ring of fractions for it) that contains the group algebra — is a division ring, so integrality is a strong form of the Kaplansky zero-divisor conjecture for group rings.

Problem Statement

Is the rate of relations τ(Γ)=inf⁡Sτ(ΓS)\tau(\Gamma) = \inf_S \tau(\Gamma_S) of a torsion-free finitely presented group Γ\Gamma always an integer, where τ(ΓS)\tau(\Gamma_S) is the von Neumann dimension of the closure of the space of 1-cycles of the Cayley graph of the marked group ΓS\Gamma_S? By Pichot's formula β1(2)(Γ)=∣S∣−1−τ(ΓS)\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S) (valid for every infinite marked group; the only torsion-free finite group is the trivial group, for which integrality is immediate) and the fact that τ(ΓS)−τ(Γ)\tau(\Gamma_S) - \tau(\Gamma) is a non-negative integer, this is equivalent to each of: (i) τ(ΓS)∈Z\tau(\Gamma_S) \in \mathbb{Z} for some, equivalently every, finite generating set SS of a torsion-free finitely presented Γ\Gamma; and (ii) the first ℓ2\ell^2-Betti number β1(2)(Γ)\beta_1^{(2)}(\Gamma) is an integer for every torsion-free finitely presented group Γ\Gamma. An affirmative answer may be given by a proof covering all torsion-free finitely presented groups; a negative answer by an explicit torsion-free finitely presented group together with a rigorous computation showing its rate of relations (equivalently its first ℓ2\ell^2-Betti number) is not an integer. This is the finitely presented, first-Betti-number instance of the Atiyah conjecture for torsion-free groups, stated as still open, due to Atiyah, in Pichot's 2006 paper.

References

  1. Mikaël Pichot, Semi-continuity of the first ℓ2\ell^2-Betti number on the space of finitely generated groups, Commentarii Mathematici Helvetici 81 (2006), no. 3, 643–652. DOI: 10.4171/cmh/67, https://doi.org/10.4171/cmh/67
  2. Michael F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, in: Colloque analyse et topologie, Astérisque 32–33, Société Mathématique de France, Paris, 1976, pp. 43–72. https://geodesic.mathdoc.fr/item/AST_1976__32-33__43_0/
  3. R. I. Grigorchuk, P. Linnell, T. Schick, A. Żuk, On a question of Atiyah, Comptes Rendus de l'Académie des Sciences 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 (preprint arXiv:math/0009175, https://arxiv.org/abs/math/0009175, is titled 'On a conjecture of Atiyah'; the published version bears the title above)
  4. Tim Austin, Rational group ring elements with kernels having irrational dimension, Proceedings of the London Mathematical Society (3) 107 (2013), no. 6, 1424–1448. DOI: 10.1112/plms/pdt029, arXiv:0909.2360, https://arxiv.org/abs/0909.2360
  5. Łukasz Grabowski, On Turing dynamical systems and the Atiyah problem, Inventiones Mathematicae 198 (2014), no. 1, 27–69. DOI: 10.1007/s00222-013-0497-5, arXiv:1004.2030, https://arxiv.org/abs/1004.2030
  6. Łukasz Grabowski, Irrational ℓ2\ell^2-invariants arising from the lamplighter group, Groups, Geometry, and Dynamics 10 (2016), no. 2, 795–817. DOI: 10.4171/ggd/366, arXiv:1009.0229, https://arxiv.org/abs/1009.0229
  7. Mikaël Pichot, Thomas Schick, Andrzej Zuk, Closed manifolds with transcendental L2L^2-Betti numbers, Journal of the London Mathematical Society (2) 92 (2015), no. 2, 371–392. DOI: 10.1112/jlms/jdv026, arXiv:1005.1147, https://arxiv.org/abs/1005.1147
  8. Peter A. Linnell, Division rings and group von Neumann algebras, Forum Mathematicum 5 (1993), no. 6, 561–576. DOI: 10.1515/form.1993.5.561, https://doi.org/10.1515/form.1993.5.561
  9. Andrei Jaikin-Zapirain, Diego López-Álvarez, The strong Atiyah and Lück approximation conjectures for one-relator groups, Mathematische Annalen 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, Andrew Ng, Outer automorphism groups and the Atiyah Conjecture, arXiv:2606.19606 (2026). https://arxiv.org/abs/2606.19606
  11. Pablo Sánchez-Peralta, Universal localizations, Atiyah conjectures and graphs of groups, arXiv:2409.12268 (2024). https://arxiv.org/abs/2409.12268
  12. Wolfgang Lück, L2L^2-invariants: theory and applications to geometry and KK-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 44, Springer-Verlag, Berlin, 2002. ISBN 3-540-43566-2, DOI: 10.1007/978-3-662-04687-6, https://link.springer.com/book/10.1007/978-3-662-04687-6 (zbMATH Zbl 1009.55001)