Integrality of the rate of relations of a torsion-free finitely presented group: the first $\ell^2$-Betti number case of the Atiyah conjecture

For a finitely generated group $\Gamma$ with a finite generating set $S$, the rate of relations $\tau(\Gamma_S)$ is the von Neumann dimension of the closure of the space of 1-cycles of the Cayley graph of the marked group $\Gamma_S$; Pichot (2006) proved that the first $\ell^2$-Betti number satisfies $\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S)$ 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 $\tau(\Gamma)$ of a torsion-free finitely presented group $\Gamma$ always an integer — equivalently, is the first $\ell^2$-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 $\Gamma$ acts by left translations on the Hilbert space $\ell^2(\Gamma)$ of square-summable complex functions on $\Gamma$. The theory of Murray and von Neumann assigns to every closed $\Gamma$-invariant subspace $H \subseteq \ell^2(\Gamma)^n$ a von Neumann dimension $\dim_\Gamma H \in [0,n]$, a normalized, additive dimension function on equivariant subspaces; for a finite group it agrees with the ordinary dimension divided by the group order. If $\Gamma$ acts freely and cocompactly on a CW complex $Y$, the $\ell^2$-Betti numbers $\beta_k^{(2)}(Y, \Gamma)$ are the von Neumann dimensions of the reduced $\ell^2$-homology spaces of $Y$, and for a finitely presented group $\Gamma$ the numbers $\beta_k^{(2)}(\Gamma)$ 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 $\ell^2$-Betti numbers of closed manifolds. The theory is developed in Lück's treatise (2002), and $\ell^2$-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 $\Gamma$ together with a finite generating set $S$; the marking turns $\Gamma$ into a labeled graph, the Cayley graph, whose vertices are the elements of $\Gamma$ and in which each vertex $g$ is joined to the vertices $gs$ and $gs^{-1}$ for every $s \in S$. Let $Y$ be this graph and $Z_1(Y) = \ker \partial_1$ the space of finitely supported 1-cycles (here $\partial_1$ is the cellular boundary map from 1-cells to 0-cells). Pichot (2006) defined the rate of relations of the marked group by $\tau(\Gamma_S) = \dim_\Gamma \overline{Z_1(Y)}$, the closure being taken in the space of $\ell^2$ 1-chains, and the rate of relations of $\Gamma$ as $\tau(\Gamma) = \inf_S \tau(\Gamma_S)$ over finite generating sets. For instance, a finite group of order $N$ has $\tau(\Gamma_S) = |S| - 1 + 1/N$, which is already non-integer — illustrating that torsion genuinely produces denominators. Pichot proved (his Proposition 2) that for every infinite marked group

$$\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S),$$

so the rate of relations is a generating-set-independent measure of 'how many relations' $\Gamma$ has, complementary to the first $\ell^2$-Betti number; he used it to prove that $\beta_1^{(2)}$ is upper semi-continuous on the space of marked groups. He also observed that $\tau(\Gamma_S) - \tau(\Gamma)$ is always a non-negative integer, so integrality of $\tau$ 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 $G$ are bounded with least common multiple $\mathrm{lcm}(G)$, then the von Neumann dimensions of kernels of matrices over $\mathbb{Q}[G]$ — equivalently the $\ell^2$-Betti numbers of free cocompact $G$-complexes — lie in $\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}$; for torsion-free groups it predicts integer values. It is known for Linnell's class $\mathcal{C}$ 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 $\mathbb{Z}$), 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 $\beta_3^{(2)} = 1/3$, 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 $\mathbb{Z}_p \wr \mathbb{Z}$, the wreath products $(\bigoplus_{i \in \mathbb{Z}} \mathbb{Z}_p) \rtimes \mathbb{Z}$), 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 $\mathbb{Z}_p \wr \mathbb{Z}$ has torsion elements of order at most $p$, but its finite subgroups $(\mathbb{Z}_p)^n$ 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 $\mathbb{C}$ 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 $\tau(\Gamma) = \inf_S \tau(\Gamma_S)$ of a torsion-free finitely presented group $\Gamma$ always an integer, where $\tau(\Gamma_S)$ is the von Neumann dimension of the closure of the space of 1-cycles of the Cayley graph of the marked group $\Gamma_S$? By Pichot's formula $\beta_1^{(2)}(\Gamma) = |S| - 1 - \tau(\Gamma_S)$ (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 $\tau(\Gamma_S) - \tau(\Gamma)$ is a non-negative integer, this is equivalent to each of: (i) $\tau(\Gamma_S) \in \mathbb{Z}$ for some, equivalently every, finite generating set $S$ of a torsion-free finitely presented $\Gamma$; and (ii) the first $\ell^2$-Betti number $\beta_1^{(2)}(\Gamma)$ is an integer for every torsion-free finitely presented group $\Gamma$. 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 $\ell^2$-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 $\ell^2$-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 $\ell^2$-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 $L^2$-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, $L^2$-invariants: theory and applications to geometry and $K$-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)