The Atiyah problem for finitely presented groups and the lamplighter group: exact sets of $\ell^2$-Betti values

Grabowski (Inventiones 2014) proved that every non-negative real number is an $\ell^2$-Betti number arising from some finitely generated group and that all reals with computable binary expansion arise from finitely presented groups, and posed four open questions on the range of values of $\ell^2$-Betti numbers (von Neumann dimensions of kernels of matrices over group rings). The open core of that problem is threefold: determine exactly which real numbers arise from finitely presented groups, in particular whether values with non-computable binary expansion can occur; determine whether the value set $\mathcal{C}(G,k)$ depends on the coefficient subfield $k\subseteq\mathbb{C}$ and whether $\mathcal{C}(G,\mathbb{C})$ is always countable, which is open even for the lamplighter group $\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}$; and determine the exact set $\mathcal{C}(\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z})$, which is known to contain $1/3$ and transcendental numbers (a concrete test question: is $\sqrt{2}$ a value?). The source paper's fourth question — rationality of the $\ell^2$-Betti numbers of groups themselves — was resolved negatively for finitely generated groups in 2025-2026 preprints (References 15 and 16); its still-open finiteness-property and torsion-free refinements concern these intrinsic group invariants and lie outside the open core.

Background

Atiyah introduced $\ell^2$-Betti numbers in 1976 to study free cocompact actions of discrete groups on non-compact manifolds. Algebraically they are described as follows. For a countable discrete group $G$, the group von Neumann algebra $\mathcal{N}(G)$ is the weak closure of the complex group ring $\mathbb{C}G$ acting by left convolution on the Hilbert space $\ell^2 G$; it carries the canonical trace $\tau(T)=\langle T\zeta_e,\zeta_e\rangle$, where $\zeta_e$ is the basis vector of the neutral element. For a finite matrix $T\in M_n(\mathbb{C}G)$ one defines the von Neumann dimension of its kernel as the trace of the spectral projection onto the kernel, a real number in $[0,n]$; this dimension is the basic finitary form of an $\ell^2$-Betti number, and the $\ell^2$-Betti numbers of any free $G$-CW-complex with compact quotient are such kernel dimensions of its cellular Laplacians.

Following Grabowski, write $\mathcal{C}(G,k)$ for the set of all von Neumann dimensions $\dim_{\mathrm{vN}}\ker T$ with $T$ ranging over finite matrices over $kG$, for a subfield $k\subseteq\mathbb{C}$, and $\mathcal{C}(G)=\mathcal{C}(G,\mathbb{Q})$; these are the $\ell^2$-Betti numbers 'arising from' $G$. The range-of-values question Atiyah asked is which non-negative reals occur. The strong Atiyah conjecture predicts $\mathcal{C}(G,\mathbb{C})\subseteq\frac{1}{\mathrm{lcm}(G)}\mathbb{Z}$, where $\mathrm{lcm}(G)$ is the least common multiple of the orders of the finite subgroups of $G$; it is proved for large classes (free groups, elementary amenable groups with bounded torsion, braid groups, right-angled Artin and Coxeter groups, one-relator groups) and implies the Kaplansky zero-divisor conjecture for torsion-free groups. The lamplighter group $\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}=(\bigoplus_{i\in\mathbb{Z}}\mathbb{Z}/2\mathbb{Z})\rtimes\mathbb{Z}$, so named because its elements are a finite 'lamp field' of bits together with the position of a 'lamplighter' moving by the shift, was the first counterexample to the unrestricted conjecture: Grigorchuk and Żuk computed an explicit element of its complex group ring with kernel dimension $1/3$, even though all its finite subgroups have order at most $2$.

Subsequent developments shape the questions. Austin (2013, announced 2009) proved non-constructively that the union of the sets $\mathcal{C}(G)$ over finitely generated groups is uncountable, giving irrational values without identifying any group. Grabowski (2014), embedding Turing machines into group rings through 'Turing dynamical systems', proved: every non-negative real lies in $\mathcal{C}(G)$ for some finitely generated group (independently also obtained by Pichot, Schick and Żuk); every real with computable binary expansion arises from a finitely presented group; and the threefold product $(\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z})^3$ produces an explicit irrational value. Here a real is said to have computable binary expansion when its fractional part is $\sum_{i\in\Sigma}2^{-i}$ for a set $\Sigma$ of natural numbers that a Turing machine can list in increasing order; algebraic numbers, $e$ and $\pi$ have such expansions, while Chaitin's halting probability does not. For sofic groups — groups locally approximable by finite permutation actions, a class containing all amenable and all residually finite groups — values in $\mathcal{C}(G)$ are computable relative to an oracle for the word problem of $G$, and the word problem of a finitely presented group is recursively enumerable, which bounds how complex finitely presented values can be; Löh and Uschold sharpened this to 1-right-computability, meaning each such value is the limit of a decreasing sequence of rationals that is computable relative to the halting problem.

One must distinguish $\ell^2$-Betti numbers arising from $G$, i.e. kernel dimensions of arbitrary matrices over $\mathbb{Q}G$, from the $\ell^2$-Betti numbers of the group $G$ itself, defined via the classifying $G$-CW-complex $EG$; the latter are a much more restricted class of values. Grabowski's paper posed four open questions, of which the first three concern the value sets above, while the fourth, on rationality of the intrinsic group invariants, was resolved negatively for finitely generated groups by the preprints in References 15 and 16; its two surviving refinements — whether a group of finite homological type ($\mathrm{FP}_{n+1}(\mathbb{Q})$, in particular a finitely presented group) can have an irrational $n$-th $\ell^2$-Betti number, and whether a torsion-free group can — concern these intrinsic invariants rather than the value sets arising from a group and lie outside the scope of this problem.

Problem Statement

Determine the range of values of $\ell^2$-Betti numbers — the Atiyah problem in Grabowski's formulation 'what is the set $\mathcal{C}(G)$ of $\ell^2$-Betti numbers arising from $G$' — on the three fronts left open by the 2014 source paper that remain open today. First, determine exactly the set of reals that arise as $\dim_{\mathrm{vN}}\ker T$ for matrices $T$ over $\mathbb{Q}G$ with $G$ finitely presented: this set contains every real with computable binary expansion and is contained in the non-negative 1-right-computable reals, and the open question is its exact characterization, in particular whether values with non-computable binary expansion can occur. Second, for a countable group $G$, determine whether $\mathcal{C}(G,k)=\mathcal{C}(G,\mathbb{C})$ for every subfield $k\subseteq\mathbb{C}$ — equivalently, whether $\mathcal{C}(G,\mathbb{Q})=\mathcal{C}(G,\mathbb{C})$ — and in particular whether $\mathcal{C}(G,\mathbb{C})$ is countable for every countable $G$; both questions are open even for the lamplighter group $G=\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}$. Third, determine the exact set $\mathcal{C}(\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z})$, which is known to contain $1/3$ and irrational and transcendental numbers but has not been characterized; a concrete form of the question posed by Grabowski is whether $\sqrt{2}\in\mathcal{C}(\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z})$. The three fronts are posed jointly; a claimed characterization requires both inclusions. The fourth question posed in the same paper — rationality of the $\ell^2$-Betti numbers of groups themselves — was resolved negatively for finitely generated groups by recent preprints (References 15 and 16) and is therefore not part of the open core; its surviving finiteness-property and torsion-free refinements, set out in the Background, concern intrinsic group invariants rather than the value sets arising from a group and are likewise excluded.

References

1. M. F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32-33 (1976) 43-72, http://www.numdam.org/item/AST_1976__32-33__43_0/
2. Rostislav I. Grigorchuk and Andrzej Żuk, The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum, Geometriae Dedicata 87 (2001) 209-244, DOI: 10.1023/A:1012061801279, https://doi.org/10.1023/A:1012061801279
3. Tim Austin, Rational group ring elements with kernels having irrational dimension, Proceedings of the London Mathematical Society 107 (2013) 1424-1448, DOI: 10.1112/plms/pdt029, https://doi.org/10.1112/plms/pdt029
4. Franz Lehner and Stephan Wagner, Free lamplighter groups and a question of Atiyah, American Journal of Mathematics 135 (2013) 835-849, DOI: 10.1353/ajm.2013.0029, https://doi.org/10.1353/ajm.2013.0029
5. Łukasz Grabowski, On Turing dynamical systems and the Atiyah problem, Inventiones mathematicae 198 (2014) 27-69, DOI: 10.1007/s00222-013-0497-5, arXiv:1004.2030, https://doi.org/10.1007/s00222-013-0497-5
6. Łukasz Grabowski, Irrational l2-invariants arising from the lamplighter group, Groups, Geometry, and Dynamics 10 (2016) 795-817, DOI: 10.4171/GGD/366, arXiv:1009.0229, https://doi.org/10.4171/GGD/366
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8. Mikaël Pichot, Thomas Schick and Andrzej Żuk, Closed manifolds with transcendental L2-Betti numbers, Journal of the London Mathematical Society 92 (2015) 371-392, DOI: 10.1112/jlms/jdv026, arXiv:1005.1147, https://doi.org/10.1112/jlms/jdv026
9. Andrei Jaikin-Zapirain, The base change in the Atiyah and the Lück approximation conjectures, Geometric and Functional Analysis 29 (2019) 464-538, DOI: 10.1007/s00039-019-00487-3, https://doi.org/10.1007/s00039-019-00487-3
10. Pere Ara and Joan Claramunt, L2-Betti numbers arising from the lamplighter group, Journal of Algebraic Combinatorics 54 (2021) 1201-1245, DOI: 10.1007/s10801-021-01044-8, arXiv:2005.12814, https://doi.org/10.1007/s10801-021-01044-8
11. Pere Ara and Joan Claramunt, Approximating the group algebra of the lamplighter by infinite matrix products, Forum Mathematicum 34 (2022) 851-891, DOI: 10.1515/forum-2020-0141, arXiv:2005.12374, https://doi.org/10.1515/forum-2020-0141
12. Clara Löh and Matthias Uschold, L2-Betti numbers and computability of reals, Computability 12 (2023) 175-201, DOI: 10.3233/COM-220416, arXiv:2202.03159, https://doi.org/10.3233/COM-220416
13. Dominik Kirstein, Christian Kremer and Wolfgang Lück, Some problems and conjectures about L2-invariants, arXiv:2311.17830 (2023), https://arxiv.org/abs/2311.17830
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15. Adrian Ioana and Robin Tucker-Drob, A continuum of non-measure equivalent groups, arXiv:2512.04531 (2025), https://arxiv.org/abs/2512.04531
16. Francesco Fournier-Facio and Roman Sauer, Kazhdan groups of dimension 16 with prescribed second l2-Betti number, arXiv:2601.00074 (2026), https://arxiv.org/abs/2601.00074