The range of $L^2$-Betti numbers of universal covers of finite CW-complexes

For a finite CW-complex $X$ (in particular a closed manifold), the $L^2$-Betti numbers of its universal covering are real-valued homotopy invariants introduced by Atiyah in 1976. Since there are only countably many finite complexes, only countably many reals can occur, and this countable set contains transcendental numbers (Pichot–Schick–Żuk 2015; Grabowski 2014) as well as, by the computability analysis of Löh–Uschold (2023), every non-negative weakly computable real (a difference of two left-computable reals) — a class that already contains non-computable reals such as Chaitin's halting probability — while being contained in the reals that are right-computable relative to the halting problem. This problem asks for an exact characterization of this set — equivalently, of the set of von Neumann dimensions of kernels of matrices over group rings of finitely presented groups — posed explicitly by Grabowski (Invent. Math. 2014, Question 1) and Pichot–Schick–Żuk (J. Lond. Math. Soc. 2015, Remark 13.4) and restated as open by Löh–Uschold (2023). Sharp sub-questions: is every non-negative real computable from the halting problem realized; is the set contained in the reals computable from the halting problem (only right-computability relative to the halting problem is known); and can an irrational value be realized by an aspherical finite complex (a group of type $FP_{n+1}(\mathbb{Q})$ with irrational $n$-th $\ell^2$-Betti number)?

Background

Let $X$ be a finite CW-complex (a cell complex with finitely many cells; every closed manifold is homotopy equivalent to one) and let $\widetilde{X} \to X$ be its universal covering, the simply connected cover corresponding to the trivial subgroup of the fundamental group $\Gamma = \pi_1(X)$. The cellular chain complex of $\widetilde{X}$ is a complex of free modules over the group ring $\mathbb{Z}\Gamma$; tensoring with $\ell^2(\Gamma)$ and taking homology with Hilbert-space dimensions measured by the group von Neumann dimension yields the $L^2$-Betti numbers $b_n^{(2)}(\widetilde{X}) \in [0,\infty)$. Here the group von Neumann algebra $\mathcal{N}(\Gamma)$ is the algebra of bounded $\Gamma$-equivariant operators on $\ell^2(\Gamma)$ with its canonical trace $\tau$, and the von Neumann dimension of a closed $\mathcal{N}(\Gamma)$-submodule is the trace of its orthogonal projection. These invariants were introduced by Atiyah (1976), who asked at the end of his paper for examples in which they are irrational. When $\Gamma$ is finitely presented (given by finitely many generators and relations — exactly the groups arising as fundamental groups of finite CW-complexes), the numbers are homotopy invariants of $X$ and a priori arbitrary reals. The expectation that they should always be rational, even integral for torsion-free $\Gamma$ (with denominators controlled by the finite subgroups otherwise), became the (strong) Atiyah conjecture; it holds for large classes of groups (Linnell's class, locally indicable, right-angled Artin and Coxeter groups, one-relator groups) and remains open, for instance, for all hyperbolic and all amenable groups (Kirstein–Kremer–Lück 2023).

The first rigorous failure of rationality came from harmonic analysis on the lamplighter group $\mathbb{Z}/2 \wr \mathbb{Z}$ (the wreath product, i.e. the semidirect product of the finitely supported lamp configurations indexed by $\mathbb{Z}$ with the group of translations): Grigorchuk–Żuk (2001) computed its full spectral measure, Dicks–Schick (2002) developed the resulting spectral-measure machinery, and Grigorchuk–Linnell–Schick–Żuk (2000) exhibited an irrational von Neumann dimension of the kernel of a group-ring element. Austin (2013) then produced uncountably many, in particular irrational, values as kernel dimensions over finitely generated groups, by a non-constructive counting argument. Two parallel 2014–2015 works made the values explicit and pushed them into the finitely presented world: Pichot–Schick–Żuk (2015), and Grabowski (2014, via his Turing dynamical systems, i.e. Turing machines encoded into group rings). Both start from a recursively presented group $G_I$ encoding a set $I \subseteq \mathbb{N}$ (for instance a group $(\mathbb{Z}/2^{\oplus\Gamma}/V) \rtimes \Gamma$ with $\Gamma$ the free group $F_2$ or $\mathbb{Z} \wr \mathbb{Z}$) for which an explicit group-ring element has kernel dimension $\sum_{n \in I} 2^{-n}$ up to explicit rational corrections; Higman's embedding theorem (1961) embeds the recursively presented group into a finitely presented one, and the kernel dimension of a fixed group-ring element is unchanged by passage to the finitely presented supergroup. A standard construction (recorded in Lück's 2002 book, Lemma 10.5) then realizes the value as the third $L^2$-Betti number of a closed 7-dimensional manifold, so transcendental $L^2$-Betti numbers of universal covers of closed manifolds exist. On the other side, Lück's approximation theorem expresses $b_n^{(2)}(\widetilde{X})$ as the limit of the ordinary rational-valued Betti numbers of finite covers, which ties the range to computability theory: a real is $\Delta^0_2$ (equivalently, the limit of a computable sequence of rationals) exactly when it is computable with an oracle for the halting problem, and the word problem of a finitely presented group is always reducible to the halting problem. Refinements of this observation — with the determinant class condition (a Fuglede–Kadison-determinant regularity condition introduced by Schick 2001) replacing residual finiteness, and with the Lück approximation conjecture for sofic groups (soficity being a finite-approximation property of groups, verified for every class of groups where it has been tested but open in general), established by Jaikin-Zapirain (2019), supplying the approximation framework — culminated in the computability analysis of Löh–Uschold (2023), which delimits the finitely presented range from both sides: every $L^2$-Betti number arising from a finitely presented group is right-computable relative to the halting problem, and conversely the range contains every non-negative weakly computable real (a difference of two left-computable reals, i.e. reals computably approximable from below) — a class that already contains non-computable reals, Chaitin's halting probability among them. The exact shape of the range nonetheless remains open. Meanwhile, for finitely generated (not necessarily finitely presented) groups the range is completely determined: Grabowski (Theorem 1.3) showed every non-negative real is a kernel dimension over some finitely generated group, and Fournier-Facio–Sauer (2026) strengthened this to the groups' own Betti numbers. The gulf between the finitely generated and finitely presented worlds is the subject of this problem.

Problem Statement

Define $S = \{ b_n^{(2)}(\widetilde{X}) : X \text{ a finite CW-complex}, \ \widetilde{X} \to X \text{ its universal covering}, \ n \geq 0 \}$, the set of real numbers occurring as $L^2$-Betti numbers of universal covers of finite CW-complexes (equivalently of closed manifolds). By the standard dictionary between topology and algebra, $S$ is the von Neumann dimension range for finitely presented groups: the values of $\dim_{\mathcal{N}(G)} \ker(A)$ for matrices $A$ over $\mathbb{Z}[G]$ with $G$ finitely presented, together with the integer combinations generated by finite cellular chain complexes. The problem is to characterize $S$ exactly — Grabowski's Question 1, PSZ's Remark 13.4, and Löh–Uschold's restatement of both. The published bounds are: every non-negative weakly computable real (a difference of two left-computable reals; this class contains non-computable reals such as Chaitin's halting probability, and in particular every real with computable binary expansion) belongs to $S$, and $S$ is contained in the non-negative reals that are right-computable relative to the halting problem (Löh–Uschold 2023). So non-computable values are known to occur, but the exact position of $S$ between the weakly computable reals and the reals right-computable relative to the halting problem is unknown. The following sharp sub-questions delimit the frontier: (1) Is every non-negative $\Delta^0_2$ real (equivalently, every real computable with an oracle for the halting problem) an element of $S$? (2) Dually, is $S$ contained in the $\Delta^0_2$ reals? The published upper bound is only right-computability relative to the halting problem, and determining the exact computability boundary of $S$ remains open. (3) Between the weakly computable reals and the $\Delta^0_2$ reals, determine exact membership: does a $\Delta^0_2$ real that is not weakly computable occur in $S$, and more generally which effectively presented classes of reals strictly containing the weakly computable reals (which are all realized) are contained in $S$? (4) Does $S$ contain an irrational value realized by an aspherical finite complex — i.e. does there exist a group of type $FP_{n+1}(\mathbb{Q})$ (meaning that the trivial $\mathbb{Q}G$-module $\mathbb{Q}$ admits a projective resolution finitely generated in degrees at most $n+1$; in particular a torsion-free group of type $F$, meaning it admits a finite classifying complex) with irrational $n$-th $\ell^2$-Betti number — the finiteness-property form of Grabowski's Question 4? A complete characterization identifies $S$ with an explicitly described, effectively presented class of reals together with proofs of both inclusions.

References

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