The paper "Some remarks on the entropy for algebraic actions of amenable groups" by Nhan-Phu Chung and Andreas Thom asks (Question 2.1): if Γ is a torsion-free amenable group, must the von Neumann rank rk(M) = dim_{L(Γ)}(L(Γ) ⊗_{ZΓ} M) of every left ZΓ-module M be a non-negative integer or infinity? The authors show that a positive answer is equivalent to Kaplansky's zero divisor conjecture for the integral group ring ZΓ and — Γ being amenable — to the Atiyah conjecture on integrality of ℓ²-Betti numbers. No proof for general torsion-free amenable groups and no torsion-free amenable counterexample is known; the problem remains open precisely for torsion-free amenable groups beyond the elementary amenable class.

Background

A countable discrete group Γ is called amenable if it admits a finitely additive, left-translation-invariant probability measure on all of its subsets; equivalently, it has Følner sets — finite sets that are almost invariant under the group action. All finite and all abelian groups are amenable, and so is the smallest class of groups built from these by extensions and directed unions, called the elementary amenable groups, but the amenable world is strictly larger (for example, it contains every group of subexponential word growth). For an amenable group Γ and a countable left module M over the integral group ring ZΓ (the ring of finite integer linear combinations of group elements), the Pontryagin dual of M — the compact abelian group of homomorphisms from M to the circle, with Γ acting by dualized module multiplication — carries an action by measure-preserving homeomorphisms called an algebraic action. Chung and Thom studied the topological entropy ρ(M) of such actions and its decomposition into p-adic contributions, and their framework made contact with several long-standing conjectures about group rings.

The primary invariant behind the entropy is the von Neumann rank. The group von Neumann algebra L(Γ) is the closure of the complex group ring CΓ acting by left multiplication on the Hilbert space ℓ²(Γ) of square-summable functions on Γ; it carries the Murray–von Neumann dimension, extended by Lück to arbitrary modules over L(Γ). For a left ZΓ-module M one defines rk(M) = dim_{L(Γ)}(L(Γ) ⊗_{ZΓ} M) ∈ [0, ∞]. The rank is additive in short exact sequences, continuous under filtered colimits, satisfies rk(ZΓ) = 1, and is positive on every non-zero submodule of ZΓ; Li and Liang proved that it coincides with the mean topological dimension of the dual algebraic action. If M is of type FL, meaning it admits a finite resolution by finitely generated free ZΓ-modules, then rk(M) automatically equals the Euler characteristic of that resolution, an integer. The integrality question therefore concerns modules admitting no such finite free resolution, where integrality is not formal.

Questions of this shape go back to Atiyah: the ℓ²-Betti numbers of a group are the von Neumann dimensions of the kernels of matrices over CΓ acting on ℓ²(Γ), and the (strong) Atiyah conjecture predicts that for a torsion-free group these numbers are integers. Closely related is Kaplansky's zero divisor conjecture, predicting that for torsion-free Γ the ring ZΓ (and more generally KΓ for any field K) has no zero divisors. Integrality implies absence of zero divisors: if a non-zero f ∈ ZΓ were a zero divisor, then rk(ZΓ·f) would lie strictly between 0 and 1, since ZΓ·f is a non-zero submodule of ZΓ and is isomorphic to a proper quotient of ZΓ. For amenable groups the converse also holds: Elek proved that for finitely generated amenable groups the classical zero divisor conjecture implies its analytic analogue, and Chung and Thom deduce (their Remark 4.5, from the argument of their Lemma 4.4) that a positive answer to their Question 2.1 is equivalent to the zero divisor conjecture with coefficients in Z; they also note that, Γ being amenable, the question is the same as the Atiyah conjecture for Γ, i.e. integrality of ℓ²-Betti numbers.

The frontier of knowledge is as follows. The zero divisor conjecture, and with it integrality of the rank, is known for torsion-free polycyclic-by-finite groups (Farkas and Snider, using that ZΓ is noetherian there) and for torsion-free elementary amenable groups (Kropholler, Linnell and Moody, via Moody's induction theorem); Linnell's division-ring construction proves the strong Atiyah conjecture for his class of groups built from free groups by directed unions and extensions with elementary amenable quotients, a class containing all elementary amenable groups with bounded torsion. On the negative side, the general Atiyah conjecture fails: Grigorchuk and Żuk's lamplighter computation already refuted the stronger version predicting that ℓ²-Betti numbers lie in the subgroup of ℚ generated by inverses of finite-subgroup orders, and Austin constructed the first group ring elements with irrational — even transcendental — von Neumann kernel dimensions, using lamplighter-like operators on wreath products of finite groups with nonabelian free groups; his groups are finitely generated but not amenable. Grabowski, and independently Pichot–Schick–Żuk, then produced such examples among amenable groups — built from lamplighter groups (the wreath product of a finite non-trivial group with Z, an amenable group containing elements of finite order) — with kernel dimensions realizing any prescribed value in [0, 1], and Grabowski showed that in fact every non-negative real number occurs as an ℓ²-Betti number. Every known counterexample to integrality involves torsion, so torsion-freeness is exactly the hypothesis that keeps the amenable case open.

Problem Statement

Question (Chung–Thom, Question 2.1). Let Γ be a torsion-free amenable (discrete, countable) group. Is it true that the von Neumann rank

rk⁡(M)=dim⁡L(Γ)(L(Γ)⊗ZΓM)\operatorname{rk}(M)=\dim_{L(\Gamma)}\bigl(L(\Gamma)\otimes_{\mathbb{Z}\Gamma}M\bigr)

belongs to N∪{∞}\mathbb{N}\cup\{\infty\} (where N={0,1,2,… }\mathbb{N}=\{0,1,2,\dots\}) for every left ZΓ\mathbb{Z}\Gamma-module MM?

The quantifier ranges over all left ZΓ-modules, not merely finitely generated ones; for amenable Γ the colimit property of the rank reduces the statement to finitely generated modules. By the results of the source paper, the question is equivalent to each of the following: (i) Kaplansky's zero divisor conjecture for the integral group ring in the amenable case — for every torsion-free amenable Γ there are no non-zero f, g ∈ ZΓ with fg = 0; and (ii) the strong Atiyah conjecture for torsion-free amenable groups — all ℓ²-Betti numbers of Γ (von Neumann dimensions of kernels of matrices over CΓ acting on ℓ²(Γ)) are integers.

References

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