The strong Atiyah conjecture for groups with bounded torsion

For a discrete group $G$ whose finite subgroups have uniformly bounded order, let $\operatorname{lcm}(G)$ be the least common multiple of the orders of its finite subgroups. The strong Atiyah conjecture predicts that for every proper cocompact $G$ -CW complex $Y$ and every degree $k$ , the $L^{2}$ -Betti number satisfies $\operatorname{lcm}(G)\cdot b_k^{(2)}(Y;G)\in\mathbb{Z}$ ; equivalently, the von Neumann rank of every finite matrix over the complex group ring $\mathbb{C}G$ lies in $\frac{1}{\operatorname{lcm}(G)}\mathbb{Z}$ . The conjecture is known for many classes (Linnell's class, elementary amenable groups, right-angled Artin and Coxeter groups, locally indicable and one-relator groups, virtually cocompact special groups, torsion-free 3-manifold groups), and irrational or transcendental $L^{2}$ -Betti numbers are known only for groups whose torsion is unbounded. Whether the conjecture holds for the entire class of groups with bounded torsion — including the torsion-free case, which would imply Kaplansky's zero-divisor conjecture in characteristic zero — remains open. The problem is posed as an open question in Kevin Schreve's 2014 paper on virtually cocompact special groups, which established the conjecture for that class and explicitly leaves the general bounded-torsion case open.

Background

Let $G$ be a discrete group acting cellularly on a CW-complex $Y$ . The action is called proper if all cell stabilizers are finite, and cocompact if the quotient $Y/G$ is a finite CW-complex; such a $Y$ is a proper cocompact $G$ -CW complex. The group von Neumann algebra $\mathcal{N}(G)$ is the algebra of bounded operators on $\ell^{2}(G)$ commuting with the left $G$ -action, equipped with its canonical trace $\operatorname{tr}_{\mathcal{N}(G)}$ , and the trace extends to a dimension $\dim_{\mathcal{N}(G)}$ for finitely generated projective $\mathcal{N}(G)$ -modules and to kernels of $\mathcal{N}(G)$ -module maps. Applying this to the cochain complex of $Y$ with coefficients in $\mathcal{N}(G)$ defines the $L^{2}$ -Betti numbers $b_k^{(2)}(Y;G)$ , which are the von Neumann dimensions of the $L^{2}$ -cohomology of $Y$ . Equivalently, for a finite matrix $A$ over the complex group ring $\mathbb{C}G$ , viewed as an operator on a finite direct sum of copies of $\ell^{2}(G)$ , one considers the von Neumann rank $\operatorname{rk}_{\mathcal{N}(G)}(A)=\dim_{\mathcal{N}(G)}\ker A$ .

Atiyah (1976), in the context of the $L^{2}$ -index theorem, asked what values these invariants can take, and the strong form of his conjecture, developed by Linnell (1993), predicts a sharp arithmetic constraint: if the orders of the finite subgroups of $G$ are bounded, then $b_k^{(2)}(Y;G)\in\frac{1}{\operatorname{lcm}(G)}\mathbb{Z}$ , where $\operatorname{lcm}(G)$ is the least common multiple of the orders of the finite subgroups of $G$ . The bound is attained (a finite subgroup of order $n$ contributes values in $\frac{1}{n}\mathbb{Z}$ ), so the conjecture, if true, is optimal.

The conjecture has been confirmed for large classes built from free groups by elementary amenable extensions (Linnell's class $\mathcal{C}$ ), for elementary amenable groups with bounded torsion, for right-angled Artin and Coxeter groups (Linnell–Okun–Schick), for locally indicable and hence all one-relator groups (Jaikin-Zapirain–López-Álvarez), for torsion-free 3-manifold groups (Kielak–Linton), and — the source of the present question — for virtually cocompact special groups (Schreve). A group is (cocompact) special if it is the fundamental group of a compact special cube complex in the sense of Haglund and Wise — a non-positively curved cube complex whose hyperplanes are two-sided and embedded without self-intersection — and every such group embeds in a right-angled Artin group; the class is central to modern geometric group theory. Schreve proved new conditions under which the conjecture lifts to finite group extensions, building on the Linnell–Schick extension criterion. Sánchez-Peralta later showed the conjecture is closed under fundamental groups of graphs of groups with finite edge groups, including free products.

The conjecture genuinely fails without the bounded-torsion hypothesis: Austin, Grabowski, and Pichot–Schick–Zuk constructed groups (necessarily with finite subgroups of unbounded order) whose $L^{2}$ -Betti numbers are irrational, and even transcendental. By contrast, no non-integral value is known for any group with bounded torsion. The stakes of the bounded-torsion case are high: for torsion-free $G$ one has $\operatorname{lcm}(G)=1$ , and the conjecture then implies Kaplansky's zero-divisor conjecture over fields of characteristic zero (if $G$ is torsion-free and satisfies the conjecture, the group algebra $\mathbb{C}G$ is a domain — for instance, the von Neumann dimension of the kernel of the multiplication map $\ell^{2}(G)\to\ell^{2}(G)$ , $b\mapsto ab$ , must then be an integer for every $a\in\mathbb{C}G$ , which rules out zero divisors; the implication is stated explicitly in Linnell–Schick and in Fisher–Ng). Even weakened forms, asserting only that the rank function takes values in some discrete subgroup $\frac{1}{n}\mathbb{Z}$ , are open in general and drive recent applications such as the coherence of one-relator groups (Jaikin-Zapirain–Linton).

Problem Statement

Let $G$ be a discrete group whose finite subgroups have uniformly bounded order, and let $\operatorname{lcm}(G)$ denote the least common multiple of the orders of the finite subgroups of $G$ . Is it true that for every proper cocompact $G$ -CW complex $Y$ — a CW-complex with a cellular $G$ -action whose cell stabilizers are all finite (proper) and whose quotient $Y/G$ is a finite CW-complex (cocompact) — and every integer $k\geq 0$ , $\operatorname{lcm}(G)\cdot b_k^{(2)}(Y;G)\in\mathbb{Z},$ where $b_k^{(2)}(Y;G)$ , the $k$ -th $L^{2}$ -Betti number, is the von Neumann dimension of the $k$ -th reduced $L^{2}$ -cohomology of $Y$ (the von Neumann dimension being the extension of the canonical trace on the group von Neumann algebra $\mathcal{N}(G)$ of bounded $G$ -equivariant operators on $\ell^{2}(G)$ )? Equivalently, is $\operatorname{rk}_{\mathcal{N}(G)}(A)\in\frac{1}{\operatorname{lcm}(G)}\mathbb{Z}$ for every finite matrix $A$ over the complex group ring $\mathbb{C}G$ , where $\operatorname{rk}_{\mathcal{N}(G)}(A)=\dim_{\mathcal{N}(G)}\ker A$ for $A$ viewed, via the left regular representation of $G$ , as a bounded operator on a finite direct sum of copies of $\ell^{2}(G)$ ? In particular, determine whether non-integral values — rational with denominator not dividing $\operatorname{lcm}(G)$ , or irrational — can occur for a group with bounded torsion, or prove that they cannot. The question includes the torsion-free case $\operatorname{lcm}(G)=1$ , where it asks whether all $L^{2}$ -Betti numbers are integers.

References

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