Diarmuid Crowley visited for a couple of days last week to talk about the Manifold Atlas Project which is a plan to produce a sort of online encyclopedia/journal of information about all sorts of manifolds. Of course we talked about other things also, and I learned from Diarmuid about a paper by Tim Austin of UCLA which gives a counterexample to a version of the “Atiyah Conjecture”.
Atiyah would want me to point out that the conjecture is misnamed. In the paper in Asterisque where he introduces the \(L^2\) Betti numbers, Atiyah asked as a problem: “Give examples where these invariants are not integers or perhaps even irrational”. The name “Atiyah conjecture” got attached to the claim that there are no such examples! And the question about the integrality of the invariants (for torsion free groups) is still open. But for groups with torsion, Zuk gave a counterexample some years ago to the claim that the denominators of the \(L^2\) Betti numbers must be generated by the torsion orders in the group; and now Austin shows that there are groups with irrational (even transcendental) \(L^2\) Betti numbers.
The argument is a non-constructive one: Austin builds an uncountable family of groups, and in the group ring of each group a particular element, such that the von Neumann dimensions of the kernels of these elements are all different. (The groups are certain “lamplighter-type” groups built on the free group.) Thus there are uncountably many different real numbers which are \(L^2\) Betti numbers, and some of them must not be rational (or algebraic). But the process doesn’t identify a particular group for which this is true.
(Added later: See Thomas’ comments below for a follow-up paper by Lukasz Grabowski, which begins “The main point of this article is to show some connections between Turing machines and von Neumann algebras”.)