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”.)
Tim's article spurred a flurry of activity at the "Rigidity semester" in Bonn last fall.In particular, tim Austin, Mikael Pichot, myself and Andrzej Zuk managed to-make the examples explicit-find among those some with finite presentation (more precisely: recursive presentation, but this gives in a notso explicit way finitely presented examples, as well).We are at the final stages of writing up the paper, which uses a lot of the techniques of Austin.More or less at the same time Lukasz Grabowski has found a slightly different method to obtain examples of thesame kind.He was more efficient and very recently posted his paper om the arXiv http://front.math.ucdavis.edu/1004.2030
PS: The reason for the "sort of" is that these examples are, of course, infinitely presented. You could still conjecture the rationality for fp groups if you wanted. Moreover the torsion free/integrality case is not addressed. Still, this is a great new idea.