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 L2 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 L2 Betti numbers must be generated by the torsion orders in the group; and now Austin shows that there are groups with irrational (even transcendental) L2 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 L2 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.
If I could figure out how to get TeX into this thing I might post more...
(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".
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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.
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 not
so 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 the
same kind.
He was more efficient and very recently posted his paper om the arXiv http://front.math.ucdavis.edu/1004.2030
Let me also shamelessly point out that my preprint gives an explicit example of a finitely generated group (i.e. a finite presentation) which gives rise to irrational l^2-Betti numbers.
Btw, it gives a rise "even" to a transcendental number, but I don't think this is the point: it seems to me that l^2-Betti numbers arising from a given group are like complexity classes. The irrational number in my preprint is from the class of numbers recognisable in linear time. Most likely every number from this class can be obtained as an l^2-Betti number arising from that group.
Natural conjecture would be that actually the class of l^2-Betti numbers arising from G \times G \times G, where G is the standard lampighter, is the class of numbers recognisable in a linear time. I believe (but am not sure) that no irrational algebraic numbers belong to that class.
So in the end I want to say that the approach "the more transcendental a number we get the more impressive it is" is not the right one. Rather "the more complex (from the point of complexity theory) a number we get the more impressive it is.
John, may I suggest that you consider moving to wordpress.com? Writing tex there is very straightforward, also in the comments section (as you may see for example at T. Tao's blog: http://terrytao.wordpress.com/ )
Sorry, I meant of course "gives an explicit example of a finitely presented group (i.e. a finite presentation)"
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