Rational group ring elements with kernels having irrational dimension. (Q2872032)

When Atiyah introduced the \(L^2\)-Betti numbers of normal coverings of a compact manifold with covering group \(\Gamma\), he asked for the possible values which can occur. In particular, he asked whether they always have to be rational numbers.NEWLINENEWLINE This problem has been popularized as the ``Atiyah conjecture on the value of \(L^2\)-Betti numbers''. Known reductions show that this is in some sense a purely algebraic question, namely on the spectrum of elements of the rational group ring \(\mathbb Q\Gamma\), acting on the Hilbert space \(l^2(\Gamma)\) via the left regular representation. This question was at the center of activity for many years, proving in many cases rationality -- but to produce examples of irrational values proved illusive. Earlier work of \textit{W. Dicks} and the reviewer [Geom. Dedicata 93, 121-137 (2002; Zbl 1021.47014)] had provided candidates of irrational \(L^2\)-Betti numbers, but nobody had every been able to verify that the values constructed there are actually irrational.NEWLINENEWLINE In the very important article at hand, Tim Austin finally manages to prove that irrational, even transcendental \(L^2\)-Betti numbers exist. Indeed, he proves that the set of \(L^2\)-Betti numbers is uncountable. In its very rough outline, the method is inspired by the paper of Dicks and Schick just cited. In particular, this is based on an explicit invariant decomposition where each summand contributes in a controlled way to the spectrum. However, Austin implements a number of very interesting new ideas.NEWLINENEWLINE The first is the use of much more general (but still lamplighter-like) operators which allow for a more flexible spectral decomposition. To be able to have more room for this flexibility, he works with a wreath product of a finite group with a nonabelian free group.NEWLINENEWLINE The third very relevant new idea is the use of quotients of the lamplighter group, so instead of one specific group a whole family of groups is considered at once. This allows to fine-tune which of the above summands in the spectral decomposition actually contribute and to ``switch off'' others.NEWLINENEWLINE Tim Austin, at this point, doesn't carry out the calculations explicitly. Indeed, he shows that the construction is sufficiently flexible to give an uncountable set of different \(L^2\)-Betti numbers. Necessarily, this requires him to use groups \(\Gamma\) which are not finitely presented.