Closed manifolds with transcendentalL2-Betti numbers:
Mikaël Pichot, Thomas Schick, Andrzej Żuk · Journal of the London Mathematical Society · 2015
In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L 2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L 2 -Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L 2 -Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below). In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L 2 -dimension of the kernel is the given number. We follow the method pioneered by Austin [‘Rational group ring elements with kernels having irrational dimension’, Proc. London Math. Soc. (3) 107 (2013) 1424–1448], but refine it to get explicit calculations which make the above statements possible.