2010/05/31 by Mikaël Pichot, Thomas Schick, Andrzej Zuk
Mathematics · #math.KT #math.RA
paper · pdf · doi:10.1112/jlms/jdv026
25 pages; v2: small corrections and improvements of the presentation. Final version to appear in Journal of the London Mathematical Society
arxiv created 2015/06/16 · arxiv updated 2017/05/17
In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L2-Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L2-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 L2-dimension of the kernel is the given number. We follow the method pioneered by Austin in "Rational group ring elements with kernels having irrational dimension" arXiv:0909.2360) but refine it to get very explicit calculations which make the above statements possible.