Closed manifolds with transcendental L2-Betti numbers

Abstract

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.

25 pages; v2: small corrections and improvements of the presentation. Final version to appear in Journal of the London Mathematical Society

References in corpus (3)

• Rational group ring elements with kernels having irrational dimension
• On Turing dynamical systems and the Atiyah problem
• Free Lamplighter Groups and a Question of Atiyah

Cited by in corpus (8)

• On Turing dynamical systems and the Atiyah problem
• The strong Atiyah conjecture for right-angled Artin and Coxeter groups
• Free Lamplighter Groups and a Question of Atiyah
• Survey on approximating L^2-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth
• Mean Dimension, Mean Rank, and von Neumann-Lück Rank
• On computing homology gradients over finite fields
• The realization problem for some wild monoids and the Atiyah problem
• The Strong Atiyah conjecture for virtually cocompact special groups