On Turing dynamical systems and the Atiyah problem

Abstract

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.

35 pages; essentially identical to the published version

References in corpus (5)

• Rational group ring elements with kernels having irrational dimension
• A spectral sequence to compute L2-Betti numbers of groups and groupoids
• Closed manifolds with transcendental L2-Betti numbers
• Free Lamplighter Groups and a Question of Atiyah
• On the spectrum of lamplighter groups and percolation clusters

Cited by in corpus (11)

• The strong Atiyah conjecture for right-angled Artin and Coxeter groups
• Closed manifolds with transcendental L2-Betti numbers
• 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
• Group ring elements with large spectral density
• Dyson's spike for random Schroedinger operators and Novikov-Shubin invariants of groups
• Mean Dimension, Mean Rank, and von Neumann-Lück Rank
• Vanishing of l^2-cohomology as a computational problem
• On computing homology gradients over finite fields
• K-theory for generalized Lamplighter groups
• The realization problem for some wild monoids and the Atiyah problem