On Turing dynamical systems and the Atiyah problem

TL;DR

This paper explores the possible values of l^2-Betti numbers, showing they encompass all non-negative reals and constructing examples with transcendental values using Turing machine embeddings into group rings.

Contribution

It introduces a novel method of embedding Turing machines into integral group rings to analyze l^2-Betti numbers of manifolds.

Findings

All non-negative real numbers are l^2-Betti numbers.

Many algebraic real numbers are realizable as l^2-Betti numbers.

Constructed an example with a transcendental l^2-Betti number.

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.