Closed manifolds with transcendental L2-Betti numbers

TL;DR

This paper constructs closed manifolds with explicitly computed irrational and transcendental L2 Betti numbers, demonstrating that a wide range of real numbers, including many transcendental ones, can appear as L2-Betti numbers of coverings.

Contribution

It introduces explicit constructions of manifolds and groups with prescribed transcendental L2 Betti numbers, refining previous methods for precise calculations.

Findings

Every non-negative real number appears as an L2-Betti number of some covering.

Many computable real numbers, including transcendental, are realized as L2-Betti numbers.

Explicit constructions of groups and elements in group rings with prescribed L2-dimension.

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.