L^2-Betti numbers arising from the lamplighter group

TL;DR

This paper develops a formula to compute $ ext{L}^2$-Betti numbers from group algebras, producing explicit irrational values for the lamplighter group and analyzing the odometer algebra to characterize its $ ext{L}^2$-Betti numbers.

Contribution

It introduces a constructive method to compute irrational $ ext{L}^2$-Betti numbers from lamplighter group algebras and characterizes $ ext{L}^2$-Betti numbers for the odometer algebra.

Findings

Explicit irrational $ ext{L}^2$-Betti numbers for lamplighter group algebra.

Complete characterization of $ ext{L}^2$-Betti numbers for odometer algebra.

Extension of previous work by Grabowski on irrational $ ext{L}^2$-Betti numbers.

Abstract

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $\ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational $\ell^2$-Betti numbers arising from the lamplighter group algebra $K[\mathbb{Z}_2 \wr \mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $\ell^2$-Betti numbers from the algebras $\mathbb{Q}[\mathbb{Z}_n \wr \mathbb{Z}]$, where $n \geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $\mathcal{O}(\overline{n})$, where $\overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $\ell^2$-Betti numbers arising from $\mathcal{O}(\overline{n})$.