On a conjecture of Atiyah

TL;DR

This paper demonstrates that the spectrum computation of the lamplighter group provides a counterexample to a strong form of Atiyah's conjectures on the possible values of $L^2$-Betti numbers for closed manifolds, challenging existing beliefs.

Contribution

It links spectral analysis of the lamplighter group to a counterexample, advancing understanding of Atiyah's conjectures and their limitations.

Findings

Counterexample to a strong Atiyah conjecture

Spectrum of lamplighter group used in topological context

Challenges previous assumptions about $L^2$-Betti numbers

Abstract

In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of $L^2$-Betti numbers of closed manifolds.