Free Lamplighter Groups and a Question of Atiyah

TL;DR

This paper calculates the von Neumann dimensions of adjacency operator kernels on free lamplighter groups, demonstrating their irrationality and addressing a question posed by Atiyah.

Contribution

It provides the first elementary constructive proof that these von Neumann dimensions can be irrational, answering Atiyah's question.

Findings

Von Neumann dimensions are irrational

Explicit computation of kernels on free lamplighter groups

Addresses Atiyah's longstanding question

Abstract

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.