Rational group ring elements with kernels having irrational dimension

TL;DR

This paper demonstrates the existence of finitely generated groups with group ring elements whose kernels have irrational von Neumann dimensions, addressing a question posed by Atiyah.

Contribution

It provides the first known examples of group ring elements with irrational kernel dimensions in finitely generated groups, advancing understanding in group ring analysis.

Findings

Existence of finitely generated groups with irrational kernel dimensions.

Construction of specific group ring elements with irrational von Neumann dimension.

Answers a long-standing question of Atiyah regarding kernel dimensions.

Abstract

We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.