The Dixmier problem, lamplighters and Burnside groups

TL;DR

This paper investigates the Dixmier problem by constructing non-unitarisable representations for certain groups, providing a new way to characterize amenability and answering questions about Burnside groups.

Contribution

It introduces non-unitarisable representations for extensions by abelian groups, offering a novel characterization of amenability and addressing open questions about Burnside groups.

Findings

Constructed non-unitarisable representations for group extensions by abelian groups.

Provided a new characterization of amenability based on these representations.

Showed certain Burnside groups are non-unitarisable, answering prior open questions.

Abstract

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.