Non-unitarisable representations and random forests

TL;DR

This paper links the unitarisability problem in group theory to properties of random forests, providing new criteria to identify non-unitarisable groups, including examples without free subgroups.

Contribution

It introduces a novel connection between Dixmier's unitarisability problem and random forests, offering new criteria for non-unitarisability in residually finite groups.

Findings

Residually finite groups with non-zero first L2-Betti number are non-unitarisable.

Groups with non-trivial cost are non-unitarisable.

Constructs the first examples of non-unitarisable groups without free subgroups.

Abstract

We establish a connection between Dixmier's unitarisability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is non-unitarisable if its first L2-Betti number is non-zero or if it is finitely generated with non-trivial cost. Our criterion also applies to torsion groups constructed by D. Osin, thus providing the first examples of non-unitarisable groups not containing a non-Abelian free subgroup.