Non-unitarisable representations and random forests

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.

References in corpus (1)

• Minimal spanning forests

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• The $M_d$-Approximation Property and Unitarisability
• On the Dixmier problem (Seminar report after Monod-Ozawa, JFA 2010)