Non-unitarisable representations and random forests

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

Nonunitarizable Representations and Random Forests

We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L2-Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroup

Modular actions and amenable representations

Consider a measure-preserving action Γ ↷ (X, μ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ ↷ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ ↷^α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones