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 \(L^2\)-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.
Let \(E_0\) denote the equivalence relation on Cantor space of eventual agreement. We present two canonization theorems concerning the structure of restrictions of Borel orders to sets on which \(E_0\) is not smooth. The first canonizes Borel linear orders on Cantor space, and the second canonizes Borel assignments of linear orders to \(E_0\) classes.