  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.
