The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary group of this von Neumann algebra contains every countable discrete amenable group as a subgroup.
I'd like to collect here some exotic (countable discrete) subgroups of the unitary group of the hyperfinite $II_{1}$ factor that aren't amenable. As subjective as it sounds, here's my question:
What is the most interesting discrete subgroup appearing as a subgroup of the unitary group of the hyperfinite $II_1$ factor?
(An invariant mean on a group G gives rise to a hypertrace for LG, but if I have a subgroup of the unitary group that gives rise to a representation not the regular representation, this no longer works.)