Andreas Thom: Maximal discrete subgroups in unitary groups of operator algebras

We show that if a group G is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing G. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for C∗-algebras. Similar results for full groups of p.m.p. equivalence relations are also discussed. This is joint work with Vadim Alekseev and Alessandro Carderi.