Pierre Emmanuel Caprace - Groups with irreducibly unfaithful subsets for unitary representations

A subset F of a group G is called irreducibly faithful if G has an irreducible unitary representation whose kernel does not contain any non-trivial element of F. We say that G has property
P(n) if every subset of size at most n is irreducibly faithful. By a classical result of Gelfand and Raikov, every group has
P(1). Walter proved that every group has P(2). The goal of this talk, based on a joint work with Pierre de la Harpe, is to provide, for each positive integer n, a purely group theoretic characterization of the countable groups satisfying P(n).