Hi, Is it true that all irreducible unitary representations of a residually finite group are finite dimensional?
Actually I suspect that it is not, but cannot find any example.
Hi, Is it true that all irreducible unitary representations of a residually finite group are finite dimensional?
Actually I suspect that it is not, but cannot find any example.
As Mark Sapir pointed out, this is of course false. There are tons of infinite dimensional irreducible representations of residually finite groups. In fact, I think that any finitely generated group for which the answer is yes, is virtually abelian.
However, you could also ask whether all irreducible representations of a given residually finite group are weakly contained in finite dimensional representations. This is true for $\mathbb F_2$ but for example not for $SL(3,\mathbb Z)$. For the group $\mathbb F_2 \times \mathbb F_2$, this problem is open and equivalent to Connes Embedding problem.