If $G$ is a countable discrete group, I'm curious if it is possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$, the collection of (equivalence classes of) strongly continuous unitary representations of G on separable Hilbert space.

If this question is too crude, consider the following refinement:

If $G$ is a countable discrete group, is it possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$ viewed as the unitary dual of $G$, the topological space of (unitary equivalence classes of) unitary representations of $G$ on separable Hilbert space, equipped with Fell's topology?