Una, due o nessuna? Unitary representations of free groups and surface groups

Some years ago M.G.Kuhn and T.Steger introduced a class of unitary representations of free groups which they called multiplicative. This class is large enough to include many examples of unitary representations constructed henceforth. Moreover, multiplicative representations have the very interesting property of being tempered, i.e. weakly contained in the regular representation. After recalling the basic notions needed to enter into the field, we outline some algebraic and analytic properties of these representations and we advance some ideas for extending such a study to surface groups and hyperbolic groups.