An analytic family of uniformly bounded representations of free products of discrete groups (Q811650)

We construct for each \(| z|<1\) a uniformly bounded representation \(\pi_ z\) of a free product group. The correspondence \(z\mapsto\pi_ z\) is proved to be analytic. The representations are irreducible if the free product factors are infinite groups. On free groups they have as coefficients block radial functions --- this gives thus a new series of representations. They can be made unitary iff \(z\in(-{1 \over N-1},1)\).