## An analytic family of uniformly bounded representations of free products of discrete groups.(English)Zbl 0742.30038

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 $$z\in(-{1 \over N-1},1)$$.

### MSC:

 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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