The Cayley transform and uniformly bounded representations

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KAN̄ and Bruhat big cell NMAN̄. Then the space G/MAN̄ may be (almost) identified with N and with K/M, and these identifications induce the (generalised) Cayley transform C : N→K/M. We show that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplication by appropriate powers of the Jacobian, induces isomorphisms of Sobolev spaces Hα(N) and Hα(K/M). We use this to construct uniformly bounded and slowly growing representations of G.