Let G be a simple Lie group of real rank one, with Iwasawa decomposition KA \bar N and Bruhat
big cell NMA\bar N: Then the space G/MA \bar N may be (almost) identi\ufb01ed with N and with K /M,
and these identi\ufb01cations induce the (generalised) Cayley transform C : N \to K /M. We show
that C is a conformal map of Carnot\u2013Caratheodory manifolds, and that composition with the
Cayley transform, combined with multiplication by appropriate powers of the Jacobian,
induces isomorphisms of Sobolev spaces on N
and on K/M. We use this to construct
uniformly bounded and slowly growing representations of G