  Let G/K be a non-compact, rank-one, Riemannian symmetric space and let G^C be
the universal complexification of G. We prove that a holomorphically separable,
G-equivariant Riemann domain over G^C / K^C is necessarily univalent, provided
that G is not a covering of SL(2, R). As a consequence of the above statement
one obtains a univalence result for holomorphically separable, G x K
-equivariant Riemann domains over G^C. Here G x K acts on G^C by left and right
translations. The proof of such results involves a detailed study of the
G-invariant complex geometry of the quotient G^C / K^C, including a complete
classification of all its Stein G-invariant subdomains.
