Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

Let G/K be a noncompact, 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, 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.