An extension theorem for embedded Riemannian symmetric spaces of non-compact type and an application to their universal property

Abstract

It is known that a geodesic Y in an abstract reflection space X in the sense of Loos, without any assumption of differential structure, canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we prove an analog of this result stating that, if X contains an embedded hyperbolic plane H, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL(2,R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type embedded in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G satisfies a universal property similar to the universal property that the group G satisfies itself.