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

Abstract

By [5] it is known that a geodesic γ 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 modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane H ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type Y lying 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, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.