By  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 split 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 split Kac–Moody group G, as defined in , satisfies a universal property similar to the universal property that the group G satisfies itself.
Communicated by: R. Löwen
The authors express their gratitude to the Deutsche Forschungsgemeinschaft for funding the research leading to this article via the project KO4323/13-1. They also thank an anonymous referee and Max Horn for several helpful comments on a preliminary version of this article.
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