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

Julius Grüning and Ralf Köhl
From the journal Advances in Geometry


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 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 [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

MSC 2010: 22.70; 20G44

  1. 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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Received: 2018-05-16
Revised: 2018-08-15
Published Online: 2019-07-04
Published in Print: 2020-10-27

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