Affiliation:
1. Mathematisches Institut , Arndtstraße 2, 35392 , Gießen , Germany
Abstract
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 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.
Reference12 articles.
1. P. Abramenko, B. Mühlherr, Présentations de certaines BN-paires jumelées comme sommes amalgamées. C. R. Acad. Sci. Paris Sér. IMath. 325 (1997), 701–706. MR1483702 Zbl 0934.20024
2. C. S. Ballantine, Products of positive definite matrices. I. Pacific J. Math. 23 (1967), 427–433. MR0219555 Zbl 0211.35302
3. P.-E. Caprace, Primitive symmetric spaces. Bull. Belg. Math. Soc. Simon Stevin12 (2005), 321–328. MR2173695 Zbl 1110.20002
4. P.-E. Caprace, On 2-spherical Kac-Moody groups and their central extensions. Forum Math. 19 (2007), 763–781. MR2350773 Zbl 1140.20028
5. W. Freyn, T. Hartnick, M. Horn, R. Köhl, Kac–Moody symmetric spaces. To appear in Münster J. Math., arXiv:1702.08426 [math.GR]