Abstract
AbstractWe generalize the natural cross ratio on the ideal boundary of a rank one symmetric space, or even CAT(−1) space, to higher rank symmetric spaces and (nonlocally compact) Euclidean buildings. We obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties of those cross ratios; for example that (under some restrictions) periods of hyperbolic isometries give back the translation vector. In addition, we show that cross ratio preserving maps on the chamber set are induced by isometries and vice versa, — motivating that the cross ratios bring the geometry of the symmetric space/Euclidean building to the boundary.
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Algebra and Number Theory
Reference33 articles.
1. P. Abramenko, K. Brown, Buildings. Theory and Applications, Graduate Texts in Mathematics, Vol. 248, Springer, Berlin, 2008.
2. P. Abramenko, H. Van Maldeghem, On opposition in spherical buildings and twin buildings, Ann. Comb. 4 (2000), no. 2, 125–137.
3. J. Beyrer, E. Fioravanti, M. Incerti-Medici, CAT(0) cube complexes are determined by their boundary cross ratio, arXiv:1805.08478v3 (2018) (to appear in Groups, Geometry and Dynamics).
4. J. Beyrer, V. Schroeder, Trees and ultrametric Möbius structures, p-Adic Numbers Ultrametric Anal. Appl. 9 (2017), no. 4, 247–256.
5. J. Bochi, R. Potrie, A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. 21 (2019), no. 11, 3343–3414.
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