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We 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.

CROSS RATIOS ON BOUNDARIES OF SYMMETRIC SPACES AND EUCLIDEAN BUILDINGS