Title: Cross ratios on boundaries of symmetric spaces and Euclidean buildings,
Abstract: We generalize the natural cross ratio on the ideal boundary of a rank one
symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric
spaces and (non-locally 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.