# Cross ratios on boundaries of symmetric spaces and Euclidean buildings

## TL;DR

This paper extends the concept of cross ratios from rank one symmetric spaces to higher rank symmetric spaces and Euclidean buildings, revealing their geometric properties and characterizations of isometries.

## Contribution

It introduces vector valued cross ratios on boundaries of higher rank symmetric spaces and Euclidean buildings, and establishes their fundamental properties and relation to isometries.

## Key findings

- Cross ratios are generalized to higher rank spaces and Euclidean buildings.
- Hyperbolic isometry periods relate to translation vectors via cross ratios.
- Cross ratio preserving maps correspond to isometries.

## 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.