A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry

TL;DR

This paper characterizes irreducible symmetric spaces and Euclidean buildings of higher rank through their asymptotic geometry, establishing conditions for symmetry and boundary equivalences.

Contribution

It provides a geometric criterion to distinguish symmetric spaces from Euclidean buildings based on geodesic branching and boundary properties.

Findings

Symmetric spaces are characterized by non-branching geodesics.

Boundary equivalences are induced by homotheties.

Extends rigidity theorems to singular nonpositively curved spaces.

Abstract

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean building by a cocompact group of isometries.