A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry (Q2726208)

The author studies rigidity phenomena for Hadamard manifolds, which are simply connected complete geodesic metric spaces of nonpositive curvature in the sense of Alexandrov. These spaces include many singular spaces such as Euclidean buildings in addition to the smooth manifolds on which attention was focused first. The author also has a number of related works that generalize important results previously known only for smooth Hadamard manifolds. References to these works are listed below. NEWLINENEWLINENEWLINEThe main result of this article is the following: Let \(X\) be a locally compact Hadamard space with extendible geodesic segments whose Tits boundary is a connected, thick, irreducible spherical building. Then \(X\) is a Riemannian symmetric space or a Euclidean building. NEWLINENEWLINENEWLINEAs a corollary one obtains the following: Let \(X\) and \(Y\) be Hadamard spaces of rank at least 2 that are either Riemannian symmetric spaces or Euclidean buildings. Let \(f\) be a homeomorphism between the Tits boundaries of \(X\) and \(Y\) that preserves the Tits metric. Then \(f\) is induced by a homothety between \(X\) and \(Y\). NEWLINENEWLINENEWLINEAnother application of the main result gives an extension of the Mostow Rigidity Theorem: Let \(X\) be a locally compact Hadamard space with extendible geodesic segments, and let \(Y\) be a Riemannian symmetric space of noncompact type or a thick Euclidean building. Suppose further that all irreducible factors of \(Y\) have rank at least 2. Let \(F\) be a finitely generated group that acts cocompactly and properly discontinuously by isometries on \(X\) and \(Y\). Then after rescaling the metric by constants on the irreducible factors of \(Y\) there exists a \(\Gamma\)-equivariant isometry between \(X\) and \(Y\). NEWLINENEWLINENEWLINEThe methods and results of this article are closely related to the following works of the author: 1) \textit{B. Kleiner} and \textit{B. Leeb}, ``Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings'', Publ. Math., Inst. Hautes Étud. Sci. 86, 115-197 (1997; Zbl 0910.53035), 2) \textit{M. Kapovich}, \textit{B. Kleiner} and \textit{B. Leeb}, ``Quasi-isometries and the de Rham decomposition'', Topology 37, 1193-1211 (1998; Zbl 0954.53027), 3) \textit{B. Kleiner} and \textit{B. Leeb}, ``Groups quasi-isometric to symmetric spaces'', Commun. Anal. Geom. 9, 239-260 (2001; Zbl 1035.53073).