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

2009/03/03 by Bernhard Leeb, Leeb, Bernhard · 7 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.DG #math.MG #msc:51E24 #msc:51K10 #msc:53C24 #msc:53C35

paper · pdf · doi:10.48550/arxiv.0903.0584

My 1997 habilitation thesis as published in Bonner Mathematische Schriften vol 326 (2000)

arxiv created 2009/03/03 · arxiv updated 2009/12/01

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.

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