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On topological Tits buildings and their classification. (English) Zbl 0643.53036
The authors introduce the notion of a topological Tits building, which can be regarded as a Tits building with a topology that makes the incidence relation closed. The topology here is usually given by a metric. The number of chambers in an apartment is always assumed to be finite. Topological Tits buildings $$\Delta$$ are studied by means of their topological automorphism groups, the groups of all homeomorphic (combinatorial) automorphisms of $$\Delta$$. The authors also consider a topological analogue of the Moufang property that ensures the existence of sufficiently many topological automorphisms. This property is automatically satisfied if $$\Delta$$ is an irreducible, compact, metric building of rank at least 3. If G is a connected semisimple Lie group without compact factors, then the set of all parabolic subgroups of G can be given a building structure and the topology of G makes this into a topological Tits building $$\Delta(G)$$. A topological Tits building $$\Delta$$ is called classical if $$\Delta =\Delta(G)$$ for some Lie group G of the type above.
Main Theorem. Let $$\Delta$$ be an infinite, irreducible, locally connected, compact, metric topologically Moufang building of rank at least 2. Then $$\Delta$$ is classical. The authors apply this result to complete, simply connected manifolds $$\tilde M$$ of bounded nonpositive sectional curvature that are Riemannian coverings of manifolds M of finite volume. The boundary sphere $$\tilde M(\infty)$$ determines a topological Tits building $$\Delta(\tilde M)$$ whose rank equals the rank of $$\tilde M,$$ an integer defined by Jacobi vector fields that measures the flatness of $$\tilde M.$$ One then obtains a result proved independently with another method by W. Ballmann. Theorem. Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. If the universal cover $$\tilde M$$ is irreducible and has rank at least 2, then $$\tilde M$$ is a symmetric space of noncompact type and rank k.
Reviewer: P.Eberlein

##### MSC:
 53C35 Differential geometry of symmetric spaces 51H20 Topological geometries on manifolds 22E46 Semisimple Lie groups and their representations
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##### References:
 [1] W. Ballmann, Nonpositively curved manifolds of higher rank,Ann. of Math.,122 (1985), 597–609. · Zbl 0585.53031 [2] W. Ballmann, M. Brin andR. Spatzier, Structure of manifolds of nonpositive curvature, II,Ann. of Math.,122 (1985), 205–235. · Zbl 0598.53046 [3] W. Ballman, M. Gromov andV. Schroeder, Manifolds of Nonpositive Curvature,Progress in Mathematics,61 (1985), Boston, Birkhäuser. · Zbl 0591.53001 [4] N. Bourbaki,Éléments de mathématique. Livre III:Topologie générale, 3e éd., Paris, Hermann (1960). [5] K. Burns andR. Spatzier, Manifolds of nonpositive curvature and their buildings,Publ. Math. I.H.E.S.,65 (1987), 35–59. · Zbl 0643.53037 [6] H. Furstenberg, Rigidity and cocycles for ergodic actions of semi-simple Lie groups,Séminaire Bourbaki, 32e année, 1979–1980, no 559,Springer Lecture Notes in Math.,842 (1981), 273–292. [7] S. Glasner, Proximal Flows,Springer Lecture Notes in Math.,517 (1976). · Zbl 0322.54017 [8] A. Gleason, Groups without small subgroups,Ann. of Math.,56 (1952), 193–212. · Zbl 0049.30105 [9] A. Kolmogoroff, Zur Begründung der projektiven Geometrie,Ann. of Math.,33 (1932), 175–176. · JFM 58.0600.01 [10] R. Moody andK. Teo, Tits’ systems with crystallographic Weyl groups,J. Algebra,21 (1972), 178–190. · Zbl 0232.20089 [11] S. Murakami, On the automorphisms of a real semisimple Lie algebra,Journal Math. Soc. Japan,4 (1952), 103–133. · Zbl 0047.03501 [12] D. Montgomery andL. Zippin,Topological Transformation Groups, New York, Interscience Publishers Inc. (1955). [13] H. Salzmann, Topological projective planes,Adv. in Math.,2 (1967), 1–60. · Zbl 0153.21601 [14] H. Salzmann, Homogene kompakte projektive Ebenen,Pacific J. Math.,60 (1975), 217–234. · Zbl 0323.50009 [15] J. Tits, Buildings of Spherical Type and Finite BN-pairs,Springer Lecture Notes in Math.,386 (1970). · Zbl 0295.20047 [16] J. Tits, Classification of buildings of spherical type and Moufang polygons: a survey,Coll. Intern. Teorie Combinatorie, Rome (1976), t. I, 229–246. · Zbl 0347.50005 [17] J. Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten,Invent. Math.,43 (1977), 283–295. · Zbl 0399.20037 [18] H. Yamabe, A generalization of a theorem of Gleason,Ann. of Math.,58 (1953), 351–365. · Zbl 0053.01602
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