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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 Symmetric spaces (differential geometry) 51H20 Topological geometries on manifolds 22E46 Semisimple Lie groups and their representations
Full Text:
##### References:
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