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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 Δ are studied by means of their topological automorphism groups, the groups of all homeomorphic (combinatorial) automorphisms of Δ. 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 Δ 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 Δ(G). A topological Tits building Δ is called classical if Δ=Δ(G) for some Lie group G of the type above.

Main Theorem. Let Δ be an infinite, irreducible, locally connected, compact, metric topologically Moufang building of rank at least 2. Then Δ is classical. The authors apply this result to complete, simply connected manifolds M ˜ of bounded nonpositive sectional curvature that are Riemannian coverings of manifolds M of finite volume. The boundary sphere M ˜() determines a topological Tits building Δ(M ˜) whose rank equals the rank of M ˜, an integer defined by Jacobi vector fields that measures the flatness of 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 M ˜ is irreducible and has rank at least 2, then M ˜ is a symmetric space of noncompact type and rank k.

Reviewer: P.Eberlein

MSC:
53C35Symmetric spaces (differential geometry)
51H20Topological geometries on manifolds
22E46Semisimple Lie groups and their representations
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