On topological Tits buildings and their classification.

*(English)*Zbl 0643.53036The 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.

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 |

##### Keywords:

topological Tits building; Moufang property; semisimple Lie group; Moufang building; nonpositive sectional curvature; symmetric space##### References:

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