Manifolds of nonpositive curvature and their buildings

Burns, Keith; Spatzier, Ralf

doi:10.1007/bf02698934

Cited by 121 publications

(80 citation statements)

References 12 publications

(7 reference statements)

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“…For example, the rank rigidity of nonpositively curved Riemannian manifolds in [3] [38] says that any irreducible nonpositively curved Riemannian manifold M of finite volume with rank at least 2 is a locally symmetric space. The proof of [38] consists of two steps:…”

Section: Mostow Strong Rigidity and Generalizationsmentioning

confidence: 99%

Curve Complexes Versus Tits Buildings: Structures and Applications

2011

Buildings, Finite Geometries and Groups

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Abstract. Tits buildings ∆ Q (G) of linear algebraic groups G defined over the field of rational numbers Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G(Q). Curve complexes C(Sg,n) of surfaces Sg,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the BruhatTits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

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Section: Mostow Strong Rigidity and Generalizationsmentioning

confidence: 99%

Curve Complexes Versus Tits Buildings: Structures and Applications

2011

Buildings, Finite Geometries and Groups

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“…Hence the conditions in Theorem 2.7.5 of rank being at least 2 and of being Moufang is weaker. This weakening to rank at least 2 is crucial for application to the rank rigidity of manifolds of nonpositive curvature in [BuS1] (see §2.9 below).…”

Section: Theorem 274 If ∆ Is An Irreducible Compact Metric Buildinmentioning

confidence: 99%

“…The solution in [BuS1] uses classification of topological Tits buildings in Theorem 2.7.5. We briefly recall the precise formulation of the problem and its proof in [BuS1].…”

Section: This Is Proved In [Mosmentioning

confidence: 99%

“…Spherical buildings with this new topology are called topological buildings by Burns and Spatzier in [BuS2] (actually, the topological buildings were defined slightly differently there) and were used by them to prove rank rigidity of complete manifolds of nonpositive curvature and finite volume in [BuS1]. They were also used by Thorbergsson [Tho2] to classify compact isoparametric submanifolds in R n of codimension at least three.…”

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confidence: 99%

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Buildings and their Applications in Geometry and Topology

Ji¹

2006

Asian Journal of Mathematics

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Abstract. Buildings were first introduced by J. Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic K-theories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and S-arithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications:1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces and also appear as limiting objects under degeneration or scaling of metrics. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups. 3. Buildings of higher rank are rigid and hence objects which contain or induce higher rank buildings tend to be rigid. 4. Additional structures on buildings, for example, topological buildings, are important in applications for infinite groups.

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“…We established a structure theory for such spaces in [12,13]. These developments culminated in the rank-rigidity theorem by W. Ballmann and independently K. Burns and myself [11,25,14]. The proof by Burns and myself is again inspired by Mostow's approach.…”

Section: Riemannian Geometrymentioning

confidence: 99%

Harmonic analysis in rigidity theory

Spatzier¹

Ergodic Theory and Its Connections With Harmonic Analysis

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Manifolds of nonpositive curvature and their buildings

Cited by 121 publications

References 12 publications

Curve Complexes Versus Tits Buildings: Structures and Applications

Curve Complexes Versus Tits Buildings: Structures and Applications

Buildings and their Applications in Geometry and Topology

Harmonic analysis in rigidity theory

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