Euclidean Buildings Research Articles

Groups acting properly on “bolic” spaces and the Novikov conjecture

We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a “bolic”, weakly geodesic metric space of bounded geometry.

Polyhedra with specified links

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$.

Inégalité de Haagerup, groupoı̈des et immeubles euclidiens

This Note presents a method of proof of the Haagerup's inequality for discrete groupoı̈ds using deformations of triangles. We apply this method to groups acting freely and by isometries on vertices of some euclidean buildings. To cite this article: M. Talbi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 233–236.

Metric characterizations of spherical and Euclidean buildings

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.

Exercices métriques immobiliers

Euclidean buildings are examples of hyperbolic spaces: their distance d verify the CAT(0) inequality. This has interesting consequences for the fixed-point-set of an isometry. Moreover such a building I is a discrete union of facets; as a consequence one proves here that if σ is an automorphism of I, with non empty fixed-point-set Iσ, there exists a constant k such that d(x, σx)/d(x, Iσ) > k > 0, for any x in I β Iσ. Some other inequalities of the same kind are also proved.

Rank rigidity of Euclidean polyhedra

A Euclidean polyhedron (a simplicial complex whose simplices are Euclidean) of nonpositive curvature (in the sense of Alexandrov) has rank ≥ 2 if every finite geodesic segment is a side of a flat rectangle. We prove that if a three-dimensional, geodesically complete, simply connected Euclidean polyhedron X of rank ≥ 2 and of nonpositive curvature admits a cocompact and properly discontinuous group of isometries, then X is either a Riemannian product or a thick Euclidean building of type [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] or [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /].

Combing Euclidean buildings

For an arbitrary Euclidean building we define a certain combing, which satisfies the `fellow traveller property' and admits a recursive definition. Using this combing we prove that any group acting freely, cocompactly and by order preserving automorphisms on a Euclidean building of one of the types A_n,B_n,C_n admits a biautomatic structure.

Diameter rigidity of spherical polyhedra

1. Introduction. The diameter rigidity question considered in this paper is motivated by the rank rigidity problem for spaces of nonpositive curvature. The rank of a complete, simply connected space Y of nonpositive curvature is greater than or equal to 2 if every geodesic segment in Y is contained in an isometrically embedded, convex Euclidean plane. The rank rigidity problem asks for a classification of such spaces, at least when the isometry group of Y is large. Recall that a topological space is called a polyhedron if it admits a triangulation. A polyhedron with a length metric is called Euclidean (respectively, spherical) if it admits a triangulation into Euclidean (respectively, spherical) simplices. Here a Euclidean (respectively, spherical) k-simplex is a k-simplex A such that A with the induced length metric is isometric to the intersection of k + 1 closed half-spaces in R k (respectively, closed hemispheres in S k ) in general position. We are interested in the rank rigidity of simply connected Euclidean polyhedra of nonpositive curvature and of rank greater than or equal to 2. We expect them to be Euclidean buildings or products. The rank rigidity for dim Y = 2 is not difficult and is contained in [BB, Section 6]. For a Euclidean polyhedron Y and p ∈ Y , we denote by SpY the link of Y at p, that is, the set of directions at p. Clearly, SpY is a spherical polyhedron, and, if Y has nonpositive curvature, then the injectivity radius of SpY is π . Furthermore, if the rank of Y is greater than or equal to 2, then SpY is geodesically complete and has diameter π . Hence, for dim Y = 3 the link X = SpY is a geodesically complete, compact, 2-dimensional spherical polyhedron of diameter and injectivity radius π . The aim of this paper is to classify such spaces X. Here are examples of geodesically complete compact spherical polyhedra of diameter and injectivity radius π . 1.1. Spherical building. If X is a spherical building, then X carries a natural metric, for which the apartments are unit spheres. For this metric, the diameter and injectivity radius of X are π .I fY is a Euclidean building of dimension n ≥ 2 with the natural metric, then every geodesic in Y is contained in an isometrically embedded, convex Euclidean n-space. The link of a vertex in Y is a spherical building of dimension n−1, which has injectivity radius and diameter π .A nn-dimensional building X

Equivariant Novikov conjecture for groups acting on Euclidean buildings

We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant F:tilsum-Skandalis methocl. We also obtain an equivariant version of the Connes-Gromov-Moscovic; th.eorem for almost flat C*-algebra bundles.

Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings

Abstract We study quasi-isometries between products of symmetric spaces and Euclidean buildings. The main results are that quasi-isometries preserve the product structure, and that in the irreducible higher rank case, quasi-isometries are at finite distance from homotheties.

Nondistorsion des horosphères dans des immeubles euclidiens et dans des espaces symétriques

We give a necessary and sufficient condition for a horosphere to be undistorted in an Euclidean building and in a symmetric space. We are using the geometry of Euclidean buildings and the asymptotic cone. As a consequence, a geometric proof of the Lubotzky-Mozes-Raghunathan theorem for \( \Bbb {Q} \)-rank one lattices is given.

Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings

We study quasi-isometries between products of symmetric spaces and Euclidean buildings. The main results are that quasi-isometries preserve the product structure, and that in the irreducible higher rank case, quasi-isometries are at finite distance from homotheties.

On the loop inequality for euclidean buildings

On the loop inequality for euclidean buildings

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense.

Buildings and their Applications in Geometry and Topology

Buildings were first introduced by J. Tits in 1950s to give systematic geometric inter- pretations 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 va- rieties, 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 isoparamet- ric 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.