Results 1  10
of
79
H.: Poincar'e inequalities and quasiconformal structure on the boundary of some hyperbolic buildings
 Proc. Amer. Math. Soc
"... Abstract. In this paper we shall show that the boundary ∂Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincaré type inequalities. Then by using HeinonenKoskela’s work, we shall prove Loewner capacity estimates for some families of curves of ∂Ip,q and the fact that every quas ..."
Abstract

Cited by 33 (1 self)
 Add to MetaCart
Abstract. In this paper we shall show that the boundary ∂Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincaré type inequalities. Then by using HeinonenKoskela’s work, we shall prove Loewner capacity estimates for some families of curves of ∂Ip,q and the fact that every quasiconformal homeomorphism f: ∂Ip,q − → ∂Ip,q is quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirtythree YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1–12) is NO. 1.
LSgalleries, the path model and MVcycles
 Duke Math. J
"... Let G be a complex semisimple algebraic group. We give an interpretation of the path model of a representation [17] in terms of the geometry of the affine Grassmannian for G. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. The ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
Let G be a complex semisimple algebraic group. We give an interpretation of the path model of a representation [17] in terms of the geometry of the affine Grassmannian for G. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. The connection with geometry is obtained as follows: consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The points of this variety can be viewed as galleries of a fixed type in the affine Tits building associated to G. The retraction with center − ∞ of the Tits building onto the affine Coxeter complex induces, in this way, a stratification of the G(C[[t]])–orbit (identified with an open subset of ˆ Σ(λ)), indexed by certain folded galleries in the Coxeter complex. Each strata can be viewed as an open subset of a Bia̷lynicki– Birula cell of ˆ Σ(λ). The connection with representation theory is given by the fact that the closures of the strata associated to LSgalleries are the MV–cycles [23].
C ∗ algebras arising from group actions on the boundary of a triangle building
 613– 637. MR1376771 (98b:46088), Zbl 0869.46035
, 1996
"... A subgroup of an amenable group is amenable. The C*algebra version of this fact is false. This was first proved by M.D. Choi [9] who proved that the nonnuclear C*algebra C*(Z2 * Z3) is a subalgebra of the nuclear Cuntz algebra €2. A. Connes provided another example, based on a crossed product co ..."
Abstract

Cited by 20 (2 self)
 Add to MetaCart
A subgroup of an amenable group is amenable. The C*algebra version of this fact is false. This was first proved by M.D. Choi [9] who proved that the nonnuclear C*algebra C*(Z2 * Z3) is a subalgebra of the nuclear Cuntz algebra €2. A. Connes provided another example, based on a crossed product construction.
Asymptotic cones of finitely presented groups
"... Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that Rrank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 2ω asymptotic cones up to homeomorphi ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that Rrank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 2ω asymptotic cones up to homeomorphism. 1.
Diagram Geometry
, 1994
"... of type fi; jg should belong to. For instance, a simple stroke ffl ffl a double stroke ffl ffl and the `null stroke' ffl ffl are normally used to denote, respectively, the class of projective planes, the class of generalized quadrangles (see [14]) and the class of generalized digons (generalized di ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
of type fi; jg should belong to. For instance, a simple stroke ffl ffl a double stroke ffl ffl and the `null stroke' ffl ffl are normally used to denote, respectively, the class of projective planes, the class of generalized quadrangles (see [14]) and the class of generalized digons (generalized digons are the trivial geometries of rank 2, where any two elements of different type are incident). The following two diagrams are drawn according to these conventions 1) ffl ffl ffl 2) ffl ffl ffl They represent geometries of rank 3, where the residues of type f1; 2g are projective planes and the residues of type f1; 3g are generalized digons. However, the residues of type f2; 3g are projective planes in 1) and generalized
On Exchange Properties for Coxeter Matroids and Oriented Matroids
"... We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as "properties" in the more general setting because they are not all equivalen ..."
Abstract

Cited by 14 (12 self)
 Add to MetaCart
We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as "properties" in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger than the others. The weaker ones constitute the definition of Coxeter matroids, and we also prove their equivalence to the matroid polytope property of Gelfand and Serganova. 2 The terminology in the present paper follows [BG, BR] (though we prefer to use the name `Coxeter matroids' rather than `WP matroids,' as used in these papers); see also the forthcoming book [BGW1]. The cited publications also contain all the necessary background material. For more detail, refer to books [We],[Wh], [O] and [R] for the systematic exposition of matroid theory and theory of Coxeter complexes. The authors wish to thank A. Kelmans for several helpful suggestions. 1 Exchange properties for matroids Matroids. The following is wellknown (see for example [O]): Theorem 1.1 Let B be a nonempty collection of subsets of E. Then the following are equivalent: (1) For every A, B # B and a # A \ B there exists b # B \ A such that A \ {a} # {b} # B (the Exchange Property). (2) For every A, B # B and a # A \ B there exists b # B \ A such that B \ {b} # {a} # B (the Dual Exchange Property). (3) For every A, B # B and a # A \ B, there exists b # B \ A such that A\{a}#{b} # B and B \{b}#{a} # B (the Symmetric Exchange Property).
Polygons in buildings and their refined side lengths, preprint, November 2005, to appear in GAFA. [KLM3] [Ka] [KN] [Ki] [Kl] [KlL] [Kly1] [Kly2] , The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra
 MR 2369545, Zbl 1140.22009. F.I. Karpelevič
"... As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber ∆euc. In addition to the metric length it contains information on the direction of the segment. We study in this paper restrictio ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber ∆euc. In addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the ∆eucvalued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set Pn(X) of possible ∆eucvalued side lengths of oriented ngons, n ≥ 3, depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of ∆eucvalued weights of semistable weighted configurations on the Tits boundary ∂T itsX. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in
Explicit constructions of Ramanujan complexes
 European J. of Combinatorics
"... Abstract. In this paper we present for every d ≥ 2 and every local field F of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the BruhatTits building Bd(F) associated with PGLd(F). 1. ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Abstract. In this paper we present for every d ≥ 2 and every local field F of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the BruhatTits building Bd(F) associated with PGLd(F). 1.
Rigidity of skewangled Coxeter groups
 ADV. GEOM. 2 (2002), 391–415
, 2002
"... A Coxeter system is called skewangled if its Coxeter matrix contains no entry equal to 2. In this paper we prove rigidity results for skewangled Coxeter groups. As a consequence of our results we obtain that skewangled Coxeter groups are rigid up to diagram twisting. ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
A Coxeter system is called skewangled if its Coxeter matrix contains no entry equal to 2. In this paper we prove rigidity results for skewangled Coxeter groups. As a consequence of our results we obtain that skewangled Coxeter groups are rigid up to diagram twisting.
Simplicity and superrigidity of twin building lattices
, 2006
"... KacMoody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of nonpositively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
KacMoody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of nonpositively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not BruhatTits buildings). In fact, many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine KacMoody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed KacMoody group is always proper. We also show that KacMoody lattices fulfill conditions implying strong superrigidity properties for isometric actions on nonpositively curved metric spaces. Most results apply to the general class of twin building lattices.