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1.
Tetsuya Hosaka 《Topology and its Applications》2006,153(11):1860-1866
H.S.M. Coxeter showed that a group Γ is a finite reflection group of an Euclidean space if and only if Γ is a finite Coxeter group. In this paper, we define reflections of geodesic spaces in general, and we prove that Γ is a cocompact discrete reflection group of some geodesic space if and only if Γ is a Coxeter group. 相似文献
2.
Daniel S. Farley 《Topology》2003,42(5):1065-1082
3.
Frédéric Haglund 《Advances in Mathematics》2010,224(5):1890-1903
In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups. 相似文献
4.
Hanspeter Fischer 《Topology》2003,42(2):423-446
All abstract reflection groups act geometrically on non-positively curved geodesic spaces. Their natural space at infinity, consisting of (bifurcating) infinite geodesic rays emanating from a fixed base point, is called a boundary of the group.We will present a condition on right-angled Coxeter groups under which they have topologically homogeneous boundaries. The condition is that they have a nerve which is a connected closed orientable PL manifold.In the event that the group is generated by the reflections of one of Davis’ exotic open contractible n-manifolds (n?4), the group will have a boundary which is a homogeneous cohomology manifold. This group boundary can then be used to equivariantly Z-compactify the Davis manifold.If the compactified manifold is doubled along the group boundary, one obtains a sphere if n?5. The system of reflections extends naturally to this sphere and can be augmented by a reflection whose fixed point set is the group boundary. It will be shown that the fixed point set of each extended original reflection on the thus formed sphere is a tame codimension-one sphere. 相似文献
5.
We construct examples of Gromov hyperbolic
Coxeter groups of arbitrarily large dimension.
We also extend Vinbergs theorem to show that if a Gromov
hyperbolic Coxeter group is a virtual Poincaré duality group
of dimension n,
then n 61.Coxeter groups acting on their associated complexes have been extremely
useful source of examples and insight into nonpositively curved spaces
over last several years. Negatively curved (or Gromov hyperbolic)
Coxeter groups were much more elusive. In particular their existence in
high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on
the virtual cohomological dimension of any Gromov hyperbolic Coxeter group.
This question was also raised by Misha Gromov [G] (who thought that perhaps
any construction of high dimensional negatively curved spaces requires
nontrivial number theory in the guise of arithmetic groups in an essential
way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter
groups do exist, and we construct them by geometric or group theoretic but
not arithmetic means. 相似文献
6.
G. Conner 《Topology and its Applications》2007,154(3):771-773
We correct lemmata in the papers in the title. 相似文献
7.
We give an example of two JSJ decompositions of a group that are not
related by conjugation, conjugation of edge-inclusions, and slide
moves. This answers the question of Rips and Sela stated in [RS].On the other hand we observe that any two JSJ decompositions of a group
are related by an elementary deformation, and that strongly slide-free
JSJ decompositions are genuinely unique. These results hold
for the decompositions of Rips and Sela, Dunwoody and Sageev, and
Fujiwara and Papasoglu, and also for accessible decompositions. 相似文献
8.
We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations. 相似文献
9.
Christopher Mooney 《Topology and its Applications》2010,157(7):1168-1181
It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also satisfy the stronger condition of being cell-like equivalent. This article describes a construction which will produce CAT(0) groups with multiple boundaries. These groups have very complicated boundaries in high dimensions. It is our hope that their study may provide insight into Bestvina's question. 相似文献
10.
A finite simplicial graph Γ determines a right-angled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance
variety of GΓ, directly from the graph Γ.
Partially supported by NSF grant DMS-0311142. 相似文献
11.
We classify those closed 3-manifolds whose universal covering space naturally admits the structure of a Lie group 相似文献
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15.
J.W. Cannon 《Topology and its Applications》2006,153(14):2648-2672
We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective. 相似文献
16.
Xuhua He 《Advances in Mathematics》2007,215(2):469-503
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570-600]. 相似文献
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18.
Julia D. Fredericks 《Journal of Pure and Applied Algebra》2003,177(3):237-251
A relative presentation is a triple where A is a group, X is a set, and R is a set of words in the free product A∗F(X) where F(X) is the free group with basis X. Under certain hypotheses on the relative presentation , we show that (1) the group presented by is locally indicable; (2) the pre-aspherical model for is aspherical; (3) the Freiheitssatz holds for . The result has applications in the computation of cohomology of groups and the field of equations over groups. 相似文献
19.
Jean-François Lafont 《Journal of Pure and Applied Algebra》2007,209(3):771-780
In this paper, we show that a class of 2-dimensional locally CAT(-1) spaces is topologically rigid: isomorphism of the fundamental groups is equivalent to the spaces being homeomorphic. An immediate application of this result is a diagram rigidity theorem for certain amalgamations of free groups. The direct limits of two such amalgamations are isomorphic if and only if there is an isomorphism between the respective diagrams. 相似文献