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1.
J. Higes 《Topology and its Applications》2010,157(17):2635-2645
In this paper we study two problems concerning Assouad-Nagata dimension:
- (1)
- Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).
- (2)
- Suppose G is a locally finite group with a proper left invariant metric dG. If dimAN(G,dG)>0, is dimAN(G,dG) infinite? (Brodskiy et al., preprint [6, Problem 5.3]).
2.
A. N. Dranishnikov 《Transactions of the American Mathematical Society》2003,355(1):155-167
We prove the following embedding theorems in the coarse geometry:
The Corollary is used in the proof of the following.
The Corollary is used in the proof of the following.
Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature.
We also prove that if a uniformly contractible manifold of bounded geometry is large scale uniformly embeddable into a Hilbert space, then is stably integrally hyperspherical.
3.
Asymptotic dimension of finitely presented groups 总被引:1,自引:0,他引:1
Thanos Gentimis 《Proceedings of the American Mathematical Society》2008,136(12):4103-4110
We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.
4.
Eva Tourís 《Journal of Mathematical Analysis and Applications》2011,380(2):865-881
In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface. 相似文献
5.
We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group. 相似文献
6.
A. Dranishnikov 《Geometriae Dedicata》2009,141(1):59-86
We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension
asdim
Z
X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology
is strictly less than its asymptotic dimension.
相似文献
7.
We prove that the dimension of any asymptotic cone over a metric space does not exceed the asymptotic Assouad-Nagata dimension of . This improves a result of Dranishnikov and Smith (2007), who showed for all separable subsets of special asymptotic cones , where is an exponential ultrafilter on natural numbers.
We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.
8.
We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of asymptotic dimension with asymptotic inductive dimension. 相似文献
9.
Robert L. Snider 《Proceedings of the American Mathematical Society》2006,134(12):3485-3486
Theorem. If is an uncountable field and is a periodic group with no elements of order the characteristic of and if all simple modules have finite central endomorphism dimension, then has an abelian subgroup of finite index.
10.
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
11.
T. Radul 《Topology and its Applications》2010,157(14):2292-2296
We prove that a transfinite extension of the asymptotic dimension asind is trivial. We introduce a transfinite extension of the asymptotic dimension asdim and give an example of a metric proper space which has transfinite infinite dimension. 相似文献
12.
Paul D. Mitchener Behnam Norouzizadeh Thomas Schick 《Mathematische Nachrichten》2020,293(8):1515-1533
In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts. 相似文献
13.
José Manuel RODRIGUEZ 《数学学报(英文版)》2014,30(2):197-212
To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs(in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r):an r-short graph G is hyperbolic if and only if S9r(G)is finite,where SR(G):=sup{L(C):C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC(x,y)≤dG(x,y)+R for every x,y∈C. 相似文献
14.
J. Smith 《Topology and its Applications》2006,153(12):2047-2054
We compute the asymptotic dimension of the rationals given with an invariant proper metric. We also show that a countable torsion Abelian group taken with an invariant proper metric has asymptotic dimension zero. 相似文献
15.
Udo Baumgartner Rögnvaldur G. Möller George A. Willis 《Israel Journal of Mathematics》2012,190(1):365-388
The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1. 相似文献
16.
Hyun Jeong Kim 《Geometriae Dedicata》2006,120(1):19-35
Warped cones were introduced by J. Roe in Geometry Topol. 9 (2005) 163–178 where he discussed Property A of these spaces. In this paper, we discuss the coarse equivalence of warped cones on the circle with the
-action by irrational rotations. First, we prove that two irrational numbers related by PSL(2,
) give coarsely equivalent warped cones. Second, we prove that there are at least countably many warped cones that are not coarsely equivalent to each other by using a ‘secondary growth function’. 相似文献
17.
18.
The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann
surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity;
however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general. 相似文献
19.
Suppose a group G acts on a Gromov-hyperbolic space X properlydiscontinuously. If the limit set L(G) of the action has atleast three points, then the second bounded cohomology groupof is infinite dimensional. For example, if M is a complete, pinched negatively curved Riemannianmanifold with finite volume, then is infinite dimensional. As an application, we show that ifG is a knot group with GZ, then is infinite dimensional. 1991 Mathematics Subject Classification:primary 20F32; secondary 53C20, 57M25. 相似文献
20.
Finite topological spaces, that is spaces with a finite number of points, have a wide range of applications in many areas such as computer graphics and image analysis. In this paper we study the covering dimension of a finite topological space. In particular, we give an algorithm for computing the covering dimension of a finite topological space using matrix algebra. 相似文献