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Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? K (G), grad(?)(? K (G)), grad(g)(exp ? K (G)), and L K (G). Let 𝔗 c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F n (𝔗 c ) be the relatively free group of rank n in 𝔗 c . We prove that ? K (F n (𝔗 c )) is relatively free in some variety of nilpotent Lie algebras, and ? K (F n (𝔗 c )) ? L K (F n (𝔗 c )) ? grad(?)(? K (F n (𝔗 c ))) ? grad(g)(exp ? K (F n (𝔗 c ))) as Lie algebras in a natural way. Furthermore, F n (𝔗 c ) is a Magnus nilpotent group. Let G 1 and G 2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G 1 and G 2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ??(H) ? L ?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? K and L K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ??(G) is not isomorphic to L ?(G) as Lie algebras. 相似文献
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S. Maillot 《Commentarii Mathematici Helvetici》2001,76(1):29-60
We give a characterization of virtual surface groups as groups quasi-isometric to complete simply-connected Riemannian surfaces.
Results on the equivalence up to quasi-isometry of various bounded geometry conditions for Riemannian surfaces are also obtained.
Received: January 18, 2000 相似文献
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Joanna Zubik 《Topology and its Applications》2010,157(18):2815-2818
Asymptotic hereditary asphericity (AHA) is a coarse property introduced by Januszkiewicz and ?wia?tkowski in the context of systolic complexes and groups. We show, that spaces of asymptotic dimension 1 are all AHA. 相似文献
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Xiangdong Xie 《Topology》2006,45(1):101-169
We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. Together with the results of Bourdon-Pajot and Kleiner, it implies the quasi-isometric rigidity for Fuchsian buildings: any quasi-isometry between two Fuchsian buildings that admit cocompact lattices must lie at a finite distance from an isomorphism. 相似文献
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O. Bogopolski 《代数通讯》2013,41(7):2494-2502
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Criteria for quasi-isometry between trees and general graphs as well as for quasi-isometries between metrically almost transitive graphs and trees are found. Thereby we use different concepts of thickness for graphs, ends and end spaces. A metrically almost transitive graph is quasi-isometric to a tree if and only if it has only thin metric ends (in the sense of Definition 3.6). If a graph is quasi-isometric to a tree then there is a one-to-one correspondence between the metric ends and those d-fibers which contain a quasi-geodesic. The graphs considered in this paper are not necessarily locally finite. 相似文献
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Jean-François Lafont 《Geometriae Dedicata》2007,124(1):143-152
In this survey paper, we outline the proofs of rigidity results (Mostow type, quasi-isometric, and Diagram rigidity) for simple,
thick, hyperbolic P-manifolds. These are stratified spaces, and are in some sense the simplest non-manifold locally CAT(−1)
spaces one can create, having codimension one singularities along embedded totally geodesic submanifolds. All the proofs depend
on the highly non-homogenous structure of the boundary at infinity of the (universal covers of the) spaces in question, and
in particular, on an understanding of the local topology of the boundary at infinity. We emphasize the similarities and differences
in the proofs of the various rigidity results. These results should be viewed as a first step towards understanding stratified
locally CAT(−1) spaces.
相似文献
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