On strong embeddability and finite decomposition complexity

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.     

模糊度量空间的强嵌入北大核心CSCD

本文定义了George和Veeramani意义下的模糊度量空间的强嵌入,证明了可强嵌入的模糊度量空间能够粗嵌入到Hilbert空间.另外还证明了强嵌入在模糊度量空间的粗范畴下是不变的,并给出了模糊度量空间强嵌入的一些等价刻画.     

Constructions preserving Hilbert space uniform embeddability of discrete groups

Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

     

Bi‐embeddability spectra and bases of spectra

We study degree spectra of structures with respect to the bi‐embeddability relation. The bi‐embeddability spectrum of a structure is the family of Turing degrees of its bi‐embeddable copies. To facilitate our study we introduce the notions of bi‐embeddable triviality and basis of a spectrum. Using bi‐embeddable triviality we show that several known families of degrees are bi‐embeddability spectra of structures. We then characterize the bi‐embeddability spectra of linear orderings and study bases of bi‐embeddability spectra of strongly locally finite graphs.     

Box spaces,group extensions and coarse embeddings into Hilbert space

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.     

Remarks on coarse embeddings of metric spaces into uniformly convex Banach spaces

We study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex Banach space. By using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex Banach space is determined by its finite subspaces.     

On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension.     

Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.     

On finitely generated soluble non-Hopfian groups

There is a continuum of 3-generator soluble non-Hopfian groups that generate pairwise distinct varieties of groups. Each countable (soluble) group is subnormally embeddable into a 3-generator (soluble) non-Hopfian group. As an illustration to a problem of Neumann, we find a continuum of nonmetanilpotent varieties that contain finitely generated non-Hopfian groups and contain uncountably many pairwise nonisomorphic finitely generated groups.     

Property A and Uniform Embeddability of Metric Spaces Under Decompositions of Finite Depth

Property A and uniform embeddability are notions of metric geometry which imply the coarse Baum-Connes conjecture and the Novikov conjecture. In this paper, the authors prove the permanence properties of property A and uniform embeddability of metric spaces under large scale decompositions of finite depth.     

Embeddable transversal designs

The embeddability of certain (group) divisible designs in symmetric 2-designs is investigated. These designs are symmetric resolvable transversal designs. It is proved that all such transversal designs with v = 2k are embeddable and some necessary and sufficient conditions for other cases are given.     

Simplicity and superrigidity of twin building lattices

Kac–Moody 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 non-positively 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 the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices. Dedicated to Jacques Tits with our admiration     

Indivisible ultrametric spaces

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces [C. Delhommé, C. Laflamme, M. Pouzet, N. Sauer, Divisibility of countable metric spaces, European J. Combin. 28 (2007) 1746-1769], we show that a countable ultrametric space is isometrically embeddable into an indivisible ultrametric space if and only if it does not contain a strictly increasing sequence of balls.     

图的上可嵌入性与独立数、非邻节点度和

本文研究了图的上可嵌入性与独立数、非邻节度点和之间的关系,得到了一些新的上可嵌入图类,推广了—个相关结果.从而,为进一步研究图的上可嵌入性提供了一定的理论基础.     

Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstnerič. The author was supported in part by the RCBS grant of Trinity College Dublin and by the Science Foundation Ireland.     

A structure of the Bang-Bang representation for 3×3 embeddable matrices

Summary Necessary and sufficient conditions are given for a 3×3 stochastic matrix to be embeddable by 6 elementary stochastic matrices (Poisson matrices). For a 3×3 embeddable matrix, a structure of the minimal Bang-Bang representation, i.e. the one that contains the smallest number of elementary matrices, is obtained. Based on the minimal Bang-Bang representation an algorithm for determining the embeddability of a 3×3 stochastic matrix is given.I would like to thank Søren Johansen for helpful comments and stimulating discussions on the subject of this paper     

Property A for groups acting on metric spaces

Gouliang Yu has introduced a property of discrete metric spaces and groups called property A which implies the coarse Baum–Connes Conjecture and hence the Novikov Higher Signature Conjecture. In this paper we extend a result of Jean-Louis Tu to conclude that a group acting by isometries on a metric space with finite asymptotic dimension whose d-stabilizers have property A, also has property A. As a result, we conclude a theorem of Tu, according to which, a fundamental group of a finite graph of groups whose vertices have property A also has property A.     

Remarks on the Operator Norm Localization Property

The author studies the metric spaces with operator norm localization property. It is proved that the operator norm localization property is coarsely invariant and is preserved under certain infinite union. In the case of finitely generated groups, the operator norm localization property is also preserved under the direct limits.     

逼近性质和广义快速衰减

将快速衰减性质推广到可以一致嵌入到Hilbert空间的离散空间, 证明了一些逼近性质和应用.     

Asymptotic properties of groups acting on complexes

We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric realization of the development has finite asymptotic dimension and the vertex groups are finitely generated and have finite asymptotic dimension. We also prove that property A is preserved by this construction provided the geometric realization of the development has finite asymptotic dimension and the vertex groups all have property A. These results naturally extend the corresponding results on preservation of these large-scale properties for fundamental groups of graphs of groups. We also use an example to show that the requirement that the development have finite asymptotic dimension cannot be relaxed.