Left invariant contact structures on Lie groups

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.     

矩阵逆半群

讨论矩阵逆半群的一些基本性质, 证明矩阵逆半群的幂等元集是有限布尔格的子半格, 从而证明等秩矩阵逆半群是群, 然后完全确定二级矩阵逆半群的结构：一个二级矩阵逆半群或者同构于二级线性群,或者同构于二级线性群添加一个零元素,或者是交换线性群的有限半格, 或者满足其他一些性质; 对于由某些二级矩阵构成的集合, 我们给出了它们成为矩阵逆半群的充分必要条件.     

ON RIESZ THEOREM

We prove a Riesz type criterion for a class of metric monoids: Local compactness implies finiteness of the Hausdorff dimension (and also of the topological dimension). We construct topological groups showing the necessity of some conditions. We finally prove that for some metric topological spaces finiteness of the algebraic dimension is equivalent to the finiteness of the Hausdorff dimension.     

The Jordan property of Cremona groups and essential dimension

We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

Decisive dimension and other related torsion theoretic dimensions

The paper is concerned with the study of the decisive dimension defined on the category of left modules over a ring R. We compare the decisive dimension with the Gabriel dimension and other dimensions recently introduced. We give module theoretic as well as lattice theoretic characterizations of rings with decisive dimension. As an application we obtain characterizations of some classes of rings.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

On finite simple groups acting on homology spheres

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each dimension n there are only finitely many finite simple groups which admit a faithful, smooth action on some homology sphere of dimension n, and in particular on the n-sphere. We discuss also the finite simple groups which admit an action on a homology sphere of dimension 3, 4 or 5.     

Homological dimensions of the extension algebras of monomial algebras

The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras — dual and trivially twisted extensions — with a unified combinatorial approach using the two combinatorial algorithms — Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.     

Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.     

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.

     

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

     

Horizontal submanifolds of groups of Heisenberg type

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution (“lagrangians”) and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence. This work was supported by CONICET, Antorchas, FONCyT and Secyt (UNC).     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

Free entropy dimension in amalgamated free products

We calculate the microstates free entropy dimension of naturalgenerators in an amalgamated free product of certain von Neumannalgebras, with amalgamation over a hyperfinite subalgebra. Inparticular, some ‘exotic’ Popa algebra generatorsof free group factors are shown to have the expected free entropydimension. We also show that microstates and non-microstatesfree entropy dimension agree for generating sets of many groups.In the appendix, the first L²-Betti number for certain amalgamatedfree products of groups is calculated.     

An algebraic version of Demailly's asymptotic Morse inequalities

We give an elementary algebraic proof of some asymptotic estimates (called by Demailly asymptotic Morse inequalities) for the dimensions of cohomology groups of the difference of two ample line bundles on a smooth complex projective variety of any dimension.

     

Dimension polynomials in the generalized difference case

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.     

Gelfand–Kirillov Dimension for Automorphism Groups of Relatively Free Algebras

Using ideas of our recent work on automorphisms of residually nilpotent relatively free groups, we introduce a new growth function for subgroups of the automorphism groups of relatively free algebras Fⁿ(V) over a field of characteristic zero and the related notion of Gelfand-Kirillov dimension, and study their behavior. We prove that, under some natural restrictions, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is equal to the Gelfand-Kirillov dimension of the algebra Fⁿ(V). We show that, in some cases, the Gelfand-Kirillov dimension of the group of tame automorphisms of Fⁿ(V) is smaller than the Gelfand-Kirillov dimension of the whole automorphism group, and calculate the Gelfand-Kirillov dimension of the automorphism group of Fⁿ(V) for some important varieties V.Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.2000 Mathematics Subject Classification: primary 16R10, 16P90; secondary 16W20, 17B01, 17B30, 17B40     

Flat manifolds isospectral on p-forms

We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that β_p(M)<β_p(M’), for 04 and ? ₂ ² , respectively.