Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures. I

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 39, Functional Analysis, 2006.     

Outer automorphism groups of certain 1-relator groups

Grossman first showed that outer automorphism groups of 1-relator groups given by orientable surface groups are residually finite, whence mapping class groups of orientable surfaces are residually finite. Allenby, Kim and Tang showed that outer automorphism groups of cyclically pinched 1-relator groups are residually finite, whence mapping class groups of orientable and non-orientable surfaces are residually finite. In this paper we show that outer automorphism groups of certain conjugacy separable 1-relator groups are residually finite.     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures,II

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism group and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 18, Functional Analysis, 2006.     

Abelian groups that are small with respect to different classes of groups

In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct products of groups small with respect to the class of slender groups are derived.     

Geometric equivalence of groups

Based on the notion of geometric equivalence of groups, new classes of groups, namely, geometric varieties of groups, are defined. Some properties of such classes, including their relation to quasi-varieties and prevarieties of groups, are studied. Examples of torsion free nilpotent groups that are geometrically nonequivalent to their minimal completions, as well as an example of centrally metabelian groups that are geometrically nonequivalent but generate equal quasi-varieties, are given.     

Finite dimensional groups of local diffeomorphisms

We are interested in classifying groups of local biholomorphisms (or even formal diffeomorphisms) that can be endowed with a canonical structure of algebraic groups and their subgroups. Such groups are called finitedimensional. We obtain that cyclic groups, virtually polycyclic groups, finitely generated virtually nilpotent groups and connected Lie groups of local biholomorphisms are finite-dimensional. We provide several methods to identify finite-dimensional groups and build examples.As a consequence we generalize results of Arnold, Seigal–Yakovenko and Binyamini on uniform estimates of local intersection multiplicities to bigger classes of groups, including for example virtually polycyclic groups and in particular finitely generated virtually nilpotent groups.     

Fuzzy幂群与Fuzzy商群的相互关系

In this paper,we extend the concept of fuzzy quotient groups.The structures of fuzzy power groups and fuzzy quotient groups are discussed.The relationship between fuzzy power groups and fuzzy quotient groups are considered.     

Tortrat groups—Groups with a nice behavior for sequences of shifted convolution powers

For a locally compact groupG a condition in terms of probability measures and conjugation is introduced, which implies that limits of shifted convolution powers are always translates of idempotent measures. Such groups are called Tortrat groups. The connection between Tortrat groups and shifted convolution powers is established by the method of tail idempotents. Some construction principles for Tortrat groups are given and applied to show that compact groups, abelian groups, and more generally SIN-groups, as well as MAP-groups and almost connected nilpotent groups are of this type. The class of Tortrat groups is compared with another class investigated by A. Tortrat.     

Quadratic algebra of square groups

Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.     

Diagram groups and directed 2-complexes: Homotopy and homology

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so-called rigidity of the R. Thompson's group F and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F_∞) each of which contains all countable diagram groups. We show how to compute minimal presentations and homology groups of a large class of diagram groups. We show that the Poincaré series of these groups are rational functions. We prove that all integer homology groups of all diagram groups are free Abelian.     

Gruppenklassen,deren kritische Gruppen minimal nicht nilpotent sind

This article is concerned with classes of groups, whose critical groups are minimal non-nilpotent, i.e. all proper subgroups are nilpotent, but the group itself is not. Since the structure of these groups is well known, the structure of the critical groups is determined as well. Thus, in a first part two theorems, which show that the critical groups for certain classes are mini=mal non-nilpotent, are proved. In the second part applications are given, concerning for example groups all of whose second maximal subgroups belong to a given class or describing the structure of groups if you have informations about certain subgroups. Also a number-theoretical characterization of all those ordinary numbers, to which belong only groups with a given ab=stract group-theoretical property, is given.     

On the minimum degree,edge-connectivity and connectivity of power graphs of finite groups

In this paper, the minimum degree of power graphs of certain cyclic groups, abelian p-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently, the minimum disconnecting sets of power graphs of the aforementioned groups are determined. In order to investigate the equality of connectivity and minimum degree of power graphs, certain necessary conditions for finite groups and a necessary and su?cient condition for finite cyclic groups are obtained. Moreover, the equality is discussed for the power graphs of abelian p-groups, dihedral groups and dicyclic groups.     

Nuclear and GP-Nuclear Groups

We define GP-nuclear groups as topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is shown that in the metrizable case only the algebraic coincidence of the mentioned groups is needed for GP-nuclearity. Some permanence properties of the class of GP-nuclear groups are obtained. Our final result asserts that nuclear groups in the sense of Banaszczyk are GP-nuclear. The validity of the converse assertion remains open.     

Groups of diffeomorphisms and geometric wraps of manifolds over ultra-normed fields

The article is devoted to the study of groups of diffeomorphisms and wraps of manifolds over ultra-metric fields of zero and positive characteristic. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies, pairwise incomparable ones are also found. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. Theorems on projective limit decompositions of these groups and their compactifications for compact manifolds are proved. Moreover, the existence of one-parameter local subgroups of diffeomorphism groups is proved.     

Splitting Obstruction Groups in Codimension 2

The splitting obstruction groups depend functorially on the square of fundamental groups. In the paper the problem of splitting along a submanifold of codimension two under some restrictions on the square of fundamental groups is considered. New exact sequences and commutative diagrams containing Wall groups, splitting obstruction groups, and surgery obstruction groups for manifold pairs are obtained. Examples of computation of splitting obstruction groups and natural maps are considered.     

Macpherson's chern classes of singular algebraic varieties

Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F₂)??? in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F₂)??? the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ?-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F₂)??? while the quasi-locally free groups are the ?-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators     

Elementarily free groups are subgroup separable

Elementarily free groups are the finitely generated groups withthe same elementary theory as free groups. We prove that elementarilyfree groups are subgroup separable, answering a question ofZlil Sela.     

初等收敛群和拟初等收敛群

作为－／R^n上的离散收敛群中的初等群的推广,本文定义了－／R^n上一般收敛群中的初等群和拟初等群,并得到了初等收敛群、拟初等收敛群和非拟初等敛群各自的一些性质和特征。     

Uniform structures and square roots in topological groups

Topological groups which are free from small subgroups and topological groups with locally uniformly continuous group multiplication are considered. Results concerning square roots, one-parameter subgroups and extensions of local groups are obtained as well as some generalisations of theorems for locally compact groups.