On the cohomology of finite soluble groups

Cohomological vanishing theorems in dimensions 1 and 2 are established for finite soluble groups. Applications are given to splitting and conjugacy theorems, and also to the structure of automorphism groups of finite soluble groups with nilpotent length at most 3.     

Separability of torsion and pretorsion in the category of right Λ -modules

In this paper, three theorems are proven on the separability of torsion and pretorsion in the category of right -modules. It follows from Theorem 2 that each non-trivial torsion and pretorsion in Abelian groups is inseparable. The theorems are accompanied by examples. The bibliography contains three titles.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 681–688, December, 1967.The author wishes to thank L. A. Skornyakov for directing this work.     

Characterization of groups with generalized chernikov periodic part

A characterization of groups with generalized Chernikov periodic part is obtained first in the class of groups without elements of order 2 and then without this restriction. Two author’s theorems for periodic groups are generalized to mixed groups. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 270–275, February, 2000.     

Abel群的一些分解定理的推广(I)

这项研究的目的是要把Abel群（有限或无限）的诸多分解定理尽可能地推广到主理想整环的模上,得到这类模上的分解定理,随后再把所得定理应用到向量空间（有限维或无限维）及其线性变换,得到向量空间的分解定理.本文是系列文章的第一篇,主要目的是建立起支撑整个研究的最基本概念,例如纯子模、有界模、局部循环模、具有minimax条件的模等.本文主要内容有： （1）确定了主理想整环上可除模、有界模、局部循环模的结构; （2）给出了主理想整环上拟循环模的生成性质,这类模在以后的研究里起着非常重要的作用; （3）描述了主理想整环上满足极小条件,minimax条件的模的结构; （4）给出了两个不同构的Z[i]-模,它们作为Abel群是同构的.     

Cohomological Properties of the Canonical Globalizations of Harish–Chandra Modules

In this note we draw consequences of theorems of Kashiwara–Schmid, Casselman, and Schneider–Stuhler. Canonical globalizations of Harish–Chandra modules can be considered as coefficient modules for cohomology groups with respect to cocompact discrete subgroups or nilpotent Lie algebras. We obtain finiteness and comparison theorems for these cohomology groups.     

Measurability of automorphisms of topological groups

Relations between the measurability and continuity of algebraic automorphisms of topological groups depending on the types of groups are examined. Various cases are considered and theorems on the continuity of measurable automorphisms are proved; for instance, such theorems are proved for separable locally compact groups and automorphisms measurable with respect to nonnegative Haar measures. On the other hand, examples of nonmetrizable nonseparable compact groups with Haar measures and of non-locally-compact separable metrizable groups with measures μ quasi-invariant with respect to dense subgroups admittings μ-measurable discontinous automorphisms are given. Translated fromMatenmaticheskie Zametki, Vol. 68, No. 1, pp. 105–112, July, 2000. An erratum to this article can be found online at .     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

Sampling theorems on locally compact groups from oscillation estimates

We present a general approach to derive sampling theorems on locally compact groups from oscillation estimates. We focus on the L ²-stability of the sampling operator by using notions from frame theory. This approach yields particularly simple and transparent reconstruction procedures. We then apply these methods to the discretization of discrete series representations and to Paley–Wiener spaces on stratified Lie groups.     

On the structure of finite perimeter sets in step 2 Carnot groups

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure. First author was supported by University of Bologna, Italy, funds for selected research topics; second and third authors were supported by MURST, Italy, and University of Trento, Italy.     

Categorical non-abelian cohomology and the Schreier theory of groupoids

By regarding the classical non-abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non-abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids, cocycles are lax 2-functors, and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand, group extensions are fibrations of categories. Furthermore, n-simplices in the nerve of a 2-category are lax 2-functors.     

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.The research of the Y. L. Xin was partially supported by NSFC of and SFECC.     

The Dolbeault-cohomology ring of a compact,even-dimensional lie group

The paper presents a classification of all homogeneous (integrable) complex structures on compact, connected lie groups of even dimension. Thereafter, using lie algebraic methods it proves theorems about the Dolbeault cohomology rings of these complex manifolds in the semisimple case and exhibits the dramatic variation of ring structure of the Dolbeault rings of groups of rank 2. Using some specific computations forSO(9), it gives a counter-example to a long-standing conjecture about the Hodge-deRham (Frohlicher) spectral sequence.     

Un theoreme sur les sommes amalgamees de groupes simpliciaux

In [1], D.M. Kan and R.F. Thurston prove two theorems on amalgamed sums of simplicial groups (cf corollaries 1 and 2 above) starting from a topological theorem of J.H.C. Whitehead. The proofs given in this paper are simplicial from the beginning; in the case where all groups are concentrated in dimension 0, one finds back J.H.C. Whitehead's theorem.     

SL(2,C)中的伪非初等子群和伪Kleinian群

定义了SL(2,C)中的伪非初等群和伪Kleinian群，得到了它们的离散准则和收敛定理．     

一类2nm(m为奇数)阶有限群的构造

本文研究了一类2nm(m为奇数)阶有限群的构造,利用解数论同余方程的方法和群的扩张理论等知识,得到了具有奇数m阶循环正规子群、其补子群为循环群的2nm阶有限群的构造及相关的计数定理.     

Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.     

Operator Ergodic Theorems for Actions of Free Semigroups and Groups

New ergodic theorems are obtained for measure-preserving actions of free semigroups and groups. These theorems are derived from ergodic theorems for Markov operators. This approach also allows one to obtain ergodic theorems for some classes of Markov semigroups. Results of the paper generalize classical ergodic theorems of Kakutani, Oseledets, and Guivarc'h, and recent ergodic theorems of Grigorchuk, Nevo, and Nevo and Stein.     

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G$G$-convex spaces properly. In this work, we show that a large number of results in KKM theory on G$G$-convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.     

L—Fuzzy群的同志和同构

本文研究了Ｌ—Ｆｕｚｚｙ群的同态和同构，给出了它们的一些性质，并把群论中的同态定理和同构定理推广到Ｌ—Ｆｕｚｚｙ群上