On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type $G/N$ where $N\subseteq G$ is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of $G/N$ when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.     

On -algebras associated with locally compact groups

Let be a locally compact group, and let denote the same group with the discrete topology. There are various associated to and We are concerned with the question of when these are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.

     

A-扩张Lie Rinehart代数

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative,associative algebra A.It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group,analogous to the well known relationship of Lie algebras and Lie groups.     

Galois cohomology of completed link groups

In this paper we compute the Galois cohomology of the pro- completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in whose linking number diagram is irreducible modulo (e.g. none of the linking numbers is divisible by ).

The result is that (with -coefficients) the Galois cohomology is naturally isomorphic to the -cohomology of the discrete link group.

The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.

     

Towards a Lie theory of locally convex groups

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.     

一类多重循环群的剩余有限性质

设A是秩为n(n≥2)的自由Abel群,A的自同构群Aut(A)= GL(n,Z).对整数m,取 α =(0 1 0…0 0 0(………)(…………)0 0 0…0 1 1 0…0 m)∈ Aut(A).记Γm(n)=A(×)〈α〉,则它是一个2元生成的多重循环群.本文给出了 Γm(n)的准确的剩余有限性质.     

The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

关于完备李群与完备李代数

孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果。本文给出完备李群与完备李代数的某些关系。     

Pure quantitative characterization of finite simple groups

Let G be a finite group, and let π _e (G) be the set of all element orders of G. In this short paper we prove that π _e (B _n (q)) ≠ π _e (C _n (q)) for all odd q.      

Computing in groups of Lie type

We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

     

On subsemigroups of semisimple Lie groups

In this paper we classify the subsemigroups of any connected semisimple Lie groupG which areK-bi-invariant, whereG=KAN is an Iwasawa decomposition ofG.     

Residually nilpotent one-relator groups with nontrivial centre

We determine explicitly the residually nilpotent one-relator groups with nontrivial centre. We show also that, if is a one-relator group, then is residually nilpotent if, and only if, its central quotient is residually nilpotent.