Extensions of lattice-ordered groups

A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension. Supported by RFBR project No. 03-01-00320. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 529–540, September–October, 2008.     

剩余有限minimax 可解群的几乎正则自同构

设G 是一个剩余有限的minimax 可解群, α 是G 的几乎正则自同构, 则G/[G, α] 是有限群, 并且(1) 当α^p = 1 时, G 有一个指数有限的幂零群其幂零类不超过h(p), 其中h(p) 是只与素数p 有关的函数.(2) 当α² = 1 时, G 有一个指数有限的Abel 特征子群且[G, α]′ 是有限群.关键词剩余有限minimax 可解群几乎正则自同构     

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given and , there exists a positive number such that if a closed -manifold satisfies and , then is almost abelian.

     

On a Generalization of the Notion of Orthogonality and on the Riesz Groups

A method for generalizing the notion of orthogonality to arbitrary partially ordered groups is considered. In the paper, the properties of almost orthogonal elements in Riesz groups are investigated. A series of results on convex directed subgroups associated with a pair of almost orthogonal elements of a Riesz group is obtained. For these subgroups, some theorems on homomorphisms are proved.     

On the fundamental group of manifolds with almost nonnegative Ricci curvature

Gromov conjectured that the fundamental group of a manifold with almost nonnegative Ricci curvature is almost nilpotent. This conjecture is proved under the additional assumption on the conjugate radius. We show that there exists a nilpotent subgroup of finite index depending on a lower bound of the conjugate radius.

     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

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.     

On large sets of almost Hamilton cycle decompositions

A large set of CS(v, k, λ), k‐cycle system of order v with index λ, is a partition of all k‐cycles of K_v into CS(v, k, λ)s, denoted by LCS(v, k, λ). A (v ? 1)‐cycle is called almost Hamilton. The completion of the existence spectrum for LCS(v, v ? 1, λ) only depends on one case: all v ≥ 4 for λ = 2. In this article, it is shown that there exists an LCS(v, v ? 1,2) for any v ≡ 0,1 (mod 4) except v = 5, and for v = 6,7,10,11. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 53–69, 2008     

On pro-unipotent groups satisfying the Golod-Shafarevich condition

We prove that a pro-unipotent group satisfying the Golod-Shafarevich condition contains a free non-abelian pro-unipotent group. Together with the result of A. Magid this implies that such a group is not linear.

     

Factorial orthogonality in the presence of covariates

Summary The present paper obtains necessary and sufficient conditions for factorial orthogonality in the presence of covariates. In particular, when interactions are absent, combinatorial characterizations of the conditions, as natural generalizations of the well-known equal and proportional frequency criteria, have been derived.     

On half cyclically ordered groups

In this paper we introduce and investigate the notion of half cyclically ordered group generalizing the notion of half partially ordered group whose study was begun by Giraudet and Lucas.     

Measuring the tameness of almost convex groups

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.

     

Pseudo almost periodic solutions of differential equations with piecewise constant arguments

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.     

Virtually Nilpotent Groups with (almost) All Localizations Trivial

A group G is generically trivial if and only if, for all prime numbers p the localization of G with respect to p is trivial. Taking off from a theorem of Casacuberta and Castellet , we prove that a virtually nilpotent group E is generically trivial if and only if E is perfect. Inspired by this result, we introduce the concept of almost generically trivial groups. Those are groups G such that, for only finitely many primes p the localization of G with respect to p is not trivial. We prove that a virtually nilpotent group E with finitely generated abelianization is almost generically trivial if and only if the abelianization of E is finite.     

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.     

Extension and separation properties of positive definite functions on locally compact groups

Continuing earlier work, we investigate two related aspects of the set of continuous positive definite functions on a locally compact group . The first one is the problem of when, for a closed subgroup of , every function in extends to some function in . The second one is the question whether elements in can be separated from by functions in which are identically one on .

     

Hilbert Transforms Along Curves in the Heisenberg Group

We consider Hilbert transforms along curves on the Heisenberggroup. In particular, necessary and sufficient conditions forthe L²-boundedness are given for (t)=(t,(t),0) when is evenor odd and convex on R⁺. 1991 Mathematics Subject Classification:42B20, 42B25.