Finitely Presented Wreath Products and Double Coset Decompositions

We characterize which permutational wreath products $G \ltimes W^{(X)}We characterize which permutational wreath products are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X ². On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.      

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

On Quasiconvex Subgroups of Word Hyperbolic Groups

We prove that a quasiconvex subgroup H of infinite index of a torsion free word hyperbolic group can be embedded in a larger quasiconvex subgroup which is the free product of H and an infinite cyclic group. Some properties of quasiconvex subgroups of word hyperbolic group are also discussed.     

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite.     

On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L₃(3) or U₃(3) (to U₃(3) or S₄(3), respectively).     

Subgroups Of Direct Products Of Elementarily Free Groups

The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass of the hyperbolic limit groups. We prove that if G ₁,...,G _n are in then a subgroup Γ ⊂ G ₁ × … × G _n is of type FP _n if and only if Γ is itself, up to finite index, the direct product of at most n groups from . This provides a partial answer to a question of Sela. This work was supported in part by Franco–British Alliance project PN 05.004. The first author is also supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Research Merit Award. Received: July 2005 Accepted: April 2006     

On the Patterson–Sullivan Measure for Geometrically Finite Groups Acting on Complex or Quaternionic Hyperbolic Space

The goal of this paper is to provide a tool, the Global Measure Formula, that will facilitate the study of the limit set of discrete geometrically finite groups of isometries of the rank one symmetric spaces. We consider the shadow of a ball from a fixed reference point onto the boundary, and prove a formula that describes the measure of the shadow in terms of the center of the shadowed ball, generalizing a result from real hyperbolic geometry.     

A Geometric Proof of Stallings' Theorem on Groups with More than One End

Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups, and separately, using somewhat different methods, generalised it to a larger class of splittings. Here we adapt the geometric method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end, and show that this new proof simultaneously establishes Dunwoody's generalisation.     

On the Isometry Groups of Hyperbolic Orbifolds

A generic, geometrically finite, hyperbolic n-orbifold is proved to have a finite group of isometries.     

关于有限群某些子群的X-半置换性

Let A be a subgroup of a group G and X a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we try to use the X-semipermutability of some subgroups to characterize the structure of finite groups.     

关于有限群不可约特征标零点的一点注记

在这篇文章中,我们分类了每一个非线性不可约特征标的零点集最多由$3$个共轭类组成的有限亚交换群的结构.     

A Class of 2-Colorable Orthogonal Double Covers of Complete Graphs by Hamiltonian Paths

K _n by a graph G is a collection ? of n spanning subgraphs of K _n , all isomorphic to G, such that any two members of ? share exactly one edge and every edge of K _n is contained in exactly two members of ?. In the 1980's Hering posed the problem to decide the existence of an ODC for the case that G is an almost-hamiltonian cycle, i.e. a cycle of length n−1. It is known that the existence of an ODC of K _n by a hamiltonian path implies the existence of ODCs of K _4n and K _16n , respectively, by almost-hamiltonian cycles. Horton and Nonay introduced 2-colorable ODCs and showed: If for n≥3 and a prime power q≥5 there are an ODC of K _n by a hamiltonian path and a 2-colorable ODC of K _q by a hamiltonian path, then there is an ODC of K _qn by a hamiltonian path. We construct 2-colorable ODCs of K _n and K _2n , respectively, by hamiltonian paths for all odd square numbers n≥9. Received: January 27, 2000     

子群的S-条件置换性对有限群结构的影响

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.     

Property (RD) for Cocompact Lattices in a Finite Product of Rank One Lie Groups with Some Rank Two Lie Groups

We apply V. Lafforgues techniques to establish property (RD) for cocompact lattices in a finite product of rank one Lie groups with Lie groups whose restricted root system is of type A ₂.     

Recognition of Finite Simple Groups S 4(q) by Their Element Orders

It is proved that among simple groups $S_4(q)$ in the class of finite groups, only the groups $S_4(3^n)$, where $n$ is an odd number greater than unity, are recognizable by a set of their element orders. It is also shown that simple groups $U_3(9)$, ${^3D}_4(2)$, $G_2(4)$, $S_6(3)$, $F_4(2)$, and ${^2E}_6(2)$ are recognizable, but $L_3(3)$ is not.     

Growth of relatively hyperbolic groups

We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.

     

On Convergence of Double Series

For the convergence of double series and iterated series, a sufficient condition is obtained. This result provides a test for the convergence of double series and iterated series.     

Boundary Amenability of Groups and Positive Scalar Curvature

We show that if a finitely presented discrete group acts amenably on the boundary of a metric compactification of itself, then the Gromov–Lawson–Rosenberg conjecture holds for . Thus, if M is a compact aspherical Spin manifold with fundamental group , then M does not admit a Riemannian metric with positive scalar curvature. The class of groups having such a property includes hyperbolic groups and amenable groups and is closed under semidirect product.     

Krull–Remak–Schmidt Fails for Artinian Modules over Local Rings

Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull–Remak–Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vámos. In order to construct such examples, one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull–Remak–Schmidt property. In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.     

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω??^N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L²‐space L_s^{2, q}(Ω), s∈?, into irrotational and solenoidal q‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright © 2008 John Wiley & Sons, Ltd.