All Groups are Outer Automorphism Groups of Simple Groups

It is shown that each group is the outer automorphism groupof a simple group. Surprisingly, the proof is mainly based onthe theory of ordered or relational structures and their symmetrygroups. By a recent result of Droste and Shelah, any group isthe outer automorphism group Out (Aut T) of the automorphismgroup Aut T of a doubly homogeneous chain (T, ). However, AutT is never simple. Following recent investigations on automorphismgroups of circles, it is possible to turn (T, ) into a circleC such that Out (Aut T) Out (Aut C). The unavoidable normalsubgroups in Aut T evaporate in Aut C, which is now simple,and the result follows.     

双曲群,自动机群与群的Dehn函数

本文第一部分对组台群论的发展史作了一个简要的回顾．然后重点介绍近20年来组合群论领域中的三个热点课题：即双曲群，自动机群和群的Dehn函数的有关概念，并对相关的重要研究成果给予概述，其中包括作者本人在群的二阶Dehn函数的研究工作中的若干成果．最后提出9个公开问题。     

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

It is known that a number of algebraic properties of the braidgroups extend to arbitrary finite Coxeter-type Artin groups.Here we show how to extend the results to more general groupsthat we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellativemonoid where the expressions of a given element have boundedlengths, and where left and right lowest common multiples exist.A Garside monoid is a Gaussian monoid in which the left andright lowest common multiples satisfy an additional symmetrycondition. A Gaussian group is the group of fractions of a Gaussianmonoid, and a Garside group is the group of fractions of a Garsidemonoid. Braid groups and, more generally, finite Coxeter-typeArtin groups are Garside groups. We determine algorithmic criteriain terms of presentations for recognizing Gaussian and Garsidemonoids and groups, and exhibit infinite families of such groups.We describe simple algorithms that solve the word problem ina Gaussian group, show that these algorithms have a quadraticcomplexity if the group is a Garside group, and prove that Garsidegroups have quadratic isoperimetric inequalities. We constructnormal forms for Gaussian groups, and prove that, in the caseof a Garside group, the language of normal forms is regular,symmetric, and geodesic, has the 5-fellow traveller property,and has the uniqueness property. This shows in particular thatGarside groups are geodesically fully biautomatic. Finally,we consider an automorphism of a finite Coxeter-type Artin groupderived from an automorphism of its defining Coxeter graph,and prove that the subgroup of elements fixed by this automorphismis also a finite Coxeter-type Artin group that can be explicitlydetermined. 1991 Mathematics Subject Classification: primary20F05, 20F36; secondary 20B40, 20M05.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.

     

Cyclic Groups as Autocommutator Groups

This article classifies the groups X whose autocommutator subgroup [X, Aut(X)] is isomorphic to ? and the finite groups X for which [X, Aut(X)] ? C _p has a prime number p of elements.     

Stable Groups and Algebraic Groups

A certain property of some type-definable subgroups of superstablegroups with finite U-rank is closely related to the Mordell–Langconjecture. This property is discussed in the context of algebraicgroups.     

Lipschitz Homotopy Groups of the Heisenberg Groups

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ${\mathbb{H}^n}$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj?asz-Lukyanenko-Tyson constructed horizontal maps from S ^k to ${\mathbb{H}^n}$ which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map ${S^k \to \mathbb{H}^1}$ factors through a tree and is thus Lipschitz null-homotopic if ${k \geq 2}$ .     

Rigidity of Coxeter Groups and Artin Groups

A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given by Bernhard Mühlherr. We show that this example belongs to a general operation of diagram twisting. We show that the Coxeter groups defined by twisted diagrams are isomorphic, and, moreover, that the Artin groups they define are also isomorphic, thus answering a question posed by Charney. Finally, we show a number of Coxeter groups are reflection rigid once twisting is taken into account.     

Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(F_n)Aut(F_m) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(F_n) to any group that does not contain an isomorphic imageof the symmetric group S_n+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of W_n = (Z/2)ⁿ S_n, or of Z^n–1. These results place constraints on howAut(F_n) can act. For example, if n 3, any action of Aut(F_n)on the circle (by homeomorphisms) factors through det : Aut(F_n)Z₂.2000 Mathematics Subject Classification 20F65, 20F28 (primary).     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

Free Groups of Outer Commutator Varieties of Groups

If F is a free group, 1 < i j 2i and i k i + j + 1 thenF/[_j(F), _i(F), _k(F)] is residually nilpotent and torsion-free.This result is extended to 1 < i j 2i and i k 2i + 2j.It is proved that the analogous Lie rings, L/[L^j, Lⁱ, L^k] whereL is a free Lie ring, are torsion-free. Candidates are foundfor torsion in L/[L^j, Lⁱ, L^k] whenever k is the least of {i,j, k}, and the existence of torsion in L/[L^j, Lⁱ, L^k] is provedwhen i, j, k 5 and k is the least of {i, j, k}.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

二秩无扭群的自同构群和只有两个自同构的二秩无扭群

本文利用Ｋｕｒǒｓ不变量理论和不定方程理论，讨论了二秩无扭群的自同构群，以及有零高元的二秩无扭群只有两个自同构的充要条件．