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.     

Group Laws and Free Subgroups in Topological Groups

A proof is given that a permutation group in which differentfinite sets have different stabilizers cannot satisfy any grouplaw. For locally compact topological groups with this property,almost all finite subsets of the group are shown to generatefree subgroups. Consequences of these theorems are derived for:Thompson's group F, weakly branch groups, automorphism groupsof regular trees, and profinite groups with alternating compositionfactors of unbounded degree. 2000 Mathematics Subject Classification20B07, 20E10, 20E18, 20P05.     

Automorphism group and dimension of ordered sets

It is shown that a finite groupG is isomorphic to the automorphism group of a two-dimensional ordered set if and only if it is a generalized wreath product of symmetric groups over an ordered index set that is a dual tree. Furthermore, every finite abelian group is isomorphic to the full automorphism group of a three-dimensional ordered set. Also every finite group is isomorphic to the automorphism group of an ordered set that does not contain an induced crown with more than four elements.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

Finite Groups with a Pre-Fixed-Point-Free Power Automorphism

A power automorphism θ of a group G is said to be pre-fixed-point-free if C_G(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

The Conjugacy Problem is Solvable in Free-By-Cyclic Groups

We show that the conjugacy problem is solvable in [finitelygenerated free]-by-cyclic groups, by using a result of O. Maslakovathat one can algorithmically find generating sets for the fixedsubgroups of free group automorphisms, and one of P. Brinkmannthat one can determine whether two cyclic words in a free groupare mapped to each other by some power of a given automorphism.We also solve the power conjugacy problem, and give an algorithmto recognize whether two given elements of a finitely generatedfree group are twisted conjugated to each other with respectto a given automorphism. 2000 Mathematics Subject Classification20F10, 20E05.     

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.     

Replacement of Factors By Subgroups in the Factorization of Abelian Groups

In his book Abelian groups, L. Fuchs raised the question asto whether, in general, in the factorization of a finite (cyclic)abelian group one factor may always be replaced by some subgroup.The answer turned out to be negative in general, but positivein certain cases. In this paper the complete answer for cyclicgroups is given. In all previously unsolved cases, the answerturns out to be positive. It is shown that a cyclic group hasthe property that in every factorization, one factor may bereplaced by a subgroup if and only if the group has order equalto the product of a prime and a square-free integer. Certainresults are also given in non-cyclic cases. 1991 MathematicsSubject Classification 20K01.     

One-Factorizations of Complete Graphs with a Doubly Transitive Automorphism Group

It is shown that a 1-factorization of K_n with a doubly transitiveautomorphism group on vertices is either the affine line-parallelismof AG(d, 2), or one of three ‘sporadic’ exampleswith n = 6, 12 or 28. The full automorphism groups are respectivelyAGL (d, 2) (the holomorph of an elementary abelian group oforder 2^d), PGL(2,5), PSL(2,11) and PL(2,8).     

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).     

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Let G be a finite nonabelian simple group and let be a connectedundirected Cayley graph for G. The possible structures for thefull automorphism group Aut are specified. Then, for certainfinite simple groups G, a sufficient condition is given underwhich G is a normal subgroup of Aut. Finally, as an applicationof these results, several new half-transitive graphs are constructed.Some of these involve the sporadic simple groups G = J₁, J₄,Ly and BM, while others fall into two infinite families andinvolve the Ree simple groups and alternating groups. The twoinfinite families contain examples of half-transitive graphsof arbitrarily large valency.     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

On Cyclic Groups of Automorphisms of Riemann Surfaces

The question of extendability of the action of a cyclic groupof automorphisms of a compact Riemann surface is considered.Particular attention is paid to those cases corresponding toSingerman's list of Fuchsian groups which are not finitely-maximal,and more generally to cases involving a Fuchsian triangle group.The results provide partial answers to the question of whichcyclic groups are the full automorphism group of some Riemannsurface of given genus g>1.     

Topology on the Spaces Of Orderings of Groups

A natural topology on the space of left orderings of an arbitrarysemi-group is introduced here. This space is proved to be compact,and for free abelian groups it is shown to be homeomorphic tothe Cantor set. An application of this result is a new proofof the existence of universal Gröbner bases. 2000 MathematicsSubject Classification 06F15, 13P10 (primary), 06F05, 20F60(secondary).     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

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.     

Marked Schottky Groups and Representations of Surface Groups

We give a complete condition for any n elements of PSL(2, R)to generate a Fuchsian group which is a Schottky group on thatset of generators, and apply it to a question of Bowditch onrepresentations of surface groups into PSL(2, R). 2000 MathematicalSubject Classification: 20H10, 32G15.     

Uncountable Homomorphic Images of Polish Groups are not N1-Free Groups

Shelah has recently proved that an uncountable free group cannotbe the automorphism group of a countable structure. In fact,he proved a more general result: an uncountable free group cannotbe a Polish group. A natural question is: can an uncountable₁-free group be a Polish group? A negative answer is given here;indeed, it is proved that an ₁-free group cannot be a homomorphicimage of a Polish group. In fact, a stronger result is proved,involving a non-commutative analogue of the notion of separablegroup. 2000 Mathematics Subject Classification 20E05.