Linear Representations of the Automorphism Group of a Free Group

Let F _n be the free group on n ≥ 2 elements and Aut(F _n ) its group of automorphisms. In this paper we present a rich collection of linear representations of Aut(F _n ) arising through the action of finite-index subgroups of it on relation modules of finite quotient groups of F _n . We show (under certain conditions) that the images of our representations are arithmetic groups. Received: November 2006, Accepted: March 2007     

The Automorphism Tower of a Free Group

It is proved that the automorphism group of any non-abelianfree group F is complete. The key technical step in the proofis that the set of all conjugations by powers of primitive elementsis first-order parameter-free definable in the group Aut(F).     

The Fixed Point Group of a Free Group Automorphism

It is proved that the fixed point group of an arbitrary automorphism of a free group of finite rank has an algorithmically computable basis.     

An Ergodic Action of the Outer Automorphism Group of a Free Group

For n > 2, the action of the outer automorphism group of the rank n free group F _n on Hom(F _n , SU(2))/SU(2) is ergodic with respect to the Lebesgue measure class. The author gratefully acknowledges support from National Science Foundation grants DMS-0405605 and DMS-0103889. Received: September 2005 Revision: January 2006 Accepted: March 2006     

On the Automorphism Group of a Smooth Schubert Variety

Let G be a simple algebraic group of adjoint type over the field \(\mathbb {C}\) of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and let X(w) be the Schubert variety in G/B corresponding to w. Let α ₀ denote the highest root of G with respect to T and B. Let P be the stabiliser of X(w) in G. In this paper, we prove that if G is simply laced and X(w) is smooth, then the connected component of the automorphism group of X(w) containing the identity automorphism equals P if and only if w ^?1(α ₀) is a negative root (see Theorem 4.2). We prove a partial result in the non simply laced case (see Theorem 6.6).     

A Countable Structure Does Not Have a Free Uncountable Automorphism Group

It is proved that the automorphism group of a countable structurecannot be a free uncountable group. The idea is that insteadof proving that every countable set of equations of a certainform has a solution, it is proved that this holds for a co-meagrefamily of appropriate countable sets of equations. This researchwas partially supported by the Israel Science Foundation.     

On the Automorphism Group of the Kohn-Nirenberg Domain

In this paper, we prove that there are no automorphism orbits of the Kohn-Nirenberg domain accumulating at the origin.     

Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

In [6] S. Shelah showed that in the endomorphism semi-groupof an infinitely generated algebra which is free in a varietyone can interpret some set theory. It follows from his resultsthat, for an algebra F which is free of infinite rank in avariety of algebras in a language L, if > |L|, then thefirst-order theory of the endomorphism semi-group of F, Th(End(F)),syntactically interprets Th(,L₂), the second-order theory ofthe cardinal . This means that for any second-order sentence of empty language there exists *, a first-order sentence ofsemi-group language, such that for any infinite cardinal >|L|, Th(,L₂)*Th(End(F)) In his paper Shelah notes that it is natural to study a similarproblem for automorphism groups instead of endomorphism semi-groups;a priori the expressive power of the first-order logic for automorphismgroups is less than the one for endomorphism semi-groups. Forinstance, according to Shelah's results on permutation groups[4, 5], one cannot interpret set theory by means of first-orderlogic in the permutation group of an infinite set, the automorphismgroup of an algebra in empty language. On the other hand, onecan do this in the endomorphism semi-group of such an algebra. In [7, 8] the author found a solution for the case of the varietyof vector spaces over a fixed field. If V is a vector spaceof an infinite dimension over a division ring D, then the theoryTh(, L₂) is interpretable in the first-order theory of GL(V),the automorphism group of V. When a field D is countable anddefinable up to isomorphism by a second-order sentence, thenthe theories Th(GL(V)) and Th(, L₂) are mutually syntacticallyinterpretable. In the general case, the formulation is a bitmore complicated. The main result of this paper states that a similar result holdsfor the variety of all groups.     

On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

无平方因子阶群的自同构群阶的上确界

群阶为素数方幂（即ｐ－ 群）时已得到该群自同构群阶的上确界,而对于其他情形的群,同样的问题要复杂得多． 本文在群阶无平方因子且为偶数时,给出了这类群的自同构群阶的上确界．     

Structure of a Conjugating Automorphism Group

We examine the automorphism group Aut(F _n ) of a free group F _n of rank n 2 on free generators x ₁, x ₂,...,x _n . It is known that Aut(F ₂) can be built from cyclic subgroups using a free and semidirect product. A question remains open as to whether this result can be extended to the case n > 2. Every automorphism of Aut(F _n ) sending a generator x _i to an element f _i ^-1 x _(i) f _i , where f _i F _n and is some permutation on a symmetric group S _n , is called a conjugating automorphism. The conjugating automorphism group is denoted C _n . A set of automorphisms for which is the identity permutation form a basis-conjugating automorphism group, denoted Cb _n . It is proved that Cb _n can be factored into a semidirect product of some groups. As a consequence we obtain a normal form for words in C _n . For n 4, C _n and Cb _n have an undecidable occurrence problem in finitely generated subgroups. It is also shown that C _n , n 2, is generated by at most four elements, and we find its respective genetic code, and that Cb _n , n 2, has no proper verbal subgroups of finite width.     

A Presentation of the Automorphism Group of the Two-Generator Free Metabelian and Nilpotent Group of Class c

We determine the structure of IA(G)/Inn(G) by giving a set of generators, and showing that IA(G)/Inn(G) is a free abelian group of rank (c – 2)(c + 3)/2. Here G = M ₂, c = x, y, c 2, is the free metabelian nilpotent group of class c.     

The Non-Finite Presentability of the Automorphism Group of the Free Z-Group of Rank Two

Let G be a group, and let F_n[G] be the free G-group of rankn. Then F_n[G] is just the natural non-abelian analogue of thefree ZG-module of rank n, and correspondingly the group _n(G)of equivariant automorphisms of F_n[G] is a natural analogueof the general linear group GL_n(ZG). The groups _n(G) have beenstudied recently in [4, 8, 5]. In particular, in [5] it wasshown that if G is not finitely presentable (f.p.) then neitheris _n(G), and conversely, that _n(G) is f.p. if G is f.p. andn2. It is a common phenomenon that for small ranks the automorphismgroups of free objects may behave unstably (see the survey article[2]), and the main aim of the present paper is to show thatthis turns out to be the case for the groups ₂(G).     

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.     

Pseudo-n-Transitivity of the Automorphism Group of a Geometric Structure

We introduce a notion of pseudo-n- transitivity which is a nontransitive counterpart of the n-transitivity. The main result states that any group of diffeomorphisms which satisfies the locality condition is pseudo-n-transitive for each n 1.     

On Automorphism-Fixed Subgroups of a Free Group

Let F be a finitely generated free group, and let n denote its rank. A subgroup H of F is said to be automorphism-fixed, or auto-fixed for short, if there exists a set S of automorphisms of F such that H is precisely the set of elements fixed by every element of S; similarly, H is 1-auto-fixed if there exists a single automorphism of F whose set of fixed elements is precisely H. We show that each auto-fixed subgroup of F is a free factor of a 1-auto-fixed subgroup of F. We show also that if (and only if) n ≥ 3, then there exist free factors of 1-auto-fixed subgroups of F which are not auto-fixed subgroups of F. A 1-auto-fixed subgroup H of F has rank at most n, by the Bestvina–Handel Theorem, and if H has rank exactly n, then H is said to be a maximum-rank 1-auto-fixed subgroup of F, and similarly for auto-fixed subgroups. Hence a maximum-rank auto-fixed subgroup of F is a (maximum-rank) 1-auto-fixed subgroup of F. We further prove that if H is a maximum-rank 1-auto-fixed subgroup of F, then the group of automorphisms of F which fix every element of H is free abelain of rank at most n − 1. All of our results apply also to endomorphisms.