Fritz Grunewald, Alexander Lubotzky
Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ 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.
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Using a unified method, we determine the structure of automorphisms and representations of arbitrary polyadic groups. More precisely, for a polyadic group $(G, f)=der_{θ, b}(G, \cdot)$, we obtain a complete description of automorphisms and representations of $(G,f)$ in terms of automorphisms and representations of the binary group $(G, \cdot)$.
Inadmissible representations of the tree automorphism group
研究树自同构群的不可接受表示，发现存在不可接受的不可约Banach表示，光滑向量子空间不含代数不可约分量。
Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.
Bounded Cohomology and Unitary Representations of Automorphism Groups of Regular Trees
本文首次计算了正则树自同构群在所有正度数下的连续有界上同调，系数来自任何不可约连续酉表示。
Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $Γ$ with respect to $H$ admit a natural semigroup structure. For any compact group $K$ the semigroup $Γ$ acts in the space $L^2$ on the product of $m$ copies of $K$
If $F$ is a finitely generated free group and $ϕ$ is a positive automorphism of $F$ then $F\rtimes_ϕZ$ satisfies a quadratic isoperimetric inequality.
We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.
We construct a linear representation of the group IA(F_n) of IA-automorphisms of a free group F_n, an extension of the Gassner representation of the pure braid group P_n. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cb_n contains a non-trivial kernel even if n=2. We construct also an extensions of the Burau representation to the group of conjugating automorphisms C_n. This representation is not faithful for n > 1.
Irreducible representations of tree automorphism groups into Pontryagin spaces
研究树自同构群在有限索引连续强非退化半双线性形式上的连续不可约表示，证明索引大于1时无更多表示，完成分类。
We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.
We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the description of involutions in automorphism groups of free groups (J. Dyer, P. Scott, 1975) we obtain a group-theoretic characterization of inner automorphisms determined by primitive elements in the automorphism group of any non-abelian free group F. It follows that the subgroup Inn(F) is characteristic in Aut(F), and hence the latter one is complete.
Complex Representations of Groups and Involutions of its Automorphisms
本文建立了不可约特征标次数和与有限群自同构相关的扭曲对合数之间的关系，并开发了内自同构和对称群情形下的算法框架，进而给出了识别群具有复（非实）不可约表示的条件并探讨了结构推论。
Generators for the Pure Symmetric Outer Automorphisms of a Free Product
本文通过构造自由积外空间的良好子复形，几何地证明了给定分裂的纯对称外自同构可由因子外自同构与相对Whitehead外自同构生成。
Periodic automorphisms of free groups are diagonalisable in free skew-fields
本文研究自由斜域上的弱周期扭乘自同构，证明由自由群的周期自同构诱导的自由斜域自同构必等价于对角自同构。
On the outer automorphism groups of free groups
该研究证明了秩可数无穷的自由群的外自同构群具有完备性，丰富了自由群外自同构群相关的代数理论结论。
In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Br…
In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.
In this paper we describe the structure of a group of conjugating automorphisms $C_n$ of free group and prove that this structure is similar to the structure of a braid group $B_n$ with $n>1$ strings. We find the linear representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of 2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the braid group $B_3(P^2)$ of the projective plane are linear. Using result of J. Dyer, E. Formanek, E. Grossman and the faithful linear representation of Lawrence--Krammer of $B_4$ we construct faithful linear representation …
We prove that if the order of a splitting automorphism of free Burnside group~$B(m,n)$ of odd period~$n\ge1003$ is a prime power, then the automorphism is inner. Thus, we give an affirmative answer to the question on the coincidence of splitting and inner automorphisms of free Burnside groups $B(m,n)$ for automorphisms of orders~$p^k$ ($p$ is a prime number). This question was posed in the Kourovka Notebook in 1990 (see 11th ed., Question 11.36.~b).
In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ …
An automorphism of a group is said to be normal if it preserves each normal subgroup. In this paper, we determine the normal automorphisms of a free metabelian nilpotent group.
Finite subgroups of automorphisms of free products
本文研究自由积外自同构的有限子群，给出了其阶的上界及特定条件下挠外自同构的阶的界。
In this work we construct free Moufang loop in the variety generated by code loops. We apply this construction for study the code loops. Moreover, we define and determine all basic representations of code loops of rank 3 and 4.
The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than $3$-generated. For the free Steiner loop with $3$ generators we describe the generator elements of the automorphism group and some relations between them.
We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.
In this paper, we are not limited to considering semi-scalar automorphisms of G as Lindsay N. Childs in 2012 which is a specific case of Aut(G). Some new results in determining the fixed point free automorphisms of G are developed.
We construct an example of an IA-automorphism of the free metabelian group of rank $n\geq 3$ without nontrivial fixed points. That gives a negative answer to the question raised by Shpilrain. By a result of Bachmuth \cite{Ba1}, such an automorphism does not exist if the rank is equal to 2.
Complete outer automorphism groups of free nilpotent groups
本研究证明了2类无限生成自由幂零群的外自同构群具有完全性，推进了无限生成自由幂零群外自同构群的结构性质研究。
The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
In this paper, we show that the abelianization of the kernel of the Magnus representation of the automorphism group of a free group is not finitely generated.
We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.
We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.
We prove that an automorphism $ϕ:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.
We consider L^p-cohomology of reflexive Banach spaces and give a spectral condition implying the vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space.
We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower…
We show that word hyperbolicity of automorphism groups of graph products $G_Γ$ and of Coxeter groups $W_Γ$ depends strongly on the shape of the defining graph $Γ$. We also characterized those $Aut(G_Γ)$ and $Aut(W_Γ)$ in terms of $Γ$ that are virtually free.
Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of finite rank which is not equal to 1, or to 3, then the automorphism group Aut(N) of N is complete. The main result of the present paper states that the automorphism group of any infinitely generated free nilpotent group of class two is also complete.
Fixed points of irreducible, displacement one automorphisms of free products
该研究探讨群自由积分解对应的G-树形变空间上的外自同构作用，证明不可约位移1自同构在协体积1切片中恰有一个不动点，为相关群论研究提供结论支撑。
We present necessary and sufficient conditions for the existence of a splitting over $\mathbb{Z}$ of the mapping torus $M_ϕ=F\rtimes \mathbb{Z}$ of a free group automorphism $ϕ$.
The goal of this paper is to introduce a new tool, called {\em long turns}, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.
We give a classification of iwip outer automorphisms of the free group, by discussing the properties of their attracting and repelling trees.
Growth and displacement of free product automorphisms
本文证明不可约自由群自同构的增长率等于其在Culler-Vogtmann空间上的最小Lipschitz位移，并将此结果推广到可约情形及自由积的相对增长率情形。
We describe a set of defining relations for automorphism groups of finitely generated free algebras of Nielsen-Schreier varieties. In particular, this gives a representation of the automorphism groups of free Lie algebras by generators and defining relations.
We show that the compressed word problem in a finitely-generated fully residually free group (F -group) is decidable in polynomial time, and use the result to show that the word problem in the automorphism group of such a group is decidable in polynomial time.
We study the automorphism group Aut(G) of a free product G of finite cyclic groups. We investigate the question in which cases Aut(G) has Serre's property FA. In the case of two or three free factors, we prove that Aut(G) does not have property FA. However, if each free factor of G occurs at least four times we show that Aut(G) does have property FA.
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
If F is a finitely generated free group and ϕis an automorphism of F then the mapping torus of ϕadmits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.
For a positive integer $n$, with $n \geq 4$, let $R_{n}$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank $n$. We show that the subgroup of Aut$(R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in Aut$(R_{n})$ with respect to the formal power series topology.
We exhibit normal subgroups of a free nilpotent group F of rank two and class three, which have isomorphic finite quotients but are not conjugate under any automorphism of F.
We prove that for any automorphism $α$ of a free group F of finite rank, one can efficiently compute a basis of the fixed point subgroup Fix(α).
This paper introduces a newly developed private key cryptosystem and a public key cryptosystem. In the first one, each letter is encrypted with a different key. Therefore, it is a kind of a one-time pad. The second one is inspired by the ElGamal cryptosystem. Both presented cryptosystems are based on automorphisms of free groups. Given a free group $F$ of finite rank, the automorphism group $Aut(F)$ can be generated by Nielsen transformations, which are the basis of a linear technique to study free groups and general infinite groups. Therefore Nielsen transformations are introduced.
We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in \cite{Erofeev}, where proper varieties of groups have been considered.
We introduce a new class of automorphisms $φ$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $φ$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of …
We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore this paper generalises the work of Leder (arXiv:1810.06287) for finite cyclic groups, as well as resolving the open case of that paper.
Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, $\textrm{Out}(F_n)$, of the free group of rank $n$ is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group $\textrm{Out}(G)$ of a finitely generated free-by-finite group $G$. We show that $\textrm{Out}(G)$ is translation discrete and that every subgroup of $\textrm{Out}(G)$ is either virtually finitely generated abelian or contains a nonabelian free group.
On the $z$-classes of Palindromic automorphisms of Free Groups
本文研究自由群回文自同构，给出其矩阵表示的充分必要条件，证明 z-类数量无限，并分类可约回文自同构的共轭类。
We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping bounded the length of the uniform current is compact (up to conjugation.) This implies that the spectrum of the length of the images of the uniform current is discrete, answering to a conjecture of I. Kapovich.
If F is a finitely generated free group and ϕis an automorphism of F then F \rtimes_ϕZ satisfties a quadratic isoperimetric inequality.
In this paper we describe the automorphism groups of the endomorphism semigroups of free Burnside groups $B(m,n)$ for odd exponents $n\ge1003$. We prove, that the groups $Aut(End(B(m,n)))$ and $Aut(B(m,n))$ are isomorphic. In particular, if the groups $Aut(End(B(m,n)))$ and $Aut(End(B(k,n)))$ are isomorphic, then $m=k$.
Let F be an infinitely generated free group and R a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F' of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group Aut(F/R') of the group F/R' is complete. In particular, the automorphism group of any infinitely generated free solvalbe group of derived length at least two is complete. This extends a result by Dyer and Formanek (1977) on finitely generated groups F_n/R' where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.
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