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.
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 $n$ …
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…
The group of isometries W of a regular rooted tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in W. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group H studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as G and H. We describe this tower for all groups acting on the binary rooted tree which decompose as infinitely iterated wreath product…
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group, and so on, iterating transfinitely. Each group maps canonically into the next using inner automorphisms, and so at limit stages one can take a direct limit and continue the iteration. The tower is said to terminate if a fixed point is reached, that is, if a group is reached which is isomorphic to its automorphism group by the natural map. This occurs if a complete group is reached, one which is centerless and has only inner automorphisms. Wielandt [1939] proved the classical…
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group and so on, iterating transfinitely. Each group maps into the next using inner automorphisms and one takes a direct limit at limit stages. The question is whether the process ever terminates in a fixed point, a group which is isomorphic to its automorphism group by the natural map. In this talk I will prove that every group has a terminating automorphism tower. After this, I will discuss the set-theoretic aspects of the height of the automorphism tower of a group, and sketch m…
We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZFC.
For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We first show that for any kappa=kappa^{aleph_0} there is a group of cardinality kappa with height>kappa^+ (improving the lower bound). Second we show that for many such kappa there is a group of height > 2^kappa, so proving that the upper bound essentially cannot be improved.
We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Th…
Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we investigate this situation and prove that if $C_G(H)$ is supersoluble and $C_{G'}(H)$ is nilpotent, then $G$ is supersoluble. Also, we show that $G$ is a Sylow tower group of a certain type if $C_{G}(H)$ is a Sylow tower group of the same type.
For a centerless group G, we can define its automorphism tower. We define G^{alpha} : G^0=G, G^{alpha +1}=Aut(G^alpha) and for limit ordinals G^delta=bigcup_{alpha < delta}G^alpha . Let tau_G be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says tau_G<2^{|G|})^{+} and more. If we consider Thomas' proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem without the axiom of choice. We attach to every element in G^alpha, the alpha-th member of the automorphism tower of G, a unique quantifier free t…
Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let F_{n,c} denote a free nilpotent group of finite rank n at least two and of nilpotency class c at least two. In 1976 Dyer and Formanek proved that the automorphism group of F_{n,2} is even complete (and hence the height of the aumorphism tower of F_{n,2} is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of F_{n,2} is three. The author proved in 2001 that the automorphism group of any infinitely generated free nilpotent …
The structure of a certain subgroup $S$ of the automorphism group of a partially commutative group (RAAG) $G$ is described in detail: namely the subgroup generated by inversions and elementary transvections. We define admissible subsets of the generators of $G$, and show that $S$ is the subgroup of automorphisms which fix all subgroups $\langle Y\rangle$ of $G$, for all admissible subsets $Y$. A decomposition of $S$ as an iterated tower of semi-direct products in given and the structure of the factors of this decomposition described. The construction allows a presentation of $S$ to be computed…
The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those groups of order 64 with an odd-order automorphism. In view of the fact that we already had determined these groups and that these automorphism groups are not given explicitly in the literature, we have appended to this report these automorphism groups. In another project we were looking for new complete groups by following automorphism group towers up to comple…
We introduce the $k$-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contractive. We analyze the limit spaces of a particular family of contracting Hanoi groups, $H_c^{(k)}$, and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.
We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.
Università di Padova（帕多瓦大学）；University of the Basque Country UPV/EHU（巴斯克地区大学）；Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学）；Università di Salerno（萨莱诺大学）
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.
Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed point group G^X to lower bounds for the growth in G. We give applications to S-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.
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.
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.
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).
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.
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.
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.
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.
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 show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: We construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.
By definition, an $ω$-residually free tower is positive-genus if all surfaces used in its construction are of positive genus. We prove that every limit group is virtually a subgroup of a positive-genus $ω$-residually free tower. By combining this with results of Gaboriau, we prove that elementarily free groups are measure equivalent to free groups.
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.
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 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.
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.
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.
In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.
