For a positive integer $n$, with $n \geq 2$, let $M_n$ be a free metabelian group of rank $n$. For $c \in \mathbb{N}$, let $γ_c(M_n)$ be the $c$-th term of the lower central series of $M_n$. For $c \geq 2$, let ${\rm I}_{c}{\rm A}(M_n)$ be the subgroup of ${\rm Aut}(M_{n})$ consisting of all automorphisms inducing the identity mapping on $M_n/γ_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}_{c}{\rm A}(M_n)/{\rm I}_{c+1}{\rm A}(M_n)$ for all $n$ and $c$. For $c \geq 2$, we show $γ_{c}({\rm IA}(M_{2})) = {\rm I}_{c+1}{\rm A}(M_{2}))$. For $n = 3$, we show $γ_{3}({\rm IA}(M_{3})) \neq {\rm I}_{4}{\rm A}(M_{3})$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n = 3$ and $c = 3$. For $n \geq 4$ and $c \geq 3$, we prove that ${\cal L}^{c}({\rm IA}(M_{n})) = γ_{c-1}({\rm IA}(M_{n})){\rm I}_{c+1}{\rm A}(M_{n})/{\rm I}_{c+1}{\rm A}(M_{n})$.
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.
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.
In this paper we study abelian and metabelian quotients of braid groups on oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell-Neuwirth fibrations.
For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.
Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.
Department of Mathematics, Wayne State University（韦恩州立大学数学系）；Center for Theoretical and Applied Algebra, Al-Farabi Kazakh National University（阿尔法拉比哈萨克国立大学理论与应用代数中心）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）
Juhasz has proved that the automorphism group of a group G of maximal class of order p^n, with p\ge 5 and n>p+1, has order divisible by $p^{\lceil(3n-2p+5)/2\rceil}$. We show that by translating the problem in terms of derivations, the result can be deduced from the case where G is metabelian. Here one can use a general result of Caranti and Scoppola concerning automorphisms of two-generator, nilpotent metabelian groups.
We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…
For each odd prime p>=5, there exist finite p-groups G with derived quotient G/D(G)=C(p)xC(p) and nearly constant transfer kernel type k(G)=(1,2,...,2) having two fixed points. It is proved that, for p=7, this type k(G) with the simplest possible case of logarithmic abelian quotient invariants t(G)=(11111,111,21,21,21,21,21,21) of the eight maximal subgroups is realized by exactly 98 non-metabelian Schur sigma-groups S of order 7^11 with fixed derived length dl(S)=3 and metabelianizations S/D(D(S)) of order 7^7. For p=5, the type k(G) with t(G)=(2111,111,21,21,21,21) leads to infinitely many n…
Assume $G$ is a finite $p$-group, and let $S$ be a Sylow $p$-subgroup of $\operatorname{Aut}(G)$ with $\exp(S)=q$. We prove that if $G$ is of class $c$, then $\exp(G)|p^{\ceil{\log_pc}}q^3$, and if $G$ is a metabelian $p$-group of class at most $2p-1$, then $\exp(G)|pq^3$.
In 2012 Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and later Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups $H$ and $G_0$. We show that the group $H$ of piecewise projective homeomorphisms of $\mathbb{R}$ has the property that $H"$ is simple and that every proper quotient of $H$ is metabelian. We establish simplicity of the commutator s…
If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and othe…
In arXiv:1303.1129, the authors provided a bound for the palindromic width of free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group ${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$. We denote by $\widetilde{G}_n = G_n / \langle \langle x_1^2, \ldots, x_n^2 \rangle \rangle$ the quotient of group $G_n = \langle x_1, \ldots, x_n \rangle$, which is free in some variety by the normal subgroup generated by $x_1^2, \ldots, x_n^2$. We prove that the palindromic width of the quotie…
Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which…
We study subgroups and quotients of outer automorphism groups of right-angled Artin groups (RAAGs). We prove that for all RAAGS, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.
Infinitely many large Schur sigma-groups G with non-elementary bicyclic commutator quotient G/G' = C(3^e) x C(3), e >= 2, are constructed as periodic sequences of vertices in descendant trees of finite 3-groups. A single root gives rise to pairs of metabelian groups G with logarithmic order lo(G) = 4+e for e >= 3. Three roots are ancestors of pairs of non-metabelian groups G with moderate rank distribution rho(G) = (2,2,3;3) and lo(G) = 7+e for e >= 5. Twentyseven roots produce sextets of non-metabelian groups G with elevated rank distribution rho(G) = (3,3,3;3) and lo(G) = 19+e for e >= 9. Th…
Given a finitely presented group $G$, we wish to explore the conditions under which automorphisms of quotients $G/N$ can be lifted to automorphisms of $G$. We discover that in the case where $N$ is a central subgroup of $G$, the question of lifting can be reduced to solving a certain matrix equation. We then use the techniques developed to show that $Inn(G)$ is not characteristic in $Aut(G)$, where $G$ is a metacyclic group of order $p^n$, $p\neq 2$.
We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Γ$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_Γ)$ virtually surjects onto some infinite Coxeter group. In particular, the group ${\rm Aut}(W_Γ)$ is virtually indicable and therefore does not have Kazhdan's property (T).
For finite metabelian 3-groups M with elementary bicyclic commutator quotient M/M' = C3*C3, coclass cc(M) in {4,6}, and transfer kernel type F, the smallest Schur sigma-groups S with second derived quotient S/S" = M are determined. Evidence is provided of arithmetical realizations of these groups by second 3-class groups M = Gal(F(3,2,K)/K), respectively 3-class field tower groups S = Gal(F(3,infty,K)/K), of imaginary quadratic number fields K=Q(sqrt{d}).
In this paper we study the quotient and "blow-up" of graph-of-groups $\cal{G}$ and of their automorphisms $H: \cal{G} \rightarrow \cal{G}$. We show that the existence of such a "blow-up" of $\bar{H}: \bar{\cal{G}} \rightarrow \bar{\cal{G}}$ relative to a given family of "local" graph-of-groups isomorphisms $H_{i}: \cal{G}_{i} \rightarrow \cal{G}_{i}$ depends crucially on the $H_{i}$-conjugacy class of the correction term $δ(\bar{E}_{i})$ for any edge $\bar{E}_{i}$ of $\bar{\cal{G}}$, where $H$-congjugacy is a new but natural concept introduced here. As an application we obtain a criterion as t…
We study the group $IAut(A)$ generated by the inertial automorphisms of an abelian group $A$, that is, automorphisms $γ$ with the property that each subgroup $H$ of $A$ has finite index in the subgroup generated by $H$ and $Hγ$. Clearly, $IAut(A)$ contains the group $FAut(A)$ of finitary automorphisms of $A$, which is known to be locally finite. In a previous paper, we showed that $IAut(A)$ is (locally finite)-by-abelian. In this paper, we show that $IAut(A)$ is also metabelian-by-(locally finite). In particular, $IAut(A)$ has a normal subgroup $Γ$ such that $IAut(A)/Γ$ is locally finite and $…
The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a group has solvable Farb growth which is at most polynomial in $\log(n)$ if and only if the group is polycyclic and virtually nilpotent. We also give new results concerning approximating oriented surface groups by nilpotent quotients. As a consequence of this, we prove that a natural number $C$ exists so that any nontrivial element of the $Ck$th term of the lo…
We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy between being virtually nilpotent and containing a non-abelian free subgroup that is determined by a graphical condition on the underlying labelled graph. Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxter groups (RACGs), providing a common framework for studying these groups.…
We give an new, elementary proof of the result that the smallest non-cyclic quotients of automorphism group of free group is the linear group over the field of two elements, and moreover all minimal quotients are obtained by the standard projection composed with an automorphism of the image. This result, originally due to Baumeister-Kielak-Pierro, proves a conjecture of Mecchia-Zimmermann.
First, we study a descending filtration of the ring of Fricke characters of a free group consisting of ideals on which the automorphism group of a free group acts. Then using it, we define a descending filtration of the IA-automorphism group of a free group, and investigate a relation between it and the Andreadakis-Johnson filtration.
By the construction of suitable non-metabelian Schur sigma-groups S of type (9,3) with log order lo(S) = 21 and nilpotency class cl(S) = 9, evidence is provided of a new class of imaginary quadratic fields K with 3-class group Cl(3,K) ~ C(9) * C(3) and punctured principalization type kappa ~ (1,4,4;4) whose 3-class field tower consists of precisely three stages. In contrast, previous examples of three-stage towers were associated with kappa in { (1,1,2;2), (1,1,2;3), (1,1,4;2), (1,2,3;1) }, lo(S) = 9 and cl(S) = 5.
The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.
We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F_n, for n = 3 and 4, using presentations of these groups.
A p-group G is p-central if the central quotient has exponent p. We prove that for a subset of finite p-central p-groups, the order of the group G divides the order of Aut(G).
It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $\text{Aut}(F_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $\text{Aut}(F_n)$ for $n \geq 4$ where $k$th powers of all transvections act trivially fo…
We prove that, for any positive integer $c$, the quotient group $γ_{c}(M_{3})/γ_{c+1}(M_{3})$ of the lower central series of the McCool group $M_{3}$ is isomorphic to two copies of the quotient group $γ_{c}(F_{3})/γ_{c+1}(F_{3})$ of the lower central series of a free group $F_{3}$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra ${\rm gr}(M_{3})$ of $M_3$ is naturally embedded into the Johnson Lie algebra ${\cal L}({\rm IA}(F_{3}))$ of the IA-automorphisms of $F_{3}$.
In this paper, we consider certain descending filtrations of the $\mathrm{SL}(m,\C)$-representation algebras of free groups and free abelian groups. By using it, we introduce analogs of the Johnson homomorphisms of the automorphism groups of free groups. We show that the first homomorphisms are extended to the automorphism groups of free groups as crossed homomorphisms. Furthermore we show that the extended crossed homomorphisms induce Kawazumi's cocycles and Morita's cocycles. This works are generalization of our previous results for the $\mathrm{SL}(2,\C)$-representation algebras.
If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of automorphisms of Delta with bounded quotient, then the center of G is trivial.
We give an upper bound for the number of ``overlattices'' in the automorphism group of a tree, containing a fixed lattice with index n. For an example of a lattice in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.
We say a group $G$ has property $R_\infty$ if the number $R(φ)$ of twisted conjugacy classes is infinite for every automorphism $φ$ of $G$. For such groups, the $R_\infty$-nilpotency degree is the least integer $c$ such that $G/γ_{c+1}(G)$ has property $R_\infty$. In this work, we compute the $R_\infty$-nilpotency degree of all Generalized Solvable Baumslag-Solitar groups $Γ_n$. Moreover, we compute the lower central series of $Γ_n$, write the nilpotent quotients $Γ_{n,c}=Γ_n/γ_{c+1}(Γ_n)$ as semidirect products of finitely generated abelian groups and classify which integer invertible matrice…
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.
Kostant constructed a section from the adjoint quotient morphism of a simple Lie algebra to the open set of regular elements, and Steinberg constructed such a section for the adjoint quotient of a simply connected and simple algebraic group. In this paper, we show that all sections of the adjoint quotient are conjugate via a morphism from the adjoint quotient to the group. In particular, the sections constructed via the parabolic construction are conjugate either to the Kostant section or to the Steinberg section. The method of proof consists in studying automorphism sheaves of certain princip…
Let $P(\mathrm{sl}_2(K))$ be the Poisson enveloping algebra of the Lie algebra $\mathrm{sl}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P(\mathrm{sl}_2(K))/(C_P-λ)$, where $C_P$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$ and $0\neq λ\in K$, are proven to be simple in \cite{UZh}. Using a result by L. Makar-Limanov \cite{ML90}, we describe generators of the automorphism group of $P(\mathrm{sl}_2(K))/(C_P-λ)$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results b…
Let $V$ be a $G$-module where $G$ is a complex reductive group. Let $Z:=\quot VG$ denote the categorical quotient and let $π\colon V\to Z$ be the morphism dual to the inclusion $Ø(V)^G\subsetØ(V)$. Let $ϕ\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $Φ\colon V\to V$ which lifts $ϕ$, i.e., $π(Φ(v))=ϕ(π(v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where $V=r\lieg$ is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\geq 2$ will do), any $ϕ$ has a lift. We consider the case of genera…
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…
For a group $G$, let $U$ be the group of units of the integral group ring $\mathbb{Z}G$. The group $G$ is said to have the normalizer property if $\text{N}_U(G)=\text{Z}(U)G$. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being nontrivial. Groups $G$ for which class-preserving automorphisms are inner automorphisms, $\text{Out}_c(G)=1$, have the normalizer property. Recently, Herman and Li have shown that $\text{Out}_c(G)=1$ for a finite Blackburn group $G$. We show that $\text{ou…
Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound…
Let $\mathfrak g_i$ be a simple complex Lie algebra, $1\leq i \leq d$, and let $G=G_1\times...\times G_d$ be the corresponding adjoint group. Consider the $G$-module $V=\oplus r_i\mathfrak g_i$ where $r_i\geq 1$ for all $i$. We say that $V$ is \emph{large} if all $r_i\geq 2$ and $r_i\geq 3$ if $G_i$ has rank 1. In [Schwarz12] we showed that when $V$ is large any algebraic automorphism $ψ$ of the quotient $Z:= V//G$ lifts to an algebraic mapping $Ψ\colon V\to V$ which sends the fiber over $z$ to the fiber over $ψ(z)$, $z\in Z$. (Most cases were already handled in [Kuttler11]). We also showed th…
A group $G$ has the $R_{\infty}$--property if the number $R(φ)$ of twisted conjugacy classes is infinite for any automorphism $φ$ of $G$. For such a group $G$, the $R_{\infty}$--nilpotency index is the least integer $c$ such that $G/γ_{c+1}(G)$ still has the $R_\infty$--property. In this paper, we determine the $R_{\infty}$--nilpotency degree of all Baumslag--Solitar groups.
Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $ρ|G|$, with $ρ\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $ρ>1/2$, then $G$ is abelian, and if $ρ>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\operatorname{Rad}(G)]\leqρ^{-1.78}$, where $\operatorname{Rad}(G)$ denotes the solvable radical of $G$. Furthermore, we generalize an example of Horoševski\uı to show that in finite groups, the quotient of the maximum automorphism orde…
We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1--1 correspondences between aperiodic necklaces and irreducible polynomials over the field $\F_2$ of two elements. We notice that this group is isomorphic to the quotient group of non-degenerate circulant matrices of size $n$ over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs.
We calculate the irreducible decomposition of the images of the Johnson homomorphisms of the automorphism group of a free group and a free metabelian group. We determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianizaton is given by the degree one part and the Morita's trace maps. We also consider twisted cohomology groups of the automorphism group of a free nilpotent group. We show that the trace map for the exterior product defines a non-trivial twisted second cohomology class of it.
