In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More precisely, for each prime number $p$ there is a family $(M_λ)_{λ=0,...,p-1}$ of indecomposable and pairwise nonisomorphic metabelian $p$-groups of the given order. In this manuscript we use recent results on the classification of possible embeddings of a subgroup in a finite abelian $p$-group to construct families of indecomposable metabelian groups, indexed by several parameters, which have upper bounds on the exponents of the center and the commutator subgroup.
We prove that all finitely generated free metabelian groups are permutation stable. This partially answers to the question asked by Levit and Lubotzky whether all finitely generated metabelian groups are permutation stable. Our proof extends the range of application of Levit-Lubotzky's method, which is used to show permutation stability of permutational wreath products of finitely generated abelian groups, to non-split and non-permutational metabelian groups.
Finitely generated (non-abelian) free metabelian pro-p groups, and wreath products of f.g. free abelian pro-p groups, are all finitely axiomatizable in the class of all profinite groups.
We prove that groups in a certain class of metabelian locally compact groups, have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups into finitely presented metabelian groups with quadratic Dehn function. Also, we prove that Baumslag's finitely presented metabelian groups, in which the lamplighter groups embed, have quadratic Dehn function.
We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.
General arguments of Baumslag and Bieri guarantee that any metabelian group of finite Prüfer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the distortions of specific embeddings.
We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian non--cyclic group has a finitely generated $p'$--isolated subgroup which is not $p$--separable.
In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.
We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.
In this note we study the Reidemeister spectrum for metabelian groups of the form ${\mathbb Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$. Particular attention is given to the $R_{\infty}$ property of a subfamily of these groups. We also define a Nielsen spectrum of a space and discuss some examples.
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 describe the two-generated limits of abelian-by-(infinite cyclic) groups in the space of marked groups using number theoretic methods. We also discuss universal equivalence of these limits.
In this paper we describe the profinite completion of the free solvable group on m generators of solvability length r>1. Then, we show that for m=r=2, the free metabelian group on two generators does not have the Congruence Subgroup Property.
We consider a class of metabelian groups first studied by Baumslag and Stammbach and we show that these groups are consistent with the Bieri-Groves conjecture which relates cohomological finiteness conditions to the Bieri-Neumann-Strebel sigma invariant.
Let $G$ be a polycyclic, metabelian or soluble of type (FP)$_{\infty}$ group such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. Then $G$ is virtually abelian. Every soluble biautomatic group is virtually abelian.
We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence (G_n) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank omega^n.
We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically a…
We consider IA-endomorphisms (i.e., Identical in Abelianization) of a free metabelian group of finite rank, and give a matrix characterization of their fixed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the fixed point groups of metabelian endomorphisms. In particular, we show that the rank of the fixed point group of an IA-endomorphism of the free metabelian group of rank $n \ge 2$ can be either equal to 0, 1, or greater than $(n…
A non-cyclic finite $p$-group $G$ is said to be thin if every normal subgroup of $G$ lies between two consecutive terms of the lower central series and $|γ_i(G):γ_{i+1}(G)|\le p^2$ for all $i\geq 1$. In this paper, we determine Beauville structures in metabelian thin $p$-groups.
We study the return probability of finitely generated metabelian groups. We give a characterization of metabelian groups whose return probability is equivalent to $\exp(-n^\frac1{3} )$ in purely algebraic terms, namely the Krull dimension of the group. Along the way, we give lower bounds on the return probability for metabelian groups with torsion derived subgroup, according to the dimension. We also establish a variation of the famous embedding theorem of Kaloujinine and Krasner for metabelian groups that respects the Krull dimension.
The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its $R$-completion induces an epimorphism of homology groups $H_2(-,R)$. This answers a problem of A.K. Bousfield for the class of metabelian groups.
We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.
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.
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 $F$.
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.
Baumslag's group is a finitely presented metabelian group with a Z \wr Z subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes C_m \wr Z. We prove that Baumslag's group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions. v.2: version for publication, incorporating referee's suggestions.
A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear on their study. The object of this paper is to describe some of the new ideas and open problems that arise.
In this note we introduce the class of $\mathcal H$-groups (or Hall groups) related to the class of $\mathcal B$-groups defined by Ph. Hall in 1950's. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely-generated metabelian groups. Hall groups allow to give a negative answer to the Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelia…
In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product $\mathbb Z\wr (\mathbb Z\times \mathbb Z)$. We also show that subsemimodule membership is undecidable for finite rank free $(\mathbb Z\times \mathbb Z)$-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.
Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in \cite{baer}. We investigate some general properties and applications of $\circ$ and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In \cite{greer14}, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ)$ is automorphic.
For a metabelian p-group G = <x,y> with two generators x and y, the annihilator A < Z[X,Y] of the main commutator [y,x] of G, as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for G. Furtwängler's isomorphism of the additive group underlying the residue class ring Z[X,Y]/A of Z[X,Y] modulo the annihilator A to the commutator subgroup G' of G admits the calculation of the abelian type of G'.
We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion, which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on metabelian groups via a comparison to the toppling of a dissipative abelian sandpile.
There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability. We show that, in a sense that can be made precise, `most' subdirect products of a finite set of finitely p…
A group has finite palindromic width if there exists $n$ such that every element can be expressed as a product of $n$ or fewer palindromic words. We show that if $G$ has finite palindromic width with respect to some generating set, then so does $G \wr \mathbb{Z}^{r}$. We also give a new, self-contained, proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.
We study the large scale geometry of the upper triangular subgroup of PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a larger class of groups which are metabelian and are higher dimensional analogues of the solvable Baumslag-Solitar groups BS(1,n).
Explicit expressions for the transfers \(V_i\) from a metabelian p-group G of coclass cc(G)=1 to its maximal normal subgroups \(M_i\) \((1\le i\le p+1)\) are derived by means of relations for generators. The expressions for the exceptional case p=2 differ significantly from the standard case of odd primes \(p\ge 3\). In both cases the transfer kernels \(Ker(V_i)\) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group \(Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\) of an algebraic number field K. For cer…
Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most exponential. For a subfamily of groups we prove that the conjugacy search problem is polynomial. We also show that for a different subfamily the conjugacy search problem reduces to the discrete logarithm problem.
Let p be an odd prime. The lattice of all normal subgroups and the terms of the lower and upper central series are determined for all metabelian p-groups with generator rank d=2 having abelianization of type (p,p) and minimal defect of commutativity k=0. It is shown that many of these groups are realized as Galois groups of second Hilbert p-class fields of an extensive set of quadratic fields which are characterized by principalization types of p-classes.
We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.
The congruence subgroup problem for a finitely generated group $Γ$ asks whether the map $\hat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left\{ 1\right\}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_ω$, where $\hat{F}_ω$ is the free profinite group on countably many generators. Considering $Φ_{n}$, the free metabelian group on $n$ generators, it was also proven that…
Bartholdi, Neuhauser and Woess proved that a family of metabelian groups including lamplighters have a striking geometric manifestation as 1-skeleta of horocyclic products of trees. The purpose of this article is to give an elementary account of this result, to widen the family addressed to include the infinite valence case (for instance $\mathbb{Z} \wr \mathbb{Z}$), and to make the translation between the algebraic and geometric descriptions explicit. In the rank-2 case, where the groups concerned include a celebrated example of Baumslag and Remeslennikov, we give the translation by means of …
Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log n}$, the degree of polynomial growth of $m_n(G)$). A condition is given for when this upper bound is attained. For $G = \mathbb{Z}^k \rtimes \mathbb{Z}$, where $A \in GL(k,\mathbb{Z})$, it is shown that $m_n(G)$ grows like a polynomial of degree equal to the number of blocks in the rational canonical form of $A$. The leading term of this polynomial is the number of distinct …
It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing $Z^2$ is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivia…
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $C\left(IA(Φ_{n}),Φ_{n}\right)$, where $Φ_{n}$ is a free metabelian group on $n\geq4$ generators, and $IA(Φ_{n})=\ker(Aut(Φ_{n})\to GL_{n}(\mathbb{Z}))$. We show that in this case $C(IA(Φ_{n}),Φ_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happen…
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 $n = 3$, w…
We study the free metabelian group $M(2,n)$ of prime power exponent $n$ on two generators by means of invariants $M(2,n)'\to \mathbb{Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb{R} \times \mathbb{Z} \cup \mathbb{Z} \times \mathbb{R}$. In particular we improve bounds found by M.F. Newman for the order of $M(2,2^k)$. We study identities in $M(2,n)$, which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$.
