Behrooz Mashayekhy, Mohsen Parvizi, Saeed Kayvanfar
In this paper we determine all finitely generated abelian groups which are varietal capable with respect to the variety of polynilpotent groups. This result is a vast generalization of the famous Baer's result about capability of finitely generated abelian groups.
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In this paper, we present an explicit formula for the Baer invariant of a finitely generated abelian group with respect to the variety of polynilpotent groups of class row $(c_1,...,c_t)$, ${\cal N}_{c_1,...,c_t}$. In particular, one can obtain an explicit structure of the $\ell$-solvable multiplier (the Baer invariant with respect to the vaiety of solvable groups of length at most $\ell\geq 1,\ {\cal S}_{\ell}$.) of a finitely generated abelian group.
In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the variety of polynilpotent groups of class row $(c,1)$, ${\cal N}_{c,1}$, for all $c > 2n-2$. In particular, an explicit structure of the Baer invariant of a free abelian group with respect to the variety of metabelian groups will be presented.
Motivated by a classic result for free groups, one says that a group $G$ has the Magnus property if the following holds: whenever two elements generate the same normal subgroup of $G$, they are conjugate or inverse-conjugate in $G$. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most $2$. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishi…
Recently, the authors gave some conditions under which a direct product of finitely many groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. In this paper, we extend this fact to any infinite direct product of groups. Moreover, we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian, nilpotent and polynilpotent groups.
We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the number of automorphism classes of elements.
We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.
We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsion part is in fact determined by the set of all growth functions of a finitely generated Abelian group.
We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $φ\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we discuss various algorithms for checking whether $b\in \imφ$ holds and if so, for computing a pre-image of $b$ in $G$ together with the kernel of $φ$.
We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, for all $c_2\leq c_1\leq 2c_2$. As a consequence we determine necessary and sufficient conditions for such groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable. We also show that if $c_1\neq 1\neq c_2$, then a finitely generated abelian group is $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable if and only if it is capable. Finally we show that $\mathfrak{S}_2$-capability implies capability but there is a finitely generat…
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…
Finitely presented kernels of right-angled Artin groups with abelian quotients
研究具有阿贝尔商的直角阿廷群的核何时有限生成及有限表示，给出明确生成集与表示，还推导出其特征标核的有限表示，推广了相关群的表示。
W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(ρ(G))$, where $ρ$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(ρ(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G…
Finitely presented kernels of homomorphisms from hyperbolic groups onto free abelian groups
本文针对每个 $m\geq 2$，构造了双曲群到自由阿贝尔群满射同态的核，该核为非双曲的有限展示子群，其有限型为 $F_2$ 而非 $F_3$。
A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We classify finitely generated unital $\ell$-groups by sequences $\mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes, where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of $W_{t}$. A simple criterion is given to recognize when two such sequences classify isomorp…
Let $G$ be an $\ell$-group (which is short for ``lattice-ordered abelian group''). Baker and Beynon proved that $G$ is finitely presented iff it is finitely generated and projective. In the category $\mathcal U$ of {\it unital} $\ell$-groups---those $\ell$-groups having a distinguished order-unit $u$---only the $(\Leftarrow)$-direction holds in general. Morphisms in $\mathcal U$ are {\it unital $\ell$-homomorphisms,} i.e., hom\-o\-mor\-phisms that preserve the order-unit and the lattice structure. We show that a unital $\ell$-group $(G,u)$ is finitely presented iff it has a basis, i.e., $G$ is…
Bounding conjugacy depth functions for wreath products of finitely generated abelian groups
本文研究了阿贝尔群圈积的共轭可分性渐近行为，完全刻画了lamplighter群的渐近类并给出了广义lamplighter群及无限基群情形的指数或超指数上下界，进而为非阿贝尔作用群的各种群圈积的共轭深度函数提供了下界。
Introducing $n$-Magic Groups and Characterizing $3$-Magic Finitely Generated Abelian Groups
本文引入$n$-魔群概念，研究$3$-魔有限生成阿贝尔群的特征，并探讨非阿贝尔群及更高阶$n$-魔群的性质。
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 extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric s…
We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier.
It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them. In fact we construct an uncountable number of infinite countable abelian groups which do have the $R_{\infty}$ property. We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group. 8 pages, no figures
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.
We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated $3$-step solvable group has finite palindromic width. More generally, we show the finiteness of palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step $\geq 3$, we prove that if $G$ is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of $G$ is finite. We also prove that the palindromic width of $\…
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 observe that abelian subgroups of Helly groups are finitely generated, and consequently, soluble subgroups of Helly groups are virtually abelian.
A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.
We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.
We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.
Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups
本文将自由群直积有限表现子群的相关结构定理推广到二维凝聚右角阿廷群（RAAGs）直积，刻画了其有限表现及$FP_n$型子群的结构，并证明这类子群的多重共轭问题与成员问题可判定。
We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.
Counting automorphic orbits in finitely generated groups
研究有限生成群中的自同构增长函数，证明其非共轭不变量，并分类了各类群的自同构增长速率。
In this article we introduce the notion of weak identities in a group and study their properties. We show that weak identities have some similar properties to ordinary ones. We use this notion to prove that any finitely generated solvable discriminating group is abelian.
A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (fo…
The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language …
We say a group is finitely annihilated if it is the set-theoretic union of all its proper normal finite index subgroups. We investigate this new property, and observe that it is independent of several other well known group properties. For finitely generated groups, we show that in many cases it is equivalent to having non-cyclic abelianisation, and at the same time construct an explicit infinite family of counterexamples to this. We show for finitely presented groups that this property is neither Markov nor co-Markov. In the context of our work we show that the weight of a non-perfect finite …
Conjugacy Problem for Dehn Twists of Free Products of Free Abelian Groups
研究有限生成自由阿贝尔群自由积的Dehn扭转自同构的共轭问题的可解性。
We show that an HNN-extension with finitely generated abelian base group is Z-linear if and only if it is residually finite.
Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.
Subgroups of the direct product of graphs of groups with free abelian vertex groups
本文将 Baumslag–Roseblade 关于自由群直积有限展示子群的结果推广到具有自由阿贝尔顶点群与循环边群的群图，并证明相关 RAAG 子群的算法问题可判定。
A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …
Let $G$ be a group. We denote by $ν(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a finitely generated group in which the set of all simple tensors $T_{\otimes}(G)$ is finite, then the non-abelian tensor square $G \otimes G$ and the group $ν(G)$ are finite. Moreover, we show that if $G$ is a locally residually finite group in which the set of simple tensors $T_{\otimes}(H)$ is finite for every proper finitely generated subgroup $H$ of $G$, then the group $ν(G)$ is locally finite.
Some torsion-free solvable groups with few subquotients
本研究构造出满足所有有限生成无挠亚阿贝尔子商均为虚阿贝尔的无限秩有限生成无挠可解群，证明了P. Kropholler的无限秩可解群刻画定理不存在无挠版本。
The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups $G$ with the following properties: (i) $G$ is finitely generated, (ii) $G$ is amenable, e.g. of intermediate growth, (iii) any finitely presented group $E$ with a quotient isomorphic to $G$ contains non-abelian free subgroups, or the stronger (iii') any finitely presented group with a quotient isomorphic to $G$ is large.
We show that the wreath product $G \wr \mathbb{Z}^n$ of any finitely generated group $G$ with $\mathbb{Z}^n$ has finite palindromic width. We also show that $C \wr A$ has finite palindromic width if $C$ has finite commutator width and $A$ is a finitely generated infinite abelian group. Further we prove that if $H$ is a non-abelian group with finite palindromic width and $G$ any finitely generated group, then every element of the subgroup $G' \wr H$ can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that $P \wr H$ has finite palindromic width if $P$ is a …
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.
It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup is abelian.
Model geometries of finitely generated groups
本文研究有限生成群的模型几何，证明了模型几何被对称空间、局部有限图或其乘积支配的条件，并刻画了可作为均匀格嵌入非紧致-by-全不连通局部紧群的有限生成群，进而分类了上同调维数为二和三的此类群。
A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We construct the JSJ decomposition of a vGBS group over abelian groups. We prove that this decomposition is explicitly computable, and may be obtained by local changes on the initial graph of groups.
A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable.
We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.
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 construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil\frac{n}{3}\rceil$.
Separability of unipotent-free abelian subgroups in linear groups
本文证明了在零特征域上的有限生成线性群中，无幂单阿贝尔子群具有可分性。
On the finite images of finitely generated perfect groups
研究猜想存在一个有限生成完全群可将所有有限 d 生成完全群作为同态像，并对具有有界长度次正规级数、因子为阿贝尔群或半单群的情形给出证明。
Intersection configurations in free and free times free-abelian groups
本文研究了自由群与自由乘自由阿贝尔群中的相交构型，证明了Howson性质是非交换自由群中的唯一障碍，并为FTFA群提供了判定多重相交有限生成性及计算基的算法，最终证明任何相交构型在足够大的FTFA群中均可实现。
For abelian groups $A, B$, $A$ is called $B$-small if the covariant functor $Hom(A,-)$ commutes with all direct sums $B^{(κ)}$ and $A$ is self-small provided it is $A$-small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.
The group A_{m} of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of A_{m}. We prove that the combined diagonal and tree-topological closure A* of A is additively a finitely presented Z_m [[x]]-module where Z_m is the ring of m-adic integers. Moreover, if A* is torsion-free then it is a finitely generated pro-m group. The group A splits over its torsion subgroup. We study in detail the case where A* corresponds to a cyclic Z_m[[x]]-module and when m is a prime number, we show A* to be conj…
In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth function is a rational function if the group has a geodesic automatic structure. We compute the growth function in the case where the group is abelian and see that the denominator of the rational function is determined from the rank of the group.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it is related to the embedding into a right-angled Coxeter group described by Davis and Januszkiewicz. We compare the approaches and also adapt the argument that the action extends to finite index supergroup that is a graph product of finite groups.
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