In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.
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 describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.
If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there exists a monomorphism of $G$ into $H$ which induces isomorphisms between the corresponding quotients of their lower central series. We first consider residually nilpotent groups and find sufficient conditions on the monomorphism so that $H$ is para-$G.$ We then prove that for certain polycyclic groups, if $H$ is para-$G$, then $G$ and $H$ have the same Hirsch l…
In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$
Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-fre…
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results e…
The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of $F_{N}(n)$ is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. Moreover, we introduce a class …
Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$ encoding, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. We show that the domains of convergence and meromorphic continuation of these zeta functions of groups $\mathbf{G}(\mathcal{O})$ are independent of the number fi…
The authors extend to the $q-$tensor square $G \otimes^q G$ of a group $G$, $q$ a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square $G \otimes G$ ($q = 0$). The results are applied to the computation of $G \otimes^q G$ for finitely generated nilpotent groups $G$, specially for free nilpotent groups of finite rank. They also generalize to all $q \geq 0$ results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square $G \otimes G$ when $G$ is a $n-$generator nilpo…
In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.
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 the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over $\mathbb{Q}$ and we deduce from this that they have rational abscissa of convergence and some meromorphic continuation. We also define Mal'cev completions of a finitely generated virtually nilpotent group and we prove that the subgroup growth and the normal subgroup growth of the latter are invariants of its $\mathbb{Q}$-Mal'cev completion.
In the paper we show that any irreducible representation of a finitely generated nilpotent group $G$ over a finitely generated field $F$ of characteristic zero is induced from a primitive representation of some subgroup of $G$.
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 prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy…
The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebra…
In this paper we consider the palindromic width of free nilpotent groups. In particular, we prove that the palindromic width of a finitely generated free nilpotent group is finite. We also prove that the palindromic width of a free abelian-by-nilpotent group is finite.
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 $\…
We obtain sufficient criteria for endomorphisms of torsion-free nilpotent groups of finite rank to be automorphisms, by considering the induced maps on the torsion-free abelianisation and the centre. Whilst these results are known in the finitely generated case removing this assumption introduces several difficulties.
We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.
We show that, for a finitely generated residually finite group $Γ$, the word $[x_1, \ldots, x_k]$ is a probabilistic identity of $Γ$ if and only if $Γ$ is virtually nilpotent of class less than $k$. Related results, generalizations and problems are also discussed.
In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of finite groups and their factor groups are given.
We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as $SL_n(\mathbb{Z})$. In the context of finite nilpotent quotients, we find a new characterization of nilpotent groups.
It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in Nickel. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented by Nickel.
We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency $|A| - |R| $ is at least $2$ then the group $G$ is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations $\langle A \mid R\ra…
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely generated we prove using a classical result of Julia Robinson that one can obtain a central series for $G$ where the action of the ring of integers $\Z$ on the quotients of each of the consecutive terms of the series except for one very specific gap, called the special gap, is interpretable in $G$. Then we use a refinement of this central series to give a c…
In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words represent the same element in a free nilpotent group of rank $r$ and class $k$ by analyzing their behavior on the Cayley graph of the free nilpotent group of rank $r$ and class $k-1$.
We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete automorphism group of the direct product.
Metric estimates are quantities that approximate the word metric of a finitely presented group up to multiplicative constants. In this paper, they are computed for some nilpotent groups and used to compute the distortion functions of several embeddings between them.
We offer in this note a self-contained proof of the fact that a finitely generated group is not virtually nilpotent if and only if it has a quotient with the infinite conjugacy class (ICC) propoerty. This proof is a modern presentation of the original proof, by McLain (1956) and Duguid and McLain (1956).
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.
We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are need to separate pairs of distinct conjugacy classes of bounded w…
In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS, a result which may be of independent interest though not directly applicable to 3-manifold topology. As a corollary, we deduce that any RFRS group cannot contain a nonabelian torsion-free nilpotent group. This result a…
Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such embeddings. In this work, we show that if $UT_n(Z)$ is embedded into $UT_N(Z)$ using Nickel's algorithm, then $N \geq 2^{n/2 -2}$ if the standard ordering of the Mal'cev basis (as in Nickel's original paper) is used. In particular, we establish an exponential worst-case running time of Nickel's algorithm. On the other hand, we also prove a general exponenti…
