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 also illustrates some of the interplay between residual torsion-free nilpotence and the RFRS condition.
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…
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.
In this paper we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these kinds of structures are often residually solvable.
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.
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 $G$
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.
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…
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 $G$.
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…
A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor…
A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces an injective endomorphism phi of the Lie algebra g associated to the Lie group in which N is a lattice. A normal form for automorphisms of nilpotent Lie algebras--permuted absolute Jordan form-- is defined and conjectured to be a quasi-isometry invariant. We show that if phi, th…
We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.
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 …
We show that the pure cactus group $Γ_{n+1}$ is residually nilpotent and exhibit a surjective homomorphism $Γ_{n+1} \to (\mathbb{Z}/2\mathbb{Z})^{2^n-n(n+1)/2-1}$ whose kernel is residually torsion-free nilpotent.
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…
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.
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…
We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent …
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.
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…
We prove that for a finitely generated infinite nilpotent group G with a first order structure (G,*,...), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals $\bigcap_{n\in\N} {g^n : g\in G^*}$. We construct a first order expansion of Z by a predicate (Z,+,P) such that the type-connected component Z*00_{\emptyset} is strictly smaller than Z*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem.
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.
A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n \times Z$)…
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…
Let $G$ be a finite group with the property that if $a,b$ are powers of $δ_1^*$-commutators such that $(|a|,|b|)=1$, then $|ab|=|a||b|$. We show that $γ_{\infty}(G)$ is nilpotent.
A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in certain families of nilpotent groups, e.g. the finitely generated Heisenberg groups, groups of unipotent matrices over the integers and the free class 2 nilpotent groups. Further it is shown that any polycyclic-by-finite…
We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.
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.
We prove that pure braid groups of closed surface are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a Corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent.
We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: $F_n \rtimes_φ \mathbb{Z}$. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for $n=2$ the length of the…
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.
Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $ζ_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $ζ_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of …
We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a finite product of limit groups, we discuss the possibility of finding an explicit set of defining equations (i.e. of expressing G as the maximal residually free quotient of an explicit finitely presented group).
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.
We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension $2$, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension $2$, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.
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…
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
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…
We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there…
Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of …
For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…
Let G be a finitely generated group with a given word metric. The asymptotic density of elements in G that have a particular property P is defined to be the limit, as r goes to infinity, of the proportion of elements in the ball of radius r which have the property P. We obtain a formula to compute the asymptotic density of finite-order elements in any virtually nilpotent group. Further, we show that the spectrum of numbers that occur as such asymptotic densities consists of exactly the rational numbers in [0,1).
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 consider the fundamental group $π$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $π$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that $π$ has infinite diameter with respect to this set. Furthermore, we give a general criterion for a finitely generated group equipped with a generating set to have this property.
We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group $G$ is bounded by $2\cdot h(G)!-1$, where $h(G)$ is the Hirsch length of $G.$ This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent grou…
