In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s) groups is Noetherian. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group, is equivalent to the problem of classification of complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism.
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.
We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotency of the free loop proved by Higman, it is proved, essentially, by the same method.
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.
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 $G$
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…
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.
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…
In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin \cite{P1, P2}. Sufficient and necessary conditions are presented for a $π$-torsion-free nilpotent group to be geometrically equivalent to its $π$-completion. We prove that a relatively free nilpotent $π$-torsion-free group and its $π$-completion define the same quasi-variety. Examples of $π$-torsion-free nilpotent groups that are geometrically equivalent to their $π$-completions are given.
The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent. We show here that the torsion factor is nilpotent too, and hence that every locally pro-p contraction group is nilpotent.
A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5.
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.
We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related issues. We comment briefly on some of the connections between these papers.
The variety of nilpotent groups is Noetherian. That is why two nilpotent class s groups are geometrically equivalent if and only if they have same quasi-identities ([Pl3]). Therefore, we can describe classes of geometrical equivalence of nilpotent groups instead of describing quasivarieties generated by single nilpotent group. The problem of geometrical equivalence can be researched by the technique of approximation of groups ([Pl2]). In this paper only nilpotent torsion free groups are considered. In the first part two sufficient conditions are presented (Theorem 1 and Theorem 2) when the nil…
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [4]. Later, this result was extended to all abelian groups [3] and, recently, to all torsion finitely quasihamiltonian groups [7]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [8]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
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…
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.
For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in class group of every $G_p$-extension. More generally, for every nilpotent group $G$ where every Sylow-$p$ subgroup $G_p\subset G$ is non-cyclic and non-quaternion, we prove a non-trivial bound on $\ell$-torsion in class group of every $G$-extension for every integer $\ell>1$.
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 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.
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 consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $Γ$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $μ$, we construct a homogeneous nilpotent Lie group $G_\bullet(Γ,μ)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on th…
To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of cl…
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…
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.
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.
In this paper we study the residual nilpotence of groups defined by basic commutators. We prove that the so-called Hydra groups as well as certain of their generalizations and quotients are, in the main, residually torsion-free nilpotent. By way of contrast we give an example of a group defined by two basic commutators which is not residually torsion-free nilpotent.
Let $G$ be a random torsion-free nilpotent group generated by two random words of length $\ell$ in $U_n(\mathbb{Z})$. Letting $\ell$ grow as a function of $n$, we analyze the step of $G$, which is bounded by the step of $U_n(\mathbb{Z})$. We prove a conjecture of Delp, Dymarz, and Schafer-Cohen, that the threshold function for full step is $\ell = n^2$.
We prove that Knapsack problem (KP) is undecidable for any group of nilpotency class two if the number of generators (without torsion) of the derived subgroup is at least 322. This result together with the fact that if KP is undecidable for a subgroup then it undecidable for the whole group allows us extend our result to certain classes of polycyclic groups, linear groups and nilpotent groups of higher nilpotency class.
We solve the following algorithmic problems using TC0 circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal'cev coordinates the algorithms run in quartic time.
We show that torsion-free finitely generated nilpotent groups are characterised by their group C*-algebras and we additionally recover their nilpotency class as well as the subquotients of the upper central series. We then use a C*-bundle decomposition and apply K-theoretic methods based on noncommutative tori to prove that every torsion-free finitely generated 2-step nilpotent group can be recovered from its group C*-algebra.
We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.
A virtual endomorphism of a group G is a homomorphism f from H into G where H is a subgroup of G of finite index m. The triple (G,H,f) produces a state-closed (or, self-similar) representation t of G on the 1-rooted m-ary tree. This paper is a study of properties of the image G^t when G is nilpotent. In particular, it is shown that if G is finitely generated, torsion-free and nilpotent then G^t has solvability degree bounded above by the number of prime divisors of m.
Let $G$ be a split reductive algebraic group defined over a complete discrete valuation ring $\mathbb{O}$, with residue field $\mathbb{F}$ and fraction field $\mathbb{K}$, where the fiber $G_{\mathbb{F}}$ is geometrically standard. A balanced nilpotent section $x \in \text{Lie}(G)$ can roughly be thought of as an $\mathbb{O}$-point in a $\mathbb{K}$ nilpotent orbit such that the corresponding orbits over $\mathbb{K}$ and $\mathbb{F}$ have the same Bala--Carter label. In this paper, we will establish a number of results on the structure of the centralizer $G^x \subseteq G$ of $x$. This includes…
We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group.
We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.
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…
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.
We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.
In this paper we study conjugacy and subgroup separability properties in the class of nilpotent $\mathbb{Q}[x]$-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over naturally to this more general class. Among other results, we offer a generalization of a theorem due to G. Baumslag. The generalized version states that if $G$ is a finitely $\mathbb{Q}[x]$-generated $\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered group and $H$ is a $\mathbb{Q}[x]$-isolated subgroup of $G,$ then for any prime $π\in \mathbb{Q}[x]…
We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing $\ell^{q,p}$ cohomology.
Let $G$ be a finitely generated torsion-free nilpotent group and $ϕ:H\rightarrow G$ be a surjective homomorphism from a subgroup $H<G$ of finite index with trivial $ϕ$-core. For every choice of coset representatives of $H$ in $G$ there is a faithful self-similar action of the group $G$ associated with $(G,ϕ)$. We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for $(G,ϕ)$. These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $\widehatϕ$ of the Lie algebra of …
In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful actions of mapping class groups of surfaces. We will draw connections between the structure of orderings on residually torsion-free nilpotent, hyperbolic groups and their Gromov boundaries, and we s…
Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. F…
Let $Γ$ be a finitely generated torsion free nilpotent group, and let $A^ω$ be the space of infinite words over a finite alphabet $A$. We investigate two types of self-similar actions of $Γ$ on $A^ω$, namely the faithfull actions with dense orbits and the free actions. A criterion for the existence of a self-similar action of each type is established. Two corollaries about the nilmanifolds are deduced. The first involves the nilmanifolds endowed with an Anosov diffeomorphism, and the second about the existence of an affine structure. Then we investigate the virtual actions of $Γ$, i.e. actions…
