Torsion-free Nilpotent Groups Research Articles

Torsion-free nilpotent groups of small Hirsch length with isomorphic finite quotients

Let T denote the class of finitely generated torsion-free nilpotent groups. For a group G let F(G) be the set of isomorphism classes of finite quotients of G. Pickel proved that if G∈T, then the set g(G) of isomorphism classes of groups H∈T with F(G)=F(H) is finite. We give an explicit description of the sets g(G) for the T-groups G of Hirsch length at most 5. Based on this, we show that for each Hirsch length n≥4 and for each m∈N there is a T-group G of Hirsch length n with |g(G)|≥m.

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the 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.

On the degree of polynomial subgroup growth of nilpotent groups

Let N be a finitely generated nilpotent group. The subgroup zeta function \(\zeta _N^{\scriptscriptstyle \leqslant }(s)\) and the normal zeta function \(\zeta _N^{\scriptscriptstyle \lhd }(s)\) of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence \(\alpha _N^{\scriptscriptstyle \leqslant }\) and \(\alpha _N^{\scriptscriptstyle \lhd }\), also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions \(N\mapsto \alpha _N^{\scriptscriptstyle \leqslant }\) and \(N\mapsto \alpha _N^{\scriptscriptstyle \lhd }\) when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic \(\mathbb {C}\)-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields.

Zeta functions enumerating normal subgroups of [formula omitted]-groups and their behavior on residue classes

Let G be a finitely generated torsion-free nilpotent class-2 group. We investigate how ζG,p◃(s), the local zeta function enumerating normal subgroups of p-power indices in G, varies on residue classes. We first show how the behaviors of local normal subgroup zeta functions ζG,p◃(s) and ζG×Zr,p◃(s) on residue classes are closely related for certain G. Then we show that for G of Hirsch length less than or equal to 7, under certain assumptions the normal subgroup zeta function of G is always a rational function on residue classes. We also show that there are examples of groups with Hirsch length 8 whose normal subgroup zeta function is not a rational function on residue classes.

Nilpotent groups with balanced presentations

Abstract We show that if a nilpotent group 𝐺 has a balanced presentation and Hirsch length h ⁢ ( G ) > 3 h(G)>3 , then β 1 ⁢ ( G ; Q ) = 2 \beta_{1}(G;\mathbb{Q})=2 . There is one such group which is torsion-free and of Hirsch length h = 4 h=4 , and none with h = 5 h=5 . We also construct a torsion-free nilpotent group 𝐺 with h ⁢ ( G ) = 6 h(G)=6 and β 2 ⁢ ( G ; F ) = β 1 ⁢ ( G ; F ) \beta_{2}(G;F)=\beta_{1}(G;F) for all fields 𝐹.

Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques.In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.

Independent axiomatizability of quasivarieties of torsion-free nilpotent groups

Independent axiomatizability of quasivarieties of torsion-free nilpotent groups

Probabilistic nilpotence in infinite groups

The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x1,…, xk+1 ∈ Gk+1 for which the simple commutator [x1, …, xk+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.

Independent Axiomatizability of Quasivarieties of Torsion-Free Nilpotent Groups

Independent Axiomatizability of Quasivarieties of Torsion-Free Nilpotent Groups

IA-automorphisms of relatively free nilpotent torsion-free groups and Lie algebras

For a free metabelian and nilpotent group G of finite rank n and class c, with and we show that for all and we extend this result to several torsion-free nilpotent varieties of groups. By our methods, which are based on a connection of automorphisms of groups with automorphisms of Lie algebras, we also show the analogous result for several relatively free nilpotent Lie algebras.

Plotkin’s Geometric Equivalence, Mal’cev’s Closure, and Incompressible Nilpotent Groups

In 1997, B. I. Plotkin introduced a concept of geometric equivalence of algebraic structures and posed a question: is it true that every nilpotent torsion-free group is geometrically equivalent to its Mal’cev’s closure? A negative answer in the form of three counterexamples was given by V. V. Bludov and B. V. Gusev in 2007. In the present paper, an infinite series of counterexamples of unbounded Hirsch rank and nilpotency degree is constructed.

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.

Free Group Algebras in Division Rings with Valuation II

AbstractWe apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

Translation-like actions of nilpotent groups

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. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension.

On torsion-free nilpotent loops

Abstract 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 nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

Free Symmetric and Unitary Pairs in the Field of Fractions of Torsion-Free Nilpotent Group Algebras

Let k be a field of characteristic different from 2 and let G be a nonabelian residually torsion-free nilpotent group. It is known that G is an orderable group. Let k(G) denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of G over k. If ∗ is an involution on G, then it extends to a unique k-involution on k(G). We show that k(G) contains pairs of symmetric elements with respect to ∗ which generate a free group inside the multiplicative group of k(G). Free unitary pairs also exist if G is torsion-free nilpotent. Finally, we consider the general case of a division ring D, with a k-involution ∗, containing a normal subgroup N in its multiplicative group, such that $G \subseteq N$, with G a nilpotent-by-finite torsion-free subgroup that is not abelian-by-finite, satisfying G∗ = G and N∗ = N. We prove that N contains a free symmetric pair.

Universal input of knapsack problem for groups

In this work we investigate the group version of the well known knapsack problem (KP) in class of nilpotent groups. We show that there exists the universal input for this problem. In other word it mean that there is the two step nilpotent group Gu and parametric input In, where n is natural parameter, such that for any torsion free nilpotent group G and input I for KP there exists a natural number m such that KP for group Gu on input Im is equivalent to KP for group G on input I.

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a1,..., an) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.

Polynomials describing the multiplication in finitely generated torsion-free 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.

A Necessary Condition for The Residual Nilpotence of HNN-Extensions

Let G be amultiple HNN-extension of a group A, and let all its associated subgroups be properly contained in some locally nilpotent subgroup of A. We prove that if G is residually nilpotent, then all the associated subgroups are p′-isolated in A for some prime p. Moreover, if q is a prime such that G is residually a q′-torsion-free nilpotent group, then p = q.