Torsion-free Nilpotent Research Articles

Matrix representations for torsion-free nilpotent groups by Deep Thought

We present an algorithm for calculating a representation by unitriangular matrices over the integers of a finitely-generated torsion-free nilpotent group given by a polycyclic presentation. The algorithm uses polynomials computed by the Deep Thought algorithm which describe the multiplication in the given group. The algorithm is more efficient than a previous method.

Wreath products and subgroups of division rings

The objective of this note is to further investigate the subgroup structure of the multiplicative group D ∗ ( G ) of the skew field, or division ring of fractions formed from the group ring of a finitely-generated torsion-free nilpotent group G. In particular, it will be shown that certain wreath products can be embedded in D ∗ ( G ) for some G.

On the Approximability by Finite p-Groups of Free Products of Groups with Normal Amalgamation

A sufficient condition for the residual p-finiteness (approximability by the class \(\mathcal{F}_p\) of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually \(\mathcal{N}\)-groups by \(\mathcal{F}_p\)-groups, where \(\mathcal{N}\) stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually \(\mathcal{F}_p\)-group), provided the quotient group G/HpH′ is residually p-finite.

On centrally orderable groups

In [V.V. Bludov, A.M.W. Glass, A.H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003) 225–239] we considered the class C of all orderable groups with all orders having central convex jumps and the class C 2 of all orderable groups all of whose two generator subgroups belong to C . Both these classes contain all locally nilpotent torsion-free groups. We proved that every soluble-by-finite group belonging to C 2 must be locally nilpotent, but there is a two-generator metabelian group belonging to C ∖ C 2 (whence it is not locally nilpotent). In this paper we consider the closure of C and C 2 under elementary equivalence, direct sums and full Cartesian products, homomorphic images, and residual properties. We further show that every metabelian group G which has a central order can be embedded in a metabelian group G ˜ belonging to C so that each central order on G can be extended to G ˜ . This is achieved by ascending HNN-extensions. We further show that a finitely generated metabelian group has a central order if and only if it is residually torsion-free nilpotent.

On [formula omitted]-orderable groups

We introduce the class of * -orderable groups as those groups G for which the complex group ring C G with the standard involution is * -orderable. In (Algebras and Representation Theory, to appear) the authors proved that residually torsion-free nilpotent groups are * -orderable and that every * -orderable group is orderable. We prove that being a * -orderable group is a local and residual property and deduce that the class of residually torsion-free nilpotent groups is strictly contained in the class of * -orderable groups. Further, it is proved that * -orderable groups are elementary (and even a quasi-variety) in the language of groups and contain certain normal series. In the last section, we give a countable family of orderable metabelian groups that are not * -orderable.

Extensions of characters of discrete torsion-free nilpotent groups

Extensions of characters of a discrete, torsion-free, nilpotent group to characters of some of its supergroups in its Mal'tsev completion are examined. Some sufficient conditions, as well as some necessary conditions, for extensibility are provided. In particular, the situation for nilpotent groups of class 2 is discussed. © de Gruyter 2005.

Generalized partition functions and subgroup growth of free products of nilpotent groups

SummaryWe estimate the growth of homomorphism numbers of a torsion-free nilpotent group

Zeta functions of groups and enumeration in Bruhat-Tits buildings

We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-groups in short. It is based on an enumeration of vertices in the Bruhat-Tits building for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. It enables us to give explicit computations for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-groups of class 2 with small centres and to derive local functional equations. Examples include formulae for nonuniform normal zeta functions.

ON RESIDUAL FINITENESS OF GRAPHS OF NILPOTENT GROUPS

Here we characterize the residually finite groups G which are the fundamental groups of a finite graph of finitely generated torsion-free nilpotent groups. Namely we show that G is residually finite if and only if for each edge group of the graph of groups the two edge monomorphisms differ essentially by an isomorphism of certain subgroups of the Mal'cev completion of the corresponding vertex groups.

Kirillov Theory for a Class of Discrete Nilpotent Groups

AbstractThis paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups G. G is assumed to be discrete, countable and π-radicable, with π containing the primes less than or equal to the nilpotence class of G. In addition, it is assumed that all of the characters of G have idempotent absolute value. Such groups are shown to be plentiful.

Characters of the discrete Heisenberg group and of its completion

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

On dimensions in Bredon homology

We define a homological and cohomological dimension of groups in the context of Bredon homology and compare the two quantities. We apply this to describe the Bredon-homological dimension of nilpotent groups in terms of the Hirsch-rank. In particular this implies that for virtually torsion-free nilpotent groups the Bredon cohomological dimension is equal to the virtual cohomological dimension.

Wreath Products and Universal Equivalence

We consider the question of preservation of universal equivalence for the cartesian and direct wreath products of lattice-ordered groups and groups. We prove that the basic rank is infinite of the quasivariety of torsion-free nilpotent groups of nilpotence length ≤c (c≥2)

ON CO-HOPFIAN NILPOTENT GROUPS

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed. 2000 Mathematics Subject Classification 17B30, 20F18 (primary).

Groups of homeomorphisms of one-manifolds III: nilpotent subgroups

This self-contained paper is part of a series (Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups. Preprint, 2001; Group actions on one-manifolds II: extensions of Hölder's Theorem. To appear in Trans. Amer. Math. Soc.) seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante and Thurston proved that every nilpotent subgroup of Diff2(S1) is abelian. One of our main results is a sharp converse: Diff1(S1) contains every finitely generated, torsion-free nilpotent group.

Valuation methods in group rings and in skew fields, I

In this first in a series of two papers we construct a family of discrete valuations in group rings of residually torsion-free nilpotent groups and extend these valuations to the Malcev–Neumann power series skew fields of these group rings. We apply these valuations to the study of these skew fields. In the second paper we will use our results and methods to study free fields and the universal fields of fractions K (( X )) of the Magnus power series ring.

Finite extensions and unipotent shadows of affine crystallographic groups

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class ⩽3, or if the polycylic rank of Γ is ⩽5, and also in some other cases. To cite this article: O. Baues, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 785–788.

Constructing Faithful Representations of Finitely-generated Torsion-free Nilpotent Groups

We formulate an algorithm for calculating a representation by unipotent matrices over the integers of a finitely-generated torsion-free nilpotent group given by a polycyclic presentation. The algorithm works along a polycyclic series of the group, each step extending a representation of an element of that series to the next element.

Isoperimetric functions of nilpotent groups amalgamated along a cyclic subgroup

We show that amalgams of finitely generated torsionfree nilpotent groups of class≤c along a cyclic subgroup satisfy a polynomial isoperimetric inequality of degree at most 4c2. The distortion of the amalgamated subgroup is bounded above by a polynomial of degree c.We also give an example of a non‐cyclic amalgam of finitely generated torsionfree nilpotent groups along an abelian, isolated and normal subgroup having an exponential isoperimetric inequality.

Continuous Quotients for Lattice Actions on Compact Spaces

Let Γ < SL n ( $$\mathbb{Z}$$ ) be a subgroup of finite index, where n ≥ 5. Suppose Γ acts continuously on a manifold M, where π1(M) = $$\mathbb{Z}$$ n , preserving a measure that is positive on open sets. Further assume that the induced Γ action on H 1(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ : M → $$\mathbb{T}$$ n that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.