On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

A special class of nilmanifolds admitting an Anosov diffeomorphism

A nilmanifold admits an Anosov diffeomorphism if and only if its fundamental group (which is finitely generated, torsion-free and nilpotent) supports an automorphism having no eigenvalues of absolute value one. Here we concentrate on nilpotency class 2 and fundamental groups whose commutator subgroup is of maximal torsion-free rank. We prove that the corresponding nilmanifold admits an Anosov diffeomorphism if and only if the torsion-free rank of the abelianization of its fundamental group is greater than or equal to 3.

     

On quasivarieties of axiomatic rank 3 of torsion-free nilpotent groups

We study the lattice of quasivarieties of axiomatic rank at most 3 of torsion-free nilpotent groups of class at most 3. We prove that this lattice has cardinality of the continuum and includes a sublattice that is order isomorphic to the set of real numbers. Also we establish that the lattice of quasivarieties of axiomatic rank at most 2 of these groups is a 5-element chain.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Anosov Diffeomorphisms on Nilmanifolds of Dimension at Most Six

A complete classification of nilmanifolds of dimension smaller than or equal to six supporting Anosov diffeomorphisms is presented. This is obtained by solving the equivalent problem of determining the torsion-free nilpotent groups of rank at most six which admit hyperbolic automorphisms.     

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.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

Isomorfismi tra reticoli di sottogruppi normali di gruppi nilpotenti senza torsione

Riassunto In questo lavoro sono studiati gli isomorfismi tra reticoli di sottogruppi normali di gruppi nilpotenti senza torsione. Si prova che ogni gruppo iperciclico con la struttura normale di gruppo nilpotente senza torsione è un gruppo nilpotente sanza torsione.

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Summary Lattice isomorphisms between the normal structures of torsion-free nilpotent groups are studied in this paper. We prove that every hypercyclic group with the normal structure of a torsion-free nilpotent group is a torsion-free nilpotent group.

     

A nilpotent Freiman dimension lemma

We prove that a K$K$-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$K$. The result can be seen as a nilpotent analogue to Freiman’s dimension lemma.     

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 (c2)     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups. Received: 3 November 2005     

Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class

By the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal nilpotency class, if a finite p-group of order p ⁿ has nilpotency class n?1, then f has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in The Kourovka Notebook. Moreover, it is shown that there are no functions r(p) and l(p) such that any finite 2-generator p-group whose all factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most l(p) with quotient of rank at most r(p). The required examples of finite p-groups are constructed as quotients of torsion-free nilpotent groups which are abstract 2-generator subgroups of torsion-free divisible nilpotent groups that are in the Mal’cev correspondence with “truncated” Witt algebras.     

On stable torsion-free nilpotent groups

Summary We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z _k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.Supported by the Deutsche Forschungsgemeinschaft     

The arithmetical hierarchy of nilpotent torsion-free groups

Previously, N. Khisamiev proved that all {ie172-1} Abelian torsion-free groups are {ie172-2}. We prove that for the class of nilpotent torsion-free groups, the situation is different: even the quotient group F of a {ie172-3} nilpotent group of class 2 by its periodic part may fail to have a {ie172-4}. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 308–313, May–June, 1996.     

Computable torsion-free nilpotent groups of finite dimension

We find criteria for the computability (constructivizability) of torsion-free nilpotent groups of finite dimension. We prove the existence of a principal computable enumeration of the class of all computable torsion-free nilpotent groups of finite dimension. An example is constructed of a subgroup in the group of all unitriangular matrices of dimension 3 over the field of rationals that is not computable but the sections of any of its central series are computable.     

Polynome de Hilbert-Samuel dans les algebres enveloppantes et les algebres de groupe

An analogue of the Hilbert-Samuel polynomial is considered with respect to the augmentation ideal of the enveloping algebra of a finite dimensional nilpotent Lie algebra and the group ring of a finitely generated, torsion-free nilpotent group. Then, the Hilbert series of finitely generated modules are rational.      

Tamely uniform orders

The following two results are proven.

(i) Let G be a finitely generated torsion-free linear group. If every torsion-free section of G is an R-group, then G is soluble of finite rank. Conversely, if G has finite rank, then it has a subgroup of finite index, in which every torsion-free section is an R-group.

Let G be a finitely generated torsion-free soluble group. If in every torsion-free section of G the normalizer of each isolated subgroup is isolated, then G has finite rank. Conversely, if G has finite rank, then it has a subgroup K of finite index such that in every torsion-free section of K the normalizer of each isolated subgroup is isolated.