Torsion-free Groups Research Articles

Semi-generalized Bassian groups

Abstract As a common non-trivial generalization of the concept of a (proper) generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of 𝑝-torsion groups, torsion-free groups and splitting mixed groups their complete description. Moreover, we classify the groups with the clearly related property that every subgroup is essential in a direct summand of the whole group.

Weak [formula omitted]-structures for some combinatorial group constructions

Bestvina [1] introduced the notion of a (weak) Z-structure and (weak) Z-boundary for a torsion-free group, motivated by the notion of boundary for hyperbolic and CAT(0) groups. Since then, some classes of groups have been shown to admit a (weak) Z-structure (see [5,20,22] for example); in fact, in all cases these groups are semistable at infinity and happen to have a pro-(finitely generated free) fundamental pro-group. The question whether or not every type F group admits such a structure remains open. In [33] it was shown that the property of admitting such a structure is closed under direct products and free products. Our main results are as follows.THEOREM: Let G be a torsion-free and semistable at infinity finitely presented group with a pro-(finitely generated free) fundamental pro-group at each end. If G has a finite graph of groups decomposition in which all the groups involved are of type F and inward tame (in particular, if they all admit a weak Z-structure) then G admits a weak Z-structure.COROLLARY: The class of those 1-ended and semistable at infinity torsion-free finitely presented groups which admit a weak Z-structure and have a pro-(finitely generated free) fundamental pro-group is closed under amalgamated products (resp. HNN-extensions) over finitely generated free groups.On the other hand, given a finitely presented group G and a monomorphism φ:G⟶G, we may consider the ascending HNN-extension G⁎φ=〈G,t;t−1gt=φ(g),g∈G〉. The results in [26] together with the Theorem above yield the following:PROPOSITION: If a finitely presented torsion-free group G is of type F and inward tame, then any (1-ended) ascending HNN-extension G⁎φ admits a weak Z-structure.In the particular case φ∈Aut(G), this ascending HNN-extension corresponds to a semidirect product G⋊φZ, and it has been shown in [18] that if G admits a Z-structure then so does G⋊φZ.

Integrals of groups. II

AbstractAn integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study: A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2). The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1). Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5). The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7). Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1). Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2). We end the paper with a number of open problems.

Units, zero-divisors and idempotents in rings graded by torsion-free groups

Abstract The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and not to have interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $ G $ is a torsion-free group or a finite group of odd order, we prove that $ G $ has a Lee metric if and only if $ G $ is cyclic. Also, if $ G $ is a group admitting Lee metrics then $ G \times \mathbb{Z}_2^k $ always has Lee metrics for every $ k \in \mathbb{N} $. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $ \le 31 $ indicating which of them have (or have not) Lee metrics and why (not).

On invariant ideals in group rings of torsion-free minimax nilpotent groups

Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.

Complicated generalized torsion elements in Seifert fibered spaces with boundary

In a group, a nontrivial element is called a generalized torsion element if some non-empty finite product of its conjugates is equal to the identity. There are various examples of torsion-free groups which contain generalized torsion elements. We can define the order of a generalized torsion element as the minimum number of its conjugates required to generate the identity. In previous works, three-manifold groups which contain a generalized torsion element of order two are determined. However, there are few previous studies that examine the order of a generalized torsion element bigger than two. In this paper, we focus on Seifert fibered spaces with boundary, including the torus knot exteriors, and construct concretely generalized torsion elements of order [Formula: see text], [Formula: see text], [Formula: see text] and others in their fundamental groups.

Small sets without unique products in torsion-free groups

In this paper, we report on a computation demonstrating that if [Formula: see text] and [Formula: see text] are nonempty subsets of a torsion-free group such that [Formula: see text] has no unique product, then [Formula: see text]. Moreover, this bound is sharp, as there are examples where [Formula: see text], and in fact [Formula: see text] for at least two such examples. More generally, when [Formula: see text] is small, we find lower bounds on [Formula: see text]. One consequence of this work is that any counterexample to Kaplansky’s zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups.

ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

AbstractThe aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$ , with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case.

On the Question of the Definability of Certain Classes of Completely Decomposable Abelian Torsion-Free Groups by Their Homomorphism Groups

On the Question of the Definability of Certain Classes of Completely Decomposable Abelian Torsion-Free Groups by Their Homomorphism Groups

ADS Abelian groups

The subgroup [Formula: see text] is absolute direct summand (ADS) if, for every [Formula: see text]-high subgroup [Formula: see text] (i.e. maximal with respect to the property [Formula: see text]), we have [Formula: see text], and [Formula: see text] itself is an ADS group if all of its summands inherit this property. In this paper, we characterize the structure of ADS abelian groups, which can be defined as groups such that members of the decomposition into two direct summands are mutually injective. In particular, an ADS abelian group is either divisible, or a direct sum of an indecomposable torsion-free group and a divisible torsion group, or a torsion group such that each [Formula: see text]-component is a direct sum of cyclic [Formula: see text]-groups of the same length or of Prüfer groups.

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

Influence of the Baer–Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem. The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.

Solving singular equations of length eight over torsion-free groups

<abstract><p>It was demonstrated by Bibi and Edjvet in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> that any equation with a length of at most seven over torsion-free group can be solvable. This corroborates Levin's <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> assertion that any equation over a torsion-free group is solvable. It is demonstrated in this article that a singular equation of length eight over torsion-free groups is solvable.</p></abstract>

Weak Z-structures and one-relator groups

Motivated by the notion of boundary for hyperbolic and C A T ( 0 ) groups, Bestvina [2] introduced the notion of a (weak) Z -structure and (weak) Z -boundary for a group G of type F (i.e., having a finite K ( G , 1 ) complex), with implications concerning the Novikov conjecture for G . Since then, some classes of groups have been shown to admit a weak Z -structure (see [15] for example), but the question whether or not every group of type F admits such a structure remains open. In this paper, we show that every torsion free one-relator group admits a weak Z -structure, by showing that they are all properly aspherical at infinity; moreover, in the 1-ended case the corresponding weak Z -boundary has the shape of either a circle or a Hawaiian earring depending on whether the group is a virtually surface group or not. Finally, we extend this result to a wider class of groups still satisfying a Freiheitssatz property.

Multiplications on torsion-free groups of finite rank

Multiplications on torsion-free groups of finite rank

On the Question of the Definability of Certain Classes of Completely Decomposable Abelian Torsion-Free Groups by Their Homomorphism Groups

Пусть $C$ - абелева группа. Класс $X$ назовем $_{C}H$-классом, если для любых групп $A,B\in X$ из изоморфизма групп $\operatorname{Hom}(C,A)$ и $\operatorname{Hom}(C,B)$ следует изоморфизм групп $A$ и $B$. В статье исследуются условия, которым должна удовлетворять вполне разложимая абелева группа $C$, чтобы класс некоторых вполне разложимых абелевых групп без кручения был $_{C}H$-классом. Библиография: 8 названий.

On Gardam's and Murray's units in group rings

We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.

On invariant von Neumann subalgebras rigidity property

We say that a countable discrete group Γ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every Γ- invariant von Neumann subalgebra M in L(Γ) is of the form L(Λ) for some normal subgroup Λ◁Γ. We show many “negatively curved” groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first L2-Betti number under a mild assumption, and certain finite direct product of them have this property. We also discuss whether the torsion-free assumption can be relaxed.

On structure analysis of finitely generated abelian groups

In this paper we give an overview of the structural analysis of finitely generated abelian groups and modules. We give an overview of recent results on the structure theory of these objects in various situations, in particular in the case of torsionfree groups of infinite rank. We also mention several open problems. We also propose a method to study the structure of finitely generated abelian groups and finitely generated nilpotent groups by using some invariants of their derived series. This method provides a new framework for studying the structure of finitely generated abelian groups and finitely generated nilpotent groups.