Elsevier

Journal of Algebra

Volume 582, 15 September 2021, Pages 1-25
Journal of Algebra

Certain residual properties of generalized Baumslag–Solitar groups

https://doi.org/10.1016/j.jalgebra.2021.05.001Get rights and content

Abstract

Let G be a generalized Baumslag–Solitar group, and let C be a class of groups containing at least one non-trivial group and closed under taking subgroups, extensions, and Cartesian products of the form yYXy, where X, YC and Xy is an isomorphic copy of X for every yY. We give a criterion for G to be residually a C-group provided C consists only of periodic groups. We also prove that G is residually a torsion-free C-group if C contains at least one non-periodic group and is closed under taking homomorphic images. These facts generalize and strengthen some known results. We also provide criteria for a GBS-group to be a) residually nilpotent; b) residually torsion-free nilpotent; c) residually free.

Introduction

A group is called a generalized Baumslag–Solitar group, or a GBS-group, if it is the fundamental group of a graph of groups with infinite cyclic vertex and edge groups. GBS-groups were actively studied recently [1], [5], [6], [7], [8], [9], [10], [16], [17]. Many of these investigations were focused on connections between the algebraic properties of GBS-groups and the structure of corresponding graphs. The aim of this paper is to describe residual properties of a GBS-group in terms of the associated graph of groups. We strengthen some known results (for example, on the residual finiteness and the residual p-finiteness of GBS-groups) and give a criterion for the residual nilpotence of a GBS-group.

Let C be a class of groups. A group G is said to be residually a C-group if, for any element gG{1}, there exists a homomorphism σ of G onto a group of C such that gσ1. The most commonly considered situation is when C is the class of all finite groups, all finite p-groups (where p is a prime number), all nilpotent groups or all solvable groups. In these cases, G is called residually finite, residually p-finite, residually nilpotent or residually solvable respectively.

We say that a class C of groups is root if it contains at least one non-trivial group and is closed under taking subgroups, extensions, and Cartesian products of the form yYXy, where X, YC and Xy is an isomorphic copy of X for every yY. The notion of a root class was introduced by Gruenberg [12], and the above definition is equivalent to that given in [12]; see [25] for details.

The classes of all finite groups, all finite p-groups, all periodic groups of finite exponent, all solvable groups, and all torsion-free groups are root. It is also easy to see that the intersection of any number of root classes is again a root class. At the same time, the classes of all nilpotent groups, all torsion-free nilpotent groups, and all finite nilpotent groups are not root because they are not closed under taking extensions.

The main goal of this paper is to get necessary and sufficient conditions for a GBS-group to be residually a C-group, where C is an arbitrary root class. The sense of studying residually C-groups, where C is an arbitrary class of groups, is to get many results at once using the same argument. This approach was originally proposed in [12], [21], and it turns out to be very fruitful in the study of free constructions of groups in the case when C is a root class; see, e.g. [2], [25], [26], [27], [28], [29], [30], [31], [32].

For a root class C and a GBS-group G, we give a criterion for G to be residually a C-group if C consists of only periodic groups (Theorem 3) and a sufficient condition for G to be residually a C-group if C contains at least one non-periodic group (Theorem 4). Using the first of these results, we prove criteria for a GBS-group to be a) residually nilpotent (Theorem 5); b) residually torsion-free nilpotent and residually free (Theorem 6). All the proofs use only the classical methods of combinatorial group theory and the basic concepts of graph theory.

Section snippets

Statement of results

Recall that an (ordinary) Baumslag–Solitar group is a group with the presentationBS(m,n)=a,b;a1bma=bn, where m and n are non-zero integers. Since BS(m,n), BS(n,m), and BS(m,n) are pairwise isomorphic, we can assume without loss of generality that |n|m>0.

We also recall that if ρ is a set of primes, then a ρ-number is an integer, all of whose prime divisors belong to ρ, and a ρ-group is a periodic group, the orders of all elements of which are ρ-numbers. If ρ consists of one number p, then

Some auxiliary statements

Throughout this section, if C is a class of groups consisting only of periodic groups, then ρ(C) denotes the set of primes defined above.

Proposition 2.1

If C is a class of groups consisting only of periodic groups and closed under taking subgroups and extensions, then any finite solvable ρ(C)-group belongs to C.

Proof

Let X be a finite solvable ρ(C)-group. Then there is a polycyclic series S in X such that the orders of all its factors belong to ρ(C). Let p be the order of some factor F. By the definition of ρ(C), p

The fundamental group of a graph of groups

Let Γ be a non-empty undirected graph with a vertex set V and an edge set E (loops and multiple edges are allowed). To turn Γ into a graph of groups, we denote the vertices of Γ that are the ends of an edge eE by e(1), e(1) and assign to each vertex vV some group Gv, to each edge eE a group He and injective homomorphisms φ+e:HeGe(1), φe:HeGe(1). We denote the resulting graph of groups by G(Γ), the subgroups Heφ+e and Heφe (eE) by H+e and He. We also call Gv (vV), He (eE), and Hεe (e

GBS-groups and their properties

It follows from the previous section that each GBS-group can be defined by a labeled graph L(Γ) for some finite connected graph Γ and vice versa, each labeled graph L(Γ) over a non-empty finite connected graph Γ defines some GBS-group. Until the end of the paper, we assume that Γ=(V,E) is a non-empty finite connected graph with a vertex set V and an edge set E, L(Γ) is a graph with labels λ(εe) (eE, ε=±1), and G is the corresponding GBS-group with the vertex groups Gv=gv (vV) and the edge

Proofs of Theorems 3, 4 and Corollaries 1–3

Proposition 5.1

Suppose that G is non-elementary, T is a maximal subtree of Γ, and L(Γ) is T-positive. Suppose also that K=eE,ε=±1Hεe and μ is the least common multiple of μ(v)=[Gv:K] (vV). If Q is the subring of Q generated by Im Δ, Q+ is the additive group of Q, A is a free abelian group with the basis {aq|qImΔ}, and X is the splitting extension of Q+ by A such that the automorphism aqˆ|Q+ acts as multiplication by q, then the mapping of the generators of G to X given by the rulegvμ/μ(v)(vV),teaΔ(te)(e

An algorithm for verifying the condition of Theorem 5

Let E be the set of paths in Γ. We define a function ξ:E{1,1} as follows. If eE, then ξ(e)=signλ(+e)λ(e). If s=(e1,e2,,en) is a path in Γ, then ξ(s)=i=1nξ(ei). In particular, if the length of s is equal to 0, then ξ(s)=1.

The algorithm given below assigns labels to the vertices of Γ. The label corresponding to a vertex v is denoted by ζ(v) and can be equal to ±1. Initially, all the vertices are unlabeled.

Algorithm

1. If the graph has no labeled vertices, take an arbitrary vertex v of Γ. Otherwise,

Proof of Theorems 5 and 6

Proposition 7.1

Suppose that G is non-solvable and L(Γ) is reduced. If G is residually nilpotent, then all the labels λ(εe) (eE, ε=±1) are p-numbers for some prime number p.

Proof

Since L(Γ) is reduced, then |λ(+e)|1|λ(e)| for each edge e that is not a loop. Let us show that these relations can be assumed to hold for all loops of L(Γ).

By Proposition 2.5, G is residually finite, and, by Theorem 3, ImΔ{1,1}. It follows that L(Γ) cannot contain a loop e such that |λ(εe)|=1|λ(εe)| for some ε=±1. Let Γ be the

Acknowledgements

The author would like to thank F. A. Dudkin (Sobolev Institute of Mathematics, Russia) for the introduction to modern studies of generalized Baumslag–Solitar groups.

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