Certain residual properties of generalized Baumslag–Solitar groups
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 be a class of groups. A group G is said to be residually a -group if, for any element , there exists a homomorphism σ of G onto a group of such that . The most commonly considered situation is when 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 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 , where X, and is an isomorphic copy of X for every . 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 -group, where is an arbitrary root class. The sense of studying residually -groups, where 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 is a root class; see, e.g. [2], [25], [26], [27], [28], [29], [30], [31], [32].
For a root class and a GBS-group G, we give a criterion for G to be residually a -group if consists of only periodic groups (Theorem 3) and a sufficient condition for G to be residually a -group if 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 presentation where m and n are non-zero integers. Since , , and are pairwise isomorphic, we can assume without loss of generality that .
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 is a class of groups consisting only of periodic groups, then denotes the set of primes defined above.
Proposition 2.1 If is a class of groups consisting only of periodic groups and closed under taking subgroups and extensions, then any finite solvable -group belongs to .
Proof Let X be a finite solvable -group. Then there is a polycyclic series in X such that the orders of all its factors belong to . Let p be the order of some factor F. By the definition of , 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 by , and assign to each vertex some group , to each edge a group and injective homomorphisms , . We denote the resulting graph of groups by , the subgroups and () by and . We also call (), (), and (
GBS-groups and their properties
It follows from the previous section that each GBS-group can be defined by a labeled graph for some finite connected graph Γ and vice versa, each labeled graph over a non-empty finite connected graph Γ defines some GBS-group. Until the end of the paper, we assume that is a non-empty finite connected graph with a vertex set V and an edge set E, is a graph with labels (, ), and G is the corresponding GBS-group with the vertex groups () 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 is T-positive. Suppose also that and μ is the least common multiple of (). If Q is the subring of generated by Im Δ, is the additive group of Q, A is a free abelian group with the basis , and X is the splitting extension of by A such that the automorphism acts as multiplication by q, then the mapping of the generators of G to X given by the rule
An algorithm for verifying the condition of Theorem 5
Let be the set of paths in Γ. We define a function as follows. If , then . If is a path in Γ, then . In particular, if the length of s is equal to 0, then .
The algorithm given below assigns labels to the vertices of Γ. The label corresponding to a vertex v is denoted by 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 is reduced. If G is residually nilpotent, then all the labels (, ) are p-numbers for some prime number p.
Proof Since is reduced, then for each edge e that is not a loop. Let us show that these relations can be assumed to hold for all loops of . By Proposition 2.5, G is residually finite, and, by Theorem 3, . It follows that cannot contain a loop e such that for some . Let 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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