FREE GROUPS

Let x be a generating subset of a group G. Certain products of members of X and their inverses will be 1 whatever X and G are; for instance, xyyz−1zy−1y−1x−1. Other products, such as xyz or xx, will be 1 for some choices of X and G but not for other choices. Those pairs G and X for which a product of elements in X ∪ X−1 is 1 only when the properties holding in all groups require it to be 1 are obviously of interest.

They are called free groups; a more formal definition will be given later. If G is such a group, any function f from x to a group H can be extended uniquely to a homomorphism from G to H. For any g ∈ G can be written as xi1ε1 … xin εn where εt = ±1 and xir ε X for r = 1, …, n. Now suppose that g can also be written as xj1δ1 … xjm, δm where δs = ±1 and xjs ε X for s − 1, …, m. Then

and our assumption on G and X then tells us we must have

Hence the element of H given by (xi1 ƒ)ε1 … (xin ƒ)εn depends only on g and not on how g is written as a product of elements of X ∪ X−1. I t follows that we can define a function φ:G → H by requiring gφ to be this element. It is easy to check that φ is a homomorphism and that xφ − xƒ for all x ε X.

In this chapter we will consider a few other topics in combinatorial group theory where the methods have a topological flavour.

SMALL CANCELLATION THEORY

Small cancellation theory is one of the major aspects of combinatorial group theory. The methods are somewhat more geometrical than topological (insofar as it is possible to make such a distinction). I have something of a blind spot in this area, so I only summarise the results. For details see Lyndon and Schupp (1977). The paper by Greendlinger and Greendlinger (1984) simplifies one of the proofs given there.

Suppose that, in F(X), we have w uiriui-1, where w and each ui and ri are reduced. Then it is possible to make a diagram in the plane, composed of regions, edges, and vertices, with each edge being given a label from X∪X-1, in such a way that the boundary of the whole diagram is a sequence of edges whose label (up to cyclic permutation) is w, while there are m regions, whose boundaries consist of sequences of edges whose labels are (up to cyclic permutation) the ri.

Let R be a subset of F= F(X), and let N-<R>F. We will assume that R is symmetrised; that is, that if r∈R then all cyclic permutations of r and of r-1 are in R.

We define a piece (of R) to be an element u of F such that there are distinct r1 and r2 in R with r1 - uv1 and r2 = uv2 both reduced as written (r2 is permitted to be r1 or a cyclic permutation of r1, or r1-1).

SOME POINT-SET TOPOLOGY

We begin with a lemma that will be frequently used, often without explicit mention.

Lemma 1 (Glueing Lemma) (i) Let X and Y be sets, let Xαbe subsets of X such that X-∪Xα, and let fα:Xα → Y be functions such that fα|Xα∩Xβ – fβ|Xα∩Xβfor all α and β. Then there is a unique function f:X→Y such that f|Xα – fαfor all α.

(ii) Let the conditions of (i) hold, and let X and Y be topological spaces. Suppose that each fαis continuous (when Xαis given the subspace topology). Suppose either that there are only finitely many sets Xαeach of which is a closed subspace of X or that each Xαis an open subspace of X. Then f is continuous.

Remark When f is a function from a set x to a set Y and A is a subset of X the notation f\A means the restriction of f to A; that is, the function from A to Y whose value on a ∈ A is fa.

Proof (i) Let S⊆XxY be {(x,y); there is α such that x∈Xα and y-xfα). Since X-⊃Xα, for every x there is at least one y with (x,y)∈S. Suppose that (x,y)∈S and (x,y)∈S. Then there are α and β with x∈Xα, y - xfα, and x∈Xβ, z - xfβ. Since fsub>α - fβ on fsub>α ∩ fβ, by hypothesis, it follows that y - z.

Combinatorial group theory can be regarded as that branch of group theory which considers groups given by generators and relations. Some of its basic results involve manipulation with words; that is, products of the generators and their inverses. It is this aspect to which the word “combinatorial” refers; it is not connected with that branch of mathematics known as combinatorics.

From its earliest stages this theory has been closely connected with topology. To any topological space there is an associated group, called the fundamental group of the space. In trying to investigate properties of certain spaces we are led to problems in combinatorial group theory. Conversely, some problems in combinatorial group theory are best solved by geometric and topological discussions of suitable fundamental groups.

It is this interplay between group theory and topology which is the theme of this book. The texts on combinatorial group theory by Magnus, Karrass, and Solitar (1966) and by Lyndon and Schupp (1977) go much deeper into the group theory, but have little to say about the topology, while the texts by Massey (1967) and Stilwell (1980) concentrate on the topology rather than the group theory.

Chapter 1 contains the main constructions of combinatorial group theory; free groups, presentations, free products, amalgamated free products, and the HNN extension. The various Normal Form Theorems are proved, with several different proofs, and applications of the constructions are made (for instance, to show the existence of a finitely generated infinite simple group). Chapter 2 has some topological preliminaries; ways of building new spaces from old ones, and a discussion of paths in spaces.

GROUPOIDS

At the end of the previous chapter we had a brief indication that it might be useful to consider objects similar to groups, but where the product of two elements is not always defined. In this section we consider these objects in detail.

A partial multiplication on a set G is a function from some subset X of G × G to G. If (x, y)∈X we denote the value of the function on (x, y) by xy or by x.y. We say that xy is defined to mean that (x, y)∈X.

The element e is an identity for a partial multiplication if ex - x whenever ex is defined and also ye - y whenever ye is defined. There may be many identities, but it is clear from the definition that if e and f are identities with ef defined then e - f.

Definition A groupoid is a set G with a partial multiplication such that:

1. (associative law) if one of (ab)c and a(bc) is defined then so is the other and they are equal; also, if both ab and bc are defined then (ab)c is defined,

2. (existence of identities) for any a, there are identities e and f with ea and af defined,

3. (existence of inverses) for any a, and e and f as in (2), there is an element a−1 such that aa−1 - eand a−1a - f.

We nearly always want our groupoids to be non-empty. I leave it to the reader to decide which properties stated should have the empty groupoid given as an exception.

DECISION PROBLEMS IN GENERAL

Can we tell whether or not an element of a free group is in the commutator subgroup? Whether or not it is a commutator? Given a finite presentation, can we tell whether or not an element of the free group is 1 in the group presented? Can we tell whether or not two finite presentations present isomorphic groups?

These questions, and other similar ones, are obviously of interest. We shall see later that the first two questions have the answer “Yes”, while the other two have the answer “No”. The techniques used are those of combinatorial group theory, but have no specific connection with the topological approach. I include this chapter because I find the material particularly interesting.

In order to make these questions precise, we need to be clearer about what is meant by “We can tell … ”. This should mean that we can tell by some purely mechanical process, not requiring thought. In other words, we want to be able to feed the data to a computer and have the computer arrive at the answer as to whether or not the required property holds for the given data.

Readers might think that a computer's ability to arrive at the answer for given data would depend very much on the computer. This turns out not to be so, though the speed and efficiency of the computer's answer will depend on the computer.

More precisely, there is a class of functions (called partial recursive functions) with the following properties.