Free Groups

Free groups are central to the study of infinite groups and provide relatively accessible examples that highlight the power of mixing algebraic and geometric approaches. Free groups can be defined in a number of ways. The standard method, which we give in Section 3.1.1, is formal and algebraic. Another common method is to use actions on trees, which is the perspective we develop in Section 3.4. While the formal definition lacks the intuition building power of the geometric perspective, it is often the easiest definition to use when trying to prove that a given group is a free group.

Free Groups of Rank n

Definition 3.1. Let S = {x1, …, xn} be a set of elements in a group, G. A word ω ∈ {S ∪ S-1}* is said to be freely reduced if it does not contain a sub word consisting of an element adjacent to its formal inverse. For example, the word ω = xyx-1y-1 is freely reduced while ω′ = xy-1yxy is not. The group G is a free group with basis S if S is a set of generators for G and no freely reduced word in the xi and their inverses represents the identity.

Gromov's Corollary, aka the Word Metric

We have seen a number of examples of groups acting on the real line. Sometimes these actions preserve the distance between points on the line, such as the action of D∞. Other actions we have considered do not preserve distances, for example, the action BS(1, 2) ↷ ℝ presented in Chapter 4. One of the most powerful insights in the study of finitely generated infinite groups is that they can always be viewed as groups acting in a distance-preserving way on a geometric object. We refer to this insight as Gromov's Corollary to Cayley's Better Theorem, in honor of a groundbreaking paper that Mikhail Gromov wrote in the 1980s [Gr87], which highlighted this perspective, introduced a number of questions motivated by the geometry of infinite groups, and introduced powerful tools that can be used to answer them.

In order to present Gromov's Corollary, we need to introduce a reasonably flexible notion of “geometric object” as well as formally define what we mean by saying that a group action “preserves distances.”

Regular Languages and Automata

We have explored the word problem about as far as is prudent without being a bit more careful with words like “decide” and “construct.” Here we introduce a standard mathematical model of computation. In thinking about modelling computation, there are a few key aspects that need to be preserved. One needs to have a way of inputing a sequence of information, and then a machine needs to execute various commands based on the input. One expects the total number of computations to be finite and possibly that there is some provision for the machine to have memory. The notion of finite-state automata, and their corresponding languages, fits these expectations.

Definition 7.1. Given an alphabet S = {x1, …, xn}, any subset of words in the free monoid on S, ℒ ⊃ S*, is a language.

There are numerous examples of languages. If S = {a}, then the set of all strings of even length, {a2n | n ∈ ℕ}, forms a language.

Changing Generators

There is a danger in working with a specific Cayley graph for a given group G. If you focus on a particular generating set, the results you get may not immediately translate into similar results when a different set of generators is used. Even worse, sometimes interesting properties hold in one Cayley graph but not in another, even though the group under consideration has not changed.

In this chapter we introduce some of the ways geometry can be imported into the study of infinite groups, which are independent of the choice of finite generating set. These are properties that always hold or always fail, no matter which finite set of generators one uses to construct a Cayley graph. Of course, such properties cannot focus too closely on a given Cayley graph, since changing generators changes the details, in other words the local structure, of the graphs. In Figure 11.1 we highlight this with two Cayley graphs for D3. The location of the vertices has been kept constant in both pictures, which makes the second picture look a bit odd. When the generators change, so do the distances between vertices, the number of cycles, and so on.

Because changing generators can have dramatic consequences for the local structure of a Cayley graph, the properties we consider here are referred to as large-scale properties; many authors also use the term geometric properties.

In this chapter we introduce another interesting infinite group described in terms of functions from ℝ to ℝ, although in this case we only use the closed interval [0, 1]. We begin by discussing dyadic divisions of [0, 1]. A dyadic division of [0, 1] is constructed by first dividing [0, 1] into [0, 1/2] and [1/2, 1], and then proceeding to “pick middles” of the resulting pieces, a finite number of times. For example,

is a dyadic division. We refer to the points of subdivision as the chosen middles, so in the example above the chosen middles are: 1/2, 3/4 and 5/8 (listed in the order they are chosen). We refer to any interval of the form as a standard dyadic interval. Exercise 1 asks you to show that any time you divide [0, 1] into standard dyadic intervals, you necessarily have a dyadic division of [0, 1].

Dyadic divisions of [0, 1] can be encoded using finite, rooted binary trees. Recall from Chapter 6 that a rooted binary tree is a rooted tree where every non-leaf vertex has two descendants.

Let a and b be reflections in parallel lines in the Euclidean plane (as in Figure 2.1). If we think of the line of reflection for a as being x = 0 and the line of reflection for b as being x = 1, then we may express a as the function a[(x, y)] = (-x, y) and b as b[(x, y)] = (2 - x, y). It follows that ab[(x, y)] = (x - 2, y) and ba[(x, y)] = (x + 2, y). Moreover, for any n ∈ ℝ, (ab)n is a horizontal translation to the left through a distance of 2n and (ba)n is a horizontal translation to the right through a distance of 2n. Thus the reflections a and b generate an infinite group.

If the two lines of reflection actually met, at an angle of π/n, then the group generated by a and b would be the dihedral group of order 2n, and the product ab would be a rotation through an angle of 2π/n. Thus the group we are considering is something like a dihedral group, except that the two reflections generate a translation, not a rotation. As the group generated by two parallel reflections is infinite, it is referred to as the infinite dihedral group, denoted D∞.

Groups are algebraic objects, consisting of a set with a binary operation that satisfies a short list of required properties: the binary operation must be associative; there is an identity element; and every element has an inverse. Presenting groups in this formal, abstract algebraic manner is both useful and powerful. Yet it avoids a wonderful geometric perspective on group theory that is also useful and powerful, particularly in the study of infinite groups. This perspective is hinted at in the combinatorial approach to finite groups that is often seen in a first course in abstract algebra. It is my intention to bring the geometric perspective forward, to establish some elementary results that indicate the utility of this perspective, and to highlight some interesting examples of particular infinite groups along the way. My own bias is that these groups are just as interesting as the theorems.

The topics covered in this book fit inside of “geometric group theory,” a field that sits in the impressively large intersection of abstract algebra, geometry, topology, formal language theory, and many other fields. I hope that this book will provide an introduction to geometric group theory at a broadly accessible level, requiring nothing more than a single-semester exposure to groups and a naive familiarity with the combinatorial theory of graphs.

The chapters alternate between those devoted to general techniques and theorems (odd numbers) and brief chapters introducing some of the standard examples of infinite groups (even numbers).

An introduction to group theory often begins with a number of examples of finite groups (symmetric, alternating, dihedral, …) and constructions for combining groups into larger groups (direct products, for example). Then one encounters Cayley's Theorem, claiming that every finite group can be viewed as a subgroup of a symmetric group. This chapter begins by recalling Cayley's Theorem, then establishes notation, terminology, and background material, and concludes with the construction and elementary exploration of Cayley graphs. This is the foundation we use throughout the rest of the text where we present a series of variations on Cayley's original insight that are particularly appropriate for the study of infinite groups.

Relative to the rest of the text, this chapter is gentle, and should contain material that is somewhat familiar to the reader. A reader who has not previously studied groups and encountered graphs will find the treatment presented here “brisk.”

Cayley's Basic Theorem

You probably already have good intuition for what it means for a group to act ona set or geometric object. For example:

• The cyclic group of order n – denoted ℤn – acts by rotations on a regular n-sided polygon.

• The dihedral group of order 2n – denoted Dn – also acts on the regular n-sided polygon, where the elements either rotate or reflect the polygon.

• […]