Real-valued functions defined on subsets of ℝ are indispensable to mathematical analysis, and as such they have been the focus of our study so far in the book. But functions with other types of domains and ranges (e.g., subsets of ℝn with n ≥ 2 or subsets of F(S, ℝ) with S ⊆ ℝn) occur abundantly in mathematics and its applications, and there is a vast body of mathematical development that is devoted to them. As with the simpler functions studied in the previous chapters, the study of these more complex types of function involves the development of such concepts as limits, convergence, continuity, differentiability, and integrability. The concept of a topological space provides a general framework in which one can study the limit, convergence, and continuity of various types of function occurring in mathematical analysis.

Before we begin our study of point set topology, we present a theorem of IST on the standardization of functions, which will be needed in this chapter. This theorem is a stronger version of Theorem 3.11.1.

Theorem (Standardization of functions II) Let X and Y be nonempty standard sets. Suppose that, for each x ∈ σX, there is a y ∈ σY such that ϕ(x, y), where ϕ is any formula, external or internal.

When we read a statement, we normally ignore its syntax (the formation pattern of sentences and phrases) and focus our attention on its meaning. But much can be gained if we also pay some attention to the syntax of mathematical statements. Thus our goal in this appendix is to direct the reader's attention to some elements of the syntax of mathematical statements. Our discussion of this subject will be exploratory and informal rather than prescriptive and formal. We will proceed by looking at some examples of symbolically written statements in the language of the real numbers. We assume that the reader already has some experience in translating English statements about the real numbers into statements written in mathematical symbols. However, as we are pursuing the main goal of this section we will provide a brief review of such translations as well.

Constituents of mathematical statements

The language of a mathematical theory is, for the most part, a fragment of a natural language (like English or Persian). In a mathematical language the words “condition” and “statement” (used synonymously in this book) refer to the same sort of syntactical structures as does the phrase “declarative sentence” in the case of a natural language.

We usually think of ℝn as a set that is equipped with the operations of vector addition and scalar multiplication, defined as follows: if α ∈ ℝ and x, y ∈ ℝn with x = (x1 … xn) and y = (y1, …, yn) then x + y = (x1 + y1, …, xn + yn) and αx = (αx1, …, αxn). These operations endow ℝn with the structure of a vector space as follows.

Definition (Vector space over ℝ) A vector space over ℝ is a triple (X, +, ·), where V is a nonempty set; (x, y) ↦ x + y with x, y ∈ X and (α, x) ↦ α · x with α ∈ ℝ and x ∈ X are two operations called vector addition and scalar multiplication, respectively.

The mathematicians of the seventeenth and eighteenth centuries used a method based on the notion of an infinitesimal (an infinitely small number) to create and develop calculus and mathematical analysis. Although their method was intuitively appealing and enabled simple arguments and calculations, by the end of the nineteenth century it had to be abandoned for lack of rigor.

A rigorous development of classical analysis requires a precise definition of the real numbers, and for this the notion of an infinite set is essential. In this introduction, we provide some background for these foundational matters of mathematical analysis.

Infinite sets and the continuum

Among the most rudimentary operations of the human mind are the acts of considering a number of entities as a unit and of regarding a single object as composed of a number of constituents. For example, when we use words such as population or nation we are regarding a number of entities as a unit. Similarly, when we regard a drop of water as made up of a number of molecules of water or a line segment as an assemblage of infinitely many points, we are thinking of a single object as composed of a number of constituents. The intuitive notion of a collection of elements occurs to us in conjunction with these basic mental activities.

In September 2007, the London Mathematical Society and the EPSRC sponsored a ‘short course for graduates’ in Oxford, under the heading ‘Asymptotic methods in infinite group theory’. This was organised by Dan Segal and consisted of three series of lectures. The present book is basically a record of these lectures, somewhat polished and expanded. It is intended to serve as an introduction, for beginning research students and for interested non-specialists, to some areas of current activity in algebra: the questions mostly originate in group theory but the methodology encompasses a wide range of mathematics, involving topology, algebraic geometry, number theory and combinatorics.

Introduction

The theory of Lie groups is highly developed and of relevance in many parts of contemporary mathematics and theoretical physics. Loosely speaking, a Lie group is a group with the additional structure of a real differentiable manifold, given by local coordinate systems, such that the group operations are smooth functions.

Historically, the study of Lie groups, over the real and complex numbers, arose toward the end of the 19th century, from the analysis of continuous symmetries of differential equations by the mathematician Sophus Lie and others. Around the middle of the 20th century, mathematicians such as Armand Borel and Claude Chevalley found that many of the foundational results concerning Lie groups could be developed completely algebraically, giving rise to the theory of algebraic groups defined over arbitrary fields. This insight opened the way for entirely new directions of investigation. Much of the theory of p-adic Lie groups was developed in the 1960s by mathematicians such as Nicolas Bourbaki, Michel Lazard and Jean-Pierre Serre. Since then the study of p-adic Lie groups and analogues of Lie groups over adele rings has largely been motivated by questions from number theory, e.g. regarding automorphic forms and Galois representations. More recently, p-adic Lie groups have also become a key tool in infinite group theory.

Throughout, let p be a prime. The real numbers ℝ form a completion of the rational numbers ℚ. Similarly, the field of p-adic numbers ℚp is obtained by completing ℚ, albeit with respect to a different, non-archimedean notion of distance.

From a purely algebraic point of view, there is not a lot one can say about infinite groups in general. Traditionally, these have been studied to good effect in combination with topology or geometry. These lectures represent an introduction to some recent developments that arise out of looking at infinite groups from a point of view inspired – in a general sense – by number theory; specifically the interaction between ‘local’ and ‘global’, where by ‘local’ properties of a group G, in this context, one means the properties of its finite quotients, or equivalently properties of its profinite completion Ĝ. The second chapter directly addresses the interplay between certain finitely generated groups and their finite images. The other two chapters are more specifically ‘local’ in emphasis: Chapter I concerns the algebraic structure of certain pro-p groups, while Chapter III introduces a way of studying the rich arithmetical data encoded in certain infinite groups and related structures.

A motivating example for all of the above is the question of ‘subgroup growth’. Say G has sn(G) subgroups of index at most n for each n; the function n ↦ sn(G) is the subgroup growth function of G, and is finite-valued if we assume that G is finitely generated. Now we can ask (inspired perhaps by Gromov's celebrated polynomial growth theorem): what does it mean for the global structure of a finitely generated group if its subgroup growth function is (bounded by a) polynomial?