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?