Traditionally the theory of modular forms has been and still is, one of the most powerful tools in number theory. Recently it has also been successfully applied to resolve some long outstanding problems in seemingly unrelated fields. Our aim in this book is to describe three such applications, developing along the way the necessary methods and material from the theory of modular forms. Briefly, the problems we examine are the following:

(A) Ruziewicz's problem.

The problem is whether the Lebesgue measure λ on the n-sphere Sn is the unique rotationally invariant mean on L∞(Sn). To put it in another context, an amenable topological group G is one which carries an invariant mean on L∞(G). Uniqueness of such a mean is a difficult question and seldom discussed. Actually, Ruziewicz in the 1920's posed the problem of the uniqueness of rotationally invariant finitely additive measures defined on Lebesgue sets on Sn. The relation between these problems is that an invariant mean on L∞(Sn) is a finitely additive measure v which is moreover absolutely continuous with respect to Lebesgue measure λ, i.e., v(E) = 0 whenever λ(E) = 0. Tarski [Tar] has remarked that it follows from the Hausdorff–Banach–Tarski paradoxical decompositions of Sn, n ≥ 2, that any rotationally invariant finitely additive measure on Sn, n ≥ 2, must be absolutely continuous with respect to λ. Hence for n ≥ 2 the invariant mean and Ruziewicz problems are equivalent.

These notes are an expanded version of the Wittemore Lectures given at Yale in November 1988. The material presented in the four chapters is more or less selfcontained. On the other hand, in the section at the end of each chapter called ‘Notes and comments,’ it is assumed that the reader is familiar with more advanced and sophisticated notions from the theory of automorphic forms. Some of the material presented here overlaps with a forthcoming book, ‘Discrete groups, expanding graphs and invariant measures’ by A. Lubotzky. The points of view, emphasis, and presentation in that book and the present notes are sufficiently different that we decided to keep the two works separate. The reader is encouraged to look at both treatments of the material.

MONOTONICITY

In this chapter, I consider the problem of determining which sequences of natural numbers can occur as (fn) or (Fn) for some oligomorphic permutation group. The problem in this generality is quite out of reach. All I can do is to illustrate by examples some of the possibilities that can occur, and to describe some of the restrictions on realisable sequences which have been established.

We saw in the last chapter that any permutation group is a dense subgroup of the automorphism group of a homogeneous relational structure, so it suffices to consider these groups. Moreover, since the group is oligomorphic, the structure is ℵ0-categorical. The results of §2.5 imply that the sequences (fn) and (Fn) realised by oligomorphic groups are precisely those which enumerate unlabelled and labelled structures respectively in a class of finite structures satisfying Fraïssé's hypotheses (notably, the amalgamation property). From this point of view, two subcases commend themselves to us:

1. (a) classes having the strong amalgamation property;

2. (b) ages of homogeneous structures over finite relational languages.

3. Other subcases are obtained by imposing model-theoretic conditions in the neighbourhood of stability on the homogeneous structure.

The most important basic restriction on realisable sequences is that they must be monotonic.

JORDAN GROUPS

Let G be a permutation group on Ω. A Jordan set for G is a subset of Ω with the property that the pointwise stabiliser of its complement acts transitively on it. (Sets consisting of just one point satisfy this condition trivially but are usually excluded for technical reasons.) If G is n-transitive, then any set containing all but n – 1 points of Ω is a Jordan set; such Jordan sets are called improper. (This needs some care in the case when n is infinite.) Then G is called a Jordan group if it has a proper Jordan set (other than the empty set).

With the exception of some recent examples constructed by Hrushovski (to appear), the known infinite Jordan groups are of three types:

(J1) Geometric examples: These are the projective group PGL(n, k),the affine group AGL(n, k), and their close relatives. The pointwise stabiliser of any subspace of a projective or affine space acts transitively on its complement. So the complements of subspaces are the Jordan sets, and the geometry can be recovered from them. In this class, it is customary now to include also the automorphism groups of algebraically closed fields (which preserve the geometry of algebraically closed subflelds). In each of these cases, the subspaces of the geometry are precisely the algebraically closed sets (in the sense of §2.7). This fact is crucial, both in their study, and in applications.

BEGINNINGS

This chapter is about subgroups of automorphism groups of various structures.

There are several aspects to note. For a start, as we saw in Chapter 2, if a group G is the automorphism group of a structure, then it is the automorphism group of a homogeneous structure; and this is equivalent to G being a closed subgroup of the symmetric group. Describing all subgroups of the symmetric group is too wide a task, so I'll restrict both the structures, and the kinds of subgroups considered. As to the first, I shall consider only

1. (a) ℵ0-categorical structures (those whose automorphism groups are oligomorphic); and

2. (b) homogeneous structures whose age has the strong amalgamation property (those for which the stabiliser of a tuple in the automorphism group fixes no additional points).

Similar results hold in other cases; a notable example of this is provided by recursively saturated structures. (The results for these structures, due to Richard Kaye, were obtained following the Durham symposium.)

The results will in the main be constructions of subgroups with various properties, but at the end of the chapter I will describe some restrictive results characterising certain kinds of subgroups (normal subgroups, subgroups of small index, etc.).

There are differing levels of detail about a subgroup. We could be concerned with its structure as abstract group, as permutation group on some subset of the domain, or as permutation group on the entire domain.

HISTORY AND NOTATION

In the summer of 1988, a London Mathematical Society symposium was held in Durham on “Model Theory and Groups”, organised by Wilfrid Hodges, Otto Kegel and Ileter Neumann. This volume of lecture notes is based on the series of lectures I gave at the symposium, but is something more: since no Proceedings of the symposium was published, I have taken the opportunity to incorporate parts of the talks given by other participants, especially David Evans, Udi Hrushovski, Dugald Macpherson, Ileter Neumann, Simon Thomas and Boris Zil'ber. (A talk by Richard Kaye revealed new horizons to me which I have not fully assimilated; but Richard's own book should appear soon.) In addition, I have made use of parts of the proceedings of the Oxford–QMC seminar on the same subject which ran weekly in 1987–8 and continues once a term (now as the Oxford–QMW seminar!); contributions by Samson Adeleke, Jacinta Covington, Angus Macintyre and John Truss have been especially valuable to me.

Why model theory and groups? In particular, why the special class of permutation groups considered here?

In the middle 1970s, when my interests were entirely finite, John McDermott asked a question about the relationship between transitivity on ordered and unordered n-tuples for infinite permutation groups. The analogous question, and more besides, had been settled for finite permutation groups by Livingstone and Wagner (1965), with techniques which were largely combinatorial and representation-theoretic, and so not likely to be useful here.