Non-standard lattices and o-minimal groups

We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced. §1. Introduction. The study of groups definable in models of a first-order theory has been a core subject in model theory spanning at least a period of thirty years. On the one hand, definable groups are present whenever non- trivial phenomena occur in the models of a theory and their study has played a prominent role in Shelah's classification theory. On the other hand, a large variety of classical groups turn out to be definable in certain structures and their study via model-theoretic methods has given rich applications to other areas of mathematics. For example, an algebraic group is definable in an algebraically closed field and a compact real Lie group is definable in some o-minimal expansion of the real field. It is the second kind of examples which we seek to embark on here. O- minimal structures provide a rich, yet tame, model-theoretic setting where definable sets enjoy many of the nice topological properties that hold for semi-algebraic sets. For example, a topological notion of dimension can be defined for every definable set. It is often said that o-minimality is the correct formalization of Grothendieck's 'topologie moderee' (8). Groups definable in an o-minimal structure, in their turn, henceforth called 'o-minimal groups', strikingly resemble real Lie groups. The starting point for the study of o-minimal groups was Pillay's theorem in (40) that every such group admits a definable manifold topology that makes it into a topological group. Since then, an increasing number of theorems have reinforced the