Groups Definable in O-minimal Structures

In this series of lectures, we will a) introduce the basics of ominimality, b) describe the manifold topology of groups definable in o-minimal structures, and c) present a structure theorem for the special case of semi-linear groups, exemplifying their relation with real Lie groups. The structure of these lectures is as follows: (1) Basics of o-minimality: definition, Cell Decomposition Theorem, dimension of definable sets. The standard reference is [vdD]. A recent survey is [Pet]. (2) Definable groups: definable manifold topology, uniqueness, questions about (a) affine embedding and (b) resemblance with real Lie groups (Pillay’s Conjecture). The standard reference is [Pi1], and a recent survey is [Ot]. (3) Semi-linear groups: Structure Theorem and answers to the above questions for semi-linear groups. Reference: [El]. 1. Basics of o-minimality Let L be a first-order language, and M an L-structure. Then X ⊆ M is called definable (in M over ā) if for some formula φ ∈ L, X = {b̄ ∈ M : M |= φ(b̄, ā)}. For example, the unit circle on the real plane {(x, y) ∈ R : x+y = 1} is definable in 〈R, +, ·〉 but not in 〈R,+〉. A function f : A ⊆ M → M is called definable if its graph Γ(f) ⊆ Mm×Mn is definable. A group G = 〈G,⊕, eG〉 is called definable if G ⊆ M and ⊕ : M → M are definable. For example, the following group is definable in 〈R, <, +〉: let S = 〈[0, 1),⊕, 0〉, with x⊕ y = { x + y if x + y < 1, x + y − 1 if x + y ≥ 1. The study of definability has been a powerful tool in stability theory. Indeed, Shelah’s classification of models of a given theory up to isomorphism turned out to be intimately related to the classification of definable sets in the models of the theory. On the other hand, the definition of an o-minimal structure was given in terms of the definable sets in the structure. The creation of o-minimality can be viewed as an attempt of developing model theory for structures that do not fall under the scope of Shelah’s classification theory. Definition 1.1 (Knight-Pillay-Steinhorn, [KPS, PiS], 1986). A structure M = 〈M, <, . . . 〉 is called o-minimal if Date: September 25, 2008. DRAFT lecture notes for the Seminário de Lógica Matemática, Universidade de Lisboa.