Model Theory and Groups

This paper is about various ways in which groups arise or are of interest in model theory. In Section I briefly introduce three important classes of first order theories, stable theories, simple theories, and NIP theories. Section 2 is about the classification of groups definable in specific theories or structures, mainly fields, and the relationship to algebraic groups. In Section 3 I study generalized stability and definable groups in more detail, giving the theory of “generic types” in the various contexts. I also discuss 1-based theories and groups. Section 4 is about the compact Hausdorff group G/G00 attached to a definable group and how it may carry information in various contexts (including approximate subgroups). In Section 5, I discuss Galois theory, including the various Galois groups attached to first order theories, various kinds of strong types, and definable groups of automorphisms. In Section 6, I study various points of interaction between topological dynamics and definable groups, in particular “Newelski’s conjecture” relating G/G00 to the “Ellis group”. And in Section 7, I touch on the model theory of the free group.

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