Model Theory and Groups

TL;DR

This paper explores the interplay between model theory and group theory, focusing on definable groups, their classifications, and applications in algebra, topology, and dynamics, providing an expository overview of recent developments.

Contribution

It offers a comprehensive survey of how groups are studied within model theory, highlighting new insights into definable groups across various theories and their applications.

Findings

Classification of definable groups in different theories

Connections between model theory and algebraic groups

Applications to topological dynamics and Galois theory

Abstract

This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups definable in stable, simple and NIP theories, definable compactifications of groups, definable Galois theory (including differential Galois theory), connections with topological dynamics, model theory of the free group.