Introduction [chapter]

2021 Groups and Model Theory  
The goal of this book is to show some directions in group theory motivated by logic and model theory. These directions include stable groups and generalizations, model theory of nonabelian free groups and rigid solvable groups, pseudofinite groups, approximate groups, topological dynamics, groups interpreting the arithmetic, decidability/undecidability of theories of groups and Diophantine problems in groups. The title (Groups and Model Theory, GAGTA book 2) reflects the fact that the book
more » ... ws the course of the GAGTA (Geometric and Asymptotic Group Theory with Applications) conference series. The first book, "Complexity and Randomness in Group Theory. GAGTA book 1," was published by De Gruyter in 2020. The model theory of groups has always held a special position within the discipline of model theory. As in many other mathematical domains, groups play an important role in model theory. There are two main points of view on the subject that in many cases cross-cut and are not always clearly distinguishable. We give a brief account of the major goals in each case. The "applied" viewpoint is concerned with particular first-order theories, usually important and well-studied in other mathematical disciplines. The objective is to investigate what kind of groups are definable in the given first-order context or ambient structure. It is important to note that groups often come equipped with extra structure, e. g., algebraic groups or Lie groups, so the above study expands the understanding of the additional structure a definable group may have. It is usually the case that one can give a structure to a group that coincides with the expected structure when studied in some ambient model. For example, any definable group in an algebraically closed field can be equipped with the structure of an algebraic group. In many cases this takes some nontrivial effort to prove. Examples of theories that have been thoroughly studied along these lines are the theory of algebraically closed fields and the theory of real closed fields. The "pure" viewpoint centers around properties usually coming from Shelah's classification theory, e. g., stability, superstability, etc., and studies classes of groups that share these properties. In particular, the study of stable groups was largely initiated by Poizat, borrowing notions, like genericity and connectedness, from algebraic geometry. This has been, indeed, a very fruitful field of research for model theorists. The interest of research on stable groups gradually moved to larger and "wilder" classes, like groups without the independence property or simple (in the sense of model theory) groups, but the guiding force has always been the results in the stable context. In yet another direction of pure model theory, motivated by results and conjectures of Zilber, model theorists studied pregeometries arising from suitable definable sets. This area is known as "geometric stability theory" and on the pure side has contributed many applications to "core" mathematics. In this context, questions of what kind of groups are definable in a first-order theory became relevant and revealed
doi:10.1515/9783110719710-201 fatcat:fswuu6z2bndd7arwiqgbgol7re