Why are model theorists so fond of definable groups?

Question

My PhD was on so called "pure" model theory, and my advisor was not very much interested in applications of model theory to algebra. Now I feel the need to fill in the gap, and I'd like to educate myself on applied model theory. One of the questions I always asked myself is about the study of groups definable (or interpretable) in some structures : it seems that model theorists are very fond of that, since a long time ago (at least since the 70's to my knowledge, and even quite recently with works about groups definable in NIP structures by Pillay). Why is it so ? I guess that the origin of all this is the fact that groups definable in ACF are precisely algebraic groups. This is indeed a nice link between model theory and algebraic geometry (by the way, does this link has brought something interesting and new to algebraic geometry, or has it always been just a slightly different point of view ?). But if it is so, why going on to study extensively groups definable on such exotic kind of theories as NIP or simple for example ? Is it only because something can be said about those groups and that groups are prestigious objects within mathematics, or are there deeper reasons ?

Answer 1

Here are a few reasons. These are just my perspectives - some people may disagree, and I'm sure there are good reasons to study definable groups that I'm missing.

1. Mathematicians in general are fond of groups. There is a general theme in mathematics (with different motivations in different contexts) of developing a theory of group objects in any category of interest. For example, algebraic groups are group objects in the category of algebraic varieties, Lie groups are group objects in the category of differentiable manifolds, topological groups are group objects in the category of topological spaces, etc. Model theorists study the category of definable sets relative to a first-order theory, and group objects in this category are definable groups. [Aside: I'm confused by your comment "model theory is not mainly about definable things in my opinion". If it's not about definable things, what is it about?] Moreover, it sometimes happens that definable groups relative to a theory $T$ correspond to groups of classical interest in mathematics, e.g. as you mention in the question, definable groups relative to the theory of algebraically closed fields are essentially the same as algebraic groups.

2. Stable groups and their generalizations. One of the most successful applications of Shelah's general stability theory has been the theory of stable groups (i.e. groups definable in stable theories). Model theorists love to generalize results, so there's a natural motivation to try to prove theorems analogous to theorems about stable groups in a wide variety of more general model theoretic contexts, or in individual (unstable) theories of interest. Extra motivation comes from the fact that the theory of stable groups, together with the kinds of connections between definable groups and groups in algebraic geometry mentioned in the previous point, led to Hrushovski's impressive applications of model theory to Mordell-Lang and related problems. This is just one example of applications of the theory of definable groups to other areas of mathematics. For others, you could look at Hrushovski's work on approximate groups, or recent applications to regularity lemmas in groups - both of these make heavy use of theorems about definable groups outside the stable context which were inspired by theorems about stable groups.

3. Binding groups and internality. Getting more technical here, internality is a key concept in geometric stability theory. Roughly speaking, if one (type-)definable set $X$ is internal to another (type-)definable set $Y$, then the automorphism group of $X$ over $Y$ in the monster model is realizable as a definable group, called the binding group. The classic example is that an $n$-dimensional vector space $V$ over $k$ is internal to the field $k$, with binding group $\text{GL}_n(k)$. If you can understand what kinds of groups are definable in your theory, then you can understand what kinds of internality relations are possible relative to your theory, which can lead to powerful structural results.

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Edit: In the comments, you ask "Why not study definable lattices, or rings, or whatever?" In fact, model theorists do study such things. Classifying definable equivalence relations (elimination of imaginaries) is extremely important - it's one of the first things you want to do when you start studying a theory. It's also very important to know whether your theory has any definable orders. And the question of the definability of a field in certain structures is the central question of the Zilber trichotomy and Zariski geometries, which has been a central motivating force in modern model theory.

My point is that definable groups get more attention than other kinds of structures (like lattices, for example) for reasons including those I outlined above. But a huge variety of instances of the general question of interpretability of certain theories in other theories come up everywhere in model theory.

Answer 2

Alex's answer is very good, but there is another reason that is worth mentioning. One of the main goals of model theory is to known when first order theories of interest are interpretable in other theories of interest and when they are not. Many structures of interest expand a group, so it is important to understand interpretable groups. For the same reason it is important to understand interpretable fields, and work on interpretable fields often builds on work on interpretable groups.

Here's a nice example: Suppose that $\mathcal{R}$ is an o-minimal expansion of a real closed field $R$. It is a theorem of Otero, Peterzil, and Pillay that any infinite field interpretable in $\mathcal{R}$ is definably isomorphic to either $R$ or $R[\sqrt{-1}]$. (O-minimal expansions of fields eliminate imaginaries, so here "interpretable" is equivalent to "definable".) It easily follows that if $\mathcal{S}$ is another o-minimal expansion of a real closed field, and $\mathcal{S}$ is interpretable in $\mathcal{R}$, then $\mathcal{S}$ is isomorphic to a reduct of $\mathcal{R}$.

This shows that there are no non-trivial interpretations between o-minimal expansions of ordered fields. So for example $(\mathbb{R},+,\cdot,\exp)$ is not interpretable in $(\mathbb{R},+,\cdot)$. The proof of their theorem uses the theory of definable groups in o-minimal structures.