I have finally gathered my thoughts enough to say something concerning Terry’s question above. In other words, this is my current (quite limited) view of some of the “secret ingredients” of the “finite-infinite” correspondence in Udi’s paper.

It seems to me that there are, in fact, two “correspondences” going on. The first is the more usual one, roughly between families of finite sets that look like subgroups and “near-subgroups” in the ultra-product (Definition 3.6). Given a near-subgroup , Udi obtains in Theorem 3.4 the “stabilizer” subgroup. This “bounds” between two subgroups (one type-definable, another ind-definable) which are very “close” (the quotient is of bounded index). This already allows him to deduce certain consequences on the “finite” world, using the first “finite-infinite correspondence” (the end of section 3 – of course, quite a bit of model theory on the “infinite side” is used just in those proofs). As I mentioned before, some of the power here is hidden in the stabilizer theorem, one of the main ingredients of which is the Independence Theorem 2.16. The techniques, as Anand was saying, generalize those of stability, simplicity, etc., and some of the main tools seem to be local stability, forking, and definable measures.

The second correspondence is possibly even more intriguing, and it is that between definable (type-definable, ind-definable…) groups in the “infinite” world (in a monster model) and compact (or locally compact) topological groups. These connections have been studied quite a bit recently in the context of dependent groups, specifically in the o-minimal context (“Pillay’s Conjecture”). In order to establish this correspondence, one again uses the fact that the stabilizer obtained in Theorem 3.4 is of bounded index. Now the tools of locally compact topological groups become available. It seems that Udi uses kind of a “transfer” principle for certain type of statements between the “Lie” world and the “infinite” world of the monster model, which then can be transferred back to the “finite” world; but I am less clear on this part at the moment.

]]>I am sorry: it was pointed out to me that I have been reading an old version of Udi’s paper. In the most recent version, the Independence Theorem is Theorem 2.16.

Also, in my previous comment I forgot to mention that if you add a predicate for membership in a subgroup A (so if A is definable), then N(A) is definable as well: you need to say that for all x in A there is x’ in A such that yx=x’y (and vice versa).

The model theory group in Lisbon is also (slowly) reading Udi’s paper. We will be very happy to participate in an online reading group (although at this point it is too early for us to be able to say or ask anything intelligent).

]]>Thanks for your comments! I will definitely be interested in understanding more about stability theory as the seminar here progresses.

]]>Actually, the last statement is not quite true.

In any case, although there is a slight set theoretic issue with saturated models, it can be easily avoided: in model theory, one only needs to work with a model U which is saturated and homogeneous with respect to some big cardinal lamda (so the properties “saturation” and “homogeneity” mentioned in Terry’s post hold over sets of cardinality less than lamda), and it is not so important that the cardinality of U is exactly lamda. Such models provably exist in ZFC.

There are other ways to deal with existence of saturated models, for example “proof by absoluteness”: most properties we are interested in in model theory are set-theoretically absolute (their truth does not depend on the model of set theory one works with), so it is harmless to assume that GCH holds for some big cardinal lamda.

Another small general remark that makes a (mild) connection between this blog and a previous online course of Terry’s on ergodic theory: types over a set A are simply ultra-filters on the Boolean algebra of A-definable sets (formulas). One can also look at a more general concept of a (finitely additive) probability measure on this space (a Keisler measure). The space of types is a compact totally disconnected topological space. Some of Keisler measures (in particular types) turn out (or can be assumed) to be definable, which means that they can be viewed as continuous functions from the type space to the reals. This allows the use of topological tools, which are often quite helpful.

Lastly a short response to Terry’s question about the power of the infinitary situation: it seems to be that some of it can be seen in Theorem 2.15, which is a variant of the so-called “Independence Theorem” (“3-Amalgamation Theorem” would be a better name), which roughly states that under certain conditions “free” (“large”) extensions of a type are compatible. This is a generalization of uniqueness of “free” (non-forking) extensions in stable theories, and has had many consequences in simple theories (generalizing results from stability). It also plays a crucial role in the proof of Theorem 3.4 in Udi’s paper, which is the central result on the model theoretic side.

]]>Here is a kind of response to Terence Tao’s question about the power of the model theoretic methods. As with the other examples he mentions (theory of topological groups, measurable dynamics,..) there is a powerful and nontrivial theory around, namely stability theory, or rather neo-stability theory. This is a specialised area of model theory, which many people in the subject may not be exposed to. Stability theory (developed originally by Shelah) includes notions of forking, stationarity, orthogonality,…and was originally developed just in the context of stable (first order) theories, for reasons I will not go into here.

In fact if you are interested in more than simply getting through the paper there would be no harm in taking a bit of time to talk about stable theories in the UCLA seminar.

Anyway, in the past 15 years or so there has been an extension of this technology outside the realm of stable theories , to e.g. “simple” theories, and NIP theories. There are new features, such as independence theorem in simple theories, and studying measures in place of types in NIP theories…I call this neo-stability theory. (Neo) Stable group theory is just (neo) stability theory in the presence of a definable group operation, which has special features like the theory of generic types, connected components, stabilizers, invariant measures,.. What Udi has done beautifully in say 3.4 is to isolate some local assumptions on a situation (and which are consequences of being an ultraproduct of finite sets or structures) and which allow arguments in the case of groups definable in SIMPLE theories to go through, with of course additional finessing. There is an additional level of generality of working with v-definable (ind-definable) groups rather than definable ones, but it doesn’t really present technical obstacles.

So my basic point is that there is a theory here, going considerably beyond compactness, which one is plugging into. One can of course ask further, what precise technical aspects or specific theorems, provide the power to encode finite combinatorics or even to say “meaningful” things at the infinitary level. I may say more about it later.

]]>This is very interesting! Thanks.

]]>Hi Andres,

Well, I can give you some indirect references only. I learned about the connection between the Group Configuration Theorem (GCT) and some combinatorial problems from my late friend and teacher Gyuri Elekes.

In 2001 Endre Szabo proved a beautiful theorem in algebraic geometry which is (partly) a corollary of GCT. (Endre Szabo wasn’t aware of this connection, but the result remained unpublished) Loosely speaking it says that if a low degree varieties contains at least points of an n x n x n Cartesian product then it is a cylinder or it is a pullback from the graph of the multiplication function of a group. My favorite application of this result is the following (due to Elekes and Szabo): the number of distinct distances between three non-collinear points and n point in the plane is at least (Much more than ) The only pointer which I can give you for these results is Elekes’ survey paper “Sums versus products in Number Theory, Algebra and Erdös Geometry” which is downloadable from the website

http://www.cs.elte.hu/~elekes/