Definably amenable NIP groups

By Artem Chernikov and Pierre Simon

Abstract

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

1. Introduction

In the same way as algebraic or Lie groups are important in algebraic or differential geometry, the understanding of groups definable in a given first-order structure (or in certain classes of first-order structures) is important for model theory as well as its applications. On the one hand, even if one is only interested in abstract classification of first-order structures (i.e., in understanding combinatorial complexity of definable sets), unavoidably one is forced to study definable groups. (This realization probably started with Zilber’s work on totally categorical structures Reference Zil93, and later it was made clear by Hrushovski’s theorem on unidimensional theories Reference Hru90.) On the other hand, some of the most striking applications of model theory are based on a detailed understanding of definable groups in certain structures. The class of stable groups is at the core of model theory, and the corresponding theory was developed in the 1970s–1980s borrowing many ideas from the study of algebraic groups over algebraically closed fields (with corresponding notions of connected components, stabilizers, generics, etc.; see, e.g., Reference Poi01). In particular, this general theory was applied to groups definable in differentially closed and separably closed fields, and it was used by Hrushovski to prove the Mordell–Lang conjecture for function fields Reference Hru96. The theory of stable groups was generalized in the 1990s to groups definable in a larger class of simple theories, centered around the model-theoretic notion of forking (see Reference Wag00), and it led to a number of results including Hrushovski’s proof of the Manin–Mumford conjecture Reference Hru01 and other applications to algebraic dynamics (e.g., Reference MS14). More recently, inspired by the ideas of stable and simple group theory, Hrushovski has obtained a general stabilizer-type theorem and found striking applications to approximate subgroups Reference Hru12, which led to a complete classification by Breuillard, Green, and Tao Reference BGT12. On the other hand, groups definable in o-minimal structures were studied extensively, generalizing the theory of real Lie groups. This study culminated in a recent resolution of Pillay’s conjecture for compact o-minimal groups Reference HPP08, and the proof has brought to light the importance of the general theory of groups definable in NIP structures (a common generalization of stable and o-minimal structures, see below) and the study of invariant measures on definable subsets of the group. In parallel, methods and objects of topological dynamics were introduced into the picture by Newelski Reference New09 and gave rise to some new invariants coming from topological dynamics and conjectures concerning their relationship to the more familiar model-theoretic invariants. This circle of ideas has rapidly become a very active research area. The present paper contributes to this direction and, continuing the work in Reference HPP08, develops the theory of groups definable in NIP structures which admit a translation invariant probability measure on the boolean algebra of definable subsets.

The NIP condition (the negation of the Independence Property) is a combinatorial tameness assumption on a first-order structure which says, in modern terms, that if is a definable set, then the family , of its fibers has finite VC-dimension (see section 2.1). Roughly speaking, it says that the collection of definable subsets of is very structured. One can think of NIP as capturing the notion of a geometric structure—as opposed to, say, arithmetic or random-like structure—and of NIP groups as groups arising in geometric settings. This condition was introduced by Shelah Reference She71, and the connection to VC-dimension was discovered later in Reference Las92. In the past 10 or 15 years, the role of NIP theories has grown to become a central notion in model theory thanks to applications to o-minimal structures, valued fields and combinatorics (see, e.g., Reference Sim15a or Reference Sta17 for a survey). Typical examples of NIP structures are given by stable structures (such as algebraically closed fields), o-minimal structures (such as real closed fields), and many Henselian valued fields. On the other hand, an ultraproduct of finite fields is an example of a structure which is not NIP, essentially because of arithmetic phenomena that enter the picture.

Let now be a group definable in an NIP structure (i.e., both the underlying set and multiplication are definable by formulas with parameters in ). Such a group comes equipped with a collection of definable subsets of cartesian powers of which is closed under boolean combinations, projection, and cartesian products. For example, if is the field of reals, then is a real semi-algebraic group and definable sets are all semi-algebraic subsets. As is typical in model theory, we prefer to work in a saturated model of our group (which, in the case of an algebraic group, corresponds to working in the universal domain in the sense of Weil). More precisely, let be a sufficiently saturated and homogeneous elementary extension of , a “monster model” for the first-order theory of . We write to denote the group obtained by evaluating in the formulas used to define in (and will refer to the set of the -points of ). So, e.g., if we start with the field of reals, and its additive group, then is the additive group of a large real closed field extending , which now contains infinitesimals, infinitesimals relatively to those infinitesimals, etc.; i.e., it satisfies a saturation condition—every small enough finitely consistent family of definable sets has nonempty intersection.

It was shown by Shelah Reference She08 that any NIP group admits a unique maximal compact quotient denoted , which plays an important role in this theory and for which we will give a dynamical interpretation below.

Our goal in this paper is to adapt techniques from stable group theory to the NIP context in order to have tools at our disposal potentially as useful as those for stable groups. However, one main difference with the stable case is that we cannot deal with all groups any more. As we show, to have any hope of having well-behaved notions of generic types and large subsets, the group must be definably amenable (this strengthens results of Hrushovski, Peterzil, and Pillay Reference HP11 who first observed this).

We say that a definable group is definably amenable if there is a finitely additive probability measure on the boolean algebra of definable subsets of which is moreover invariant under the group action (this property holds for if and only if it holds for ; see the remark after Definition 3.1). This notion has been introduced and studied in Reference HPP08 and Reference HP11. The emphasis in those papers is on the special case of the so-called fsg groups, which will not be relevant to us here. Of course, if is amenable as a discrete group, then it is definably amenable since we have such a measure on all subsets, not just the definable ones, but the converse need not hold (e.g., deep work of Sela Reference Sel13 demonstrates that any noncommutative free group, viewed as a first-order structure in the group language, is stable, hence definably amenable; but of course it is not amenable). Here are some important examples of definably amenable NIP groups:

• stable groups;

• definable compact groups in o-minimal theories or in -adics (e.g., );

• solvable NIP groups or, more generally, any NIP group such that is amenable as a discrete group.

Examples of definable NIP groups which are not definably amenable are or (see Reference HPP08).

It is classical in topological dynamics to consider the action of a discrete group on the compact space of ultrafilters on , or in other words ultrafilters on the boolean algebra of all subsets of . In the definable setting, given a definable group , we let denote the space of ultrafilters on the boolean algebra of definable subsets of , hence the space (called the space of types of ) is a “tame” analogue of the Stone–Čech compactification of the discrete group . Then acts on by homeomorphisms. The same construction applies to giving the space of ultrafilters on the definable subsets of . Our main objects of study in this paper are the dynamical systems and and related objects. In this context, we classify regular ergodic measures and show in particular that minimal flows are uniquely ergodic. We also give various characterizations of definable subsets of which have positive measure for some (resp., for all) invariant measures, connecting topological dynamics of the system with Shelah’s model-theoretic notion of forking.

A starting point of this theory is a theorem of Shelah stating that any NIP group admits a maximal compact quotient (the kernel is characterized as the smallest subgroup of which is an intersection of definable subsets and has small index in ). We give a dynamical interpretation of this compact quotient by establishing an isomorphism between the ideal subgroup of the Ellis semigroup of a certain extension of and . Those results settle several questions in the area.

Now we state the main results more precisely. In the case of stable groups, a natural notion of a generic set (or type) was given by Poizat (generalizing the notion of a generic point in an algebraic group), and a very satisfactory theory of such generics was developed in Reference Poi87. In a nonstable group, however, generic types need not exist, and several substitutes were suggested in the literature, either motivated by the theory of forking as in simple groups (Reference HPP08Reference HP11) or by topological dynamics (Reference NP06). First we show that in a definably amenable NIP group all these notions coincide and that in fact nice behavior of these notions characterizes definable amenability.

Theorem 1.1.

Let be a definable NIP group with a sufficiently saturated model. Then the following are equivalent:

(1)

is definably amenable (i.e., admits a -invariant measure on its definable subsets).

(2)

The action of on admits a small orbit.

The proof is contained in Theorem 3.12. It confirms a conjecture of Petrykowski in the case of NIP groups Reference New12, Conjecture 0.1 and solves Conjecture 4.13 of Reference CP12.

Theorem 1.2.

Let be a definably amenable NIP group. Then the following are equivalent for a definable set

(1)

does not -divide (i.e., there is no infinite sequence of elements of and natural number such that any sets in have empty intersection, see Definition 3.2

(2)

is weakly generic (i.e., there is some nongeneric such that is generic, see Definition 3.28

(3)

for some -invariant measure

(4)

is -generic (meaning that for any small model over which is defined, no -translate of forks over see Definition 3.2

Moreover, for a global type the following are equivalent:

(1)

is -generic (i.e., every formula in is -generic

(2)

has a small -orbit;

(3)

This is given by Theorem 3.35 and Proposition 3.8 and, combined with Theorem 1.1, solves in particular Reference CP12, Problem 4.13.

We continue by studying the space of -invariant measures using VC-theory, culminating with a characterization of regular ergodic measures (section 4) and unique ergodicity (section 3.4). Generalizing slightly a construction from Reference HP11, we associate to every generic type a measure , which is a lifting of the Haar measure on the compact group via (see Definition 3.16). It follows from Theorem 1.2 that the supports of the measures are exactly the minimal subflows of (see Proposition 3.31).

Theorem 1.3.

Let be a definably amenable NIP group. Then regular ergodic measures on are precisely the measures of the form , for an -generic type in . If two such measures have the same support, then they are equal (i.e., minimal subflows of are uniquely ergodic).

The first statement is Theorem 4.5 and the second follows from Proposition 3.24. The following is Theorem 3.36.

Theorem 1.4.

Let be a definably amenable NIP group. Then has a unique invariant measure if and only if it admits a unique minimal subflow if and only if it admits a global generic type. Moreover, in such a group all the notions in Theorem 1.2 coincide with is generic”, and in the moreover part we can add is almost periodic”.

Next we study enveloping semigroups. This notion from topological dynamics (see Reference Gla07a) was introduced in model theory by Newelski Reference New09. He observed that it behaved better when one replaced the dynamical system with an extension of it: The set embeds into as realized types, and we let be its closure. Then acts on and this flow admits as a factor. We consider the enveloping semigroup of the dynamical system . In view of the results in Reference CPS14, can be identified with , where is a naturally defined operation extending multiplication on (see section 5.3 for details).

Fix a minimal flow in (i.e., a closed -invariant set), and an idempotent . Then the general theory of Ellis semigroups implies that is a subgroup of , which we call the Ellis group. The canonical surjective homomorphism factors naturally through the space , so we have a well-defined continuous surjection , and the restriction of to the group is a surjective homomorphism. Newelski asked whether under certain model-theoretic assumptions this map could be shown to be an isomorphism. Pillay later formulated a precise conjecture that we are able to prove here.

Theorem 1.5 (Ellis group conjecture).

Let be definably amenable and NIP. Then is an isomorphism.

In particular, this demonstrates that the Ellis group is indeed a model theoretic object, i.e., it only depends on the first-order theory of the group and does not depend on the choice of a small model over which it is computed. Some special cases of the conjecture were previously known (see Reference CPS14). For the proof, we establish a form of generic compact domination for minimal flows in definably amenable groups with respect to the Baire ideal; see Theorem 5.3.

Remark 1.6.

We remark that the study of NIP definably amenable groups can be thought of as a model-theoretic version of tame dynamics as studied by Glasner, Megrelishvili, and others, see Reference Gla07b (in fact, we discovered the connection only after having essentially completed this work). The NIP assumption implies that the dynamical system is tame—and even null—in the sense of Reference Gla07b, Reference KL07, but it is not equivalent to it. Nullness of this system is equivalent to the fact that the definable family of translates of any given definable set has finite VC-dimension (see Reference KL07, Proposition 5.4(2)), whereas the NIP condition implies that any uniformly defined family of sets has finite VC-dimension.

2. Preliminaries

In this section we summarize some of the context for our results, including the theory of forking and groups in NIP, along with some general results about families of sets of finite VC-dimension.

2.1. Combinatorics of VC-families

Let be a set, finite or infinite, and let be a family of subsets of . Given , we say that it is shattered by if for every there is some such that . A family is said to have finite VC-dimension if there is some such that no subset of of size is shattered by . If this is the case, we let be the largest integer such that some subset of of size is shattered by it.

If is a subset and , we let

Similarly, if is a set of truth values, we let True.

Later in the paper, we will often write for .

A fundamental fact about families of finite VC-dimension is the following uniform version of the weak law of large numbers (Reference VČ71, see also Reference HP11, Section 4 for a discussion).

Fact 2.1.

For any and there is satisfying the following.

Let be a probability space, and let be a family of subsets of of VC-dimension such that:

(1)

every set from is measurable;

(2)

for each , the function given by

is measurable;

(3)

for each , the function

is measurable.

Then there is some tuple such that for any we have .

The assumptions (2) and (3) are necessary in general (but follow from (1) if the family is countable).

Another fundamental fact about VC-families that we will need is the following theorem about transversal sets due to Matousek. It uses the following definition: a family of subsets of some set has the -property if among any sets in , some have nonempty intersection.

Fact 2.2 (Reference Mat04).

Let be a family of subsets of some set . Assume that has finite VC-dimension. Then there is some such that for every , there is an integer such that for every finite subfamily , if has the -property, then there is an -point set intersecting all members of .

2.2. Forking in NIP theories

We will use standard notation. We work with a complete theory in a language . We fix a monster model which is -saturated and -strongly homogeneous for a sufficiently large strong limit cardinal.

Recall that a formula is NIP if the family of subsets has finite VC-dimension. The theory is NIP if all formulas are NIP. In this paper, we always assume that is NIP unless explicitly stated otherwise.

We summarize some facts about forking in NIP theories. Recall that a set is an extension base if every type has a global extension nonforking over . In particular, any model of an arbitrary theory is an extension base, and every set is an extension base in o-minimal theories, algebraically closed valued fields, or -adics.

Definition 2.3 (Reference CK12).
(1)

A global type is strictly nonforking over a small model if does not fork over , and for every and , does not fork over .

(2)

Given , we say that is a strict Morley sequence in if there is some global extension of strictly nonforking over satisfying for all .

Fact 2.4 (Reference CK12).

Assume that is NIP, and let be an extension base.

(1)

A formula forks over if and only if it divides over , i.e., the set of formulas dividing over forms an ideal.

(2)

Every admits a global extension strictly nonforking over .

(3)

Assume that forks (equivalently, divides) over , and let in be an infinite strict Morley sequence in . Then is inconsistent.

From now on, we will freely use the equivalence of forking and dividing over models in NIP theories.

Fact 2.5 (See, e.g., Reference HP11, Proposition).

Assume that is NIP and . A global type does not fork (equivalently, does not divide) over if and only if it is -invariant. This is, for every and , we have .

Remark 2.6.

In particular, in view of Fact 2.4, if is a partial type that does not divide over (e.g., if is -invariant), then it extends to a global -invariant type.

Now let , be global types invariant over . For any set , let . Then by invariance of and , the type does not depend on the choice of . Call this type , and let

Then is a well-defined, global invariant type over .

Let be a global type invariant over . Then one defines

For any small set and , the sequence is indiscernible over .

We now discuss Borel definability. Let be a global -invariant type, pick a formula , and consider the set . By invariance, this set is a union of types over . In fact, it can be written as a finite boolean combination of -type-definable sets (Reference HP11). Specifically, let and let and be the type-definable subsets of defined by

and

respectively.

Then for some , .

Note that the set of all global -invariant types is a closed subset of . We now consider the local situation. Let be a fixed formula, and let be the space of all global -types (i.e., maximal consistent collections of formulas of the form ). Let be the set of all global -invariant -types—a closed subset of , which we equip with the induced topology.

Fact 2.7 (Reference Sim15b).

Let be a countable model, and let be NIP. For any set and , if (i.e., in the topological closure of ), then is the limit of a countable sequence of elements of .

2.3. Keisler measures

Now we introduce some terminology and basic results around the study of measures in model theory. A Keisler measure (or ) over a set of parameters is a finitely additive probability measure on the boolean algebra of -definable subsets of in the variable . Alternatively, a Keisler measure may be viewed as assigning a measure to the clopen basis of the space of types . A standard argument shows that it can be extended in a unique way to a countably additive regular probability measure on all Borel subsets of (see, e.g., Reference Sim15a, Chapter 7 for details). From now on we will just say “measure” unless it could create some confusion.

For a measure over we denote by its support: the set of types weakly random for , i.e., the closed set of all such that for any , implies .

Remark 2.8.

Let denote the set of measures over in variable ; it is naturally equipped with a compact topology as a closed subset of with the product topology. Every type over can be identified with the -measure concentrating on it; thus is identified with a closed subset of .

The following implication of Fact 2.1 was observed in Reference HP11, Section 4.

Fact 2.9.

Let be NIP. Let a measure over , let be a finite set of -formulas, and let be arbitrary. Then there are some types such that for every and , we have

Furthermore, we may assume that , the support of , for all .

Corollary 2.10.

Let be an NIP theory in a countable language , and let be a measure. Then the support is separable (with respect to the topology induced from .

Proof.

By Fact 2.9 for any finite and , we can find some such that for any and any we have

Let . Then is a countable subset of , and we claim that it is dense. Let be a nonempty open subset of . Then there is some formula such that . In particular , hence for some and large enough, we have by the construction of that necessarily for at least one .

A measure is nonforking over a small model if for every formula with , does not fork over . A theory of forking for measures in NIP generalizing the previous section from types to measures is developed in Reference HP11Reference HPS13. In particular, a global measure nonforking over a small model is in fact -invariant. Moreover, using Fact 2.9 along with results in section 2.2, one shows that a global measure invariant over is Borel definable over , i.e., for any the map is Borel (and it is well-defined by -invariance of ). This allows us to define a tensor product of -invariant measures: Given -invariant and , let be some small model over which is defined. We define by taking , where is viewed as a Borel measure on . Then is a global -invariant measure.

We will need the following basic combinatorial fact about measures (see Reference HPP08 or Reference Sim15a, Lemma 7.5).

Fact 2.11.

Let be a Keisler measure, let be a formula, and let be an indiscernible sequence. Assume that for some we have for every . Then the partial type is consistent.

2.4. Model-theoretic connected components

Now let be a definable group. Let be a small subset of . We say that has bounded index if is smaller than the saturation of , and we define the following:

.

.

.

Of course for any and these are all normal -invariant subgroups of .

Fact 2.12 (See, e.g., Reference Sim15a, Chapter 8 and references therein).

Let be NIP. Then for every small set we have , , . Moreover, .

We will be omitting in the subscript and write, for instance, for .

Remark 2.13.

It follows that is equal to the subgroup of generated by the set for any small model .

Let be the canonical projection map.

The quotient can be equipped with a natural “logic” topology: a set is closed iff is type-definable over some (equivalently, any) small model .

Fact 2.14 (Reference Pil04).

The group equipped with the logic topology is a compact topological group.

Remark 2.15.

If is countable, then is a Polish space with respect to the logic topology. Indeed, there is a countable model such that every closed set is a projection of a partial type over , and is a countable basis of the topology.

In particular, admits an invariant normalized Haar probability measure . Furthermore, is the unique left--invariant Borel probability measure on (see, e.g., Reference Hal50, Section 60), as well as simultaneously the unique right--invariant Borel probability measure on .

The usual completion procedure for a measure preserves -invariance, so we may take to be complete.

3. Generic sets and measures

3.1. -dividing, bounded orbits, and definable amenability

Context: We work in an NIP theory , and let be an -definable group.

We will consider as acting on itself on the left. For any model , this action extends to an action of on the space of types concentrating on . Hence, if and , we have where . The group also acts on -definable subsets of by and on measures by .

One could also consider the right action of on itself and obtain corresponding notions. Contrary to the theory of stable groups, this would not yield equivalent definitions (see section 6.1 for a discussion).

Definition 3.1.

The group is definably amenable if it admits a global Keisler measure on definable subsets of which is invariant under (left-) translation by elements of .

As explained for example in Reference Sim15a, 8.2, if for some model , there is a -invariant Keisler measure on -definable subsets of , then is definably amenable (it can be seen by taking an elementary extension expanded by predicates for the invariant measure).

Definition 3.2.
(1)

Let be a subset of defined over some model . We say that (left-) -divides if there is an -indiscernible sequence such that is inconsistent.

(2)

The formula is (left-) -generic over if no translate of forks over . We say that is -generic if it is -generic over some small . A (partial) type is -generic if every formula implied by it is -generic.

(3)

A global type is called (left-) strongly f-generic over if no -translate of forks over . A global type is strongly -generic if it is strongly -generic over some small model .

Note that we change the usual terminology: our notion of strongly -generic corresponds to what was previously called -generic in the literature (see, e.g., Reference HP11). We feel that this change is justified by the development of the theory presented here.

Note that if is a global -invariant and -invariant measure and , then is strongly -generic over since all its translates are weakly random for . It is shown in Reference HP11 how to conversely obtain a measure from a strongly -generic type . We summarize some of the results from Reference HP11 in the following fact.

Recall that the stabilizer of is .

Fact 3.3.
(1)

If admits a strongly -generic type over some small model , then it admits a strongly -generic type over any model .

(2)

If is strongly -generic, then for any small model ).

(3)

The group admits a -invariant measure if and only if there is a global strongly -generic type in .

Our first task is to understand basic properties of -generic formulas and types.

Proposition 3.4.

Let be a definably amenable group, and let . Let also be strongly -generic, -invariant and take . Then the following are equivalent:

(1)

is -generic over ;

(2)

does not -divide;

(3)

does not fork over .

Proof.

: Assume that some translate forks over . Then it divides over , and as is over , we obtain an -indiscernible sequence such that is inconsistent. This shows that -divides.

: This is clear.

: Assume that does -divide, and let be an -indiscernible sequence witnessing it; i.e., is -inconsistent for some . By indiscernibility, all of the ’s are in the same -coset, and replacing by , we may assume that for all .

Let realize over . Then by -invariance of . As the set is inconsistent, so is . Then the sequence is an -indiscernible sequence in (as is -invariant). Therefore, divides over .

Note that we do not say -divides over ”, because the model does not matter in the definition: for any , an -definable -divides over if and only if it -divides over . Therefore, the same is true for -genericity (i.e., if is both -definable and -definable, then it is -generic over if and only if it is -generic over ), and from now on we will just say -generic, without specifying the base.

Corollary 3.5.

Let be definably amenable. The family of nonf-generic formulas (equivalently, -dividing formulas) forms an ideal. In particular, every partial -generic type extends to a global one.

Proof.

Assume that are not -generic, and let be some small model over which both formulas are defined. Also let be a global type strongly -generic over (exists by Fact 3.3) and take . Then by Fact 3.4(3) we have that both fork over , in which case also forks over . Applying Fact 3.4(3) again, it follows that is not -generic.

The “in particular” statement follows by compactness.

Lemma 3.6.

Let be definably amenable, let be a formula, and let . Then is not -generic (and hence it -divides by Proposition 3.4).

Proof.

Let be a model over which and are defined. Let be a global strongly -generic type which is -invariant (exists by Fact 3.3(1)), and let realize over . Then . Since (as by Fact 3.3(2)), the latter formula cannot belong to any global -invariant type, and so it must fork over by Remark 2.6. Hence is not -generic.

Definition 3.7.

A global type has a bounded orbit if for some strong limit cardinal such that is -saturated.

Proposition 3.8.

Let be definably amenable. For , the following are equivalent:

(1)

is -generic;

(2)

is -invariant (and

(3)

has a bounded orbit.

Proof.

(1) (2): If is not -invariant, then for some , and so is not -generic by Lemma 3.6. Hence, . Given an arbitrary , let be a small model containing and let . Then , hence and . By Fact 3.3(2) it follows that , hence .

(2) (3): If is -invariant, then the size of its orbit is bounded by the index of (which is ).

(3) (1): If is not -generic, then some must -divide (by Proposition 3.4). Then, as in the proof of Proposition 3.4, we can find an arbitrarily long indiscernible sequence in such that is -inconsistent for some , which implies that the -orbit of is unbounded.

Next we clarify the relationship between -generic and strongly -generic types in definably amenable groups.

Proposition 3.9.

Let be definably amenable. A type is strongly -generic if and only if it is -generic and -invariant over some small model .

Proof.

Strongly -generic implies -generic is clear.

Conversely, assume that is -invariant but not strongly -generic over . Then divides over for some . It follows that there is some such that for any there is some -indiscernible sequence with and such that is -inconsistent for some . By -invariance of we have that , so is -inconsistent. This implies that the orbit of is unbounded and that is not -generic in view of Proposition 3.8.

Example 3.10.

There are -generic types which are not strongly -generic. Let be a saturated model of RCF. We give an example of a -invariant (and so -generic by Proposition 3.8) type in which is not invariant over any small model (and so not strongly -generic by Proposition 3.9). Let denote the definable 1-type at , and let denote a global 1-type which is not invariant over any small model (hence corresponds to a cut of maximal cofinality from both sides). Then and are weakly orthogonal types. Let (in some bigger model), and consider . Then is a -invariant type which is not invariant over any small model.

The following lemma is standard.

Lemma 3.11.

Let be -saturated, and let be such that does not fork over for every . Then extends to a global type strongly -generic over .

Proof.

It is enough to show that

is consistent. Assume not. Then for some , and such that forks over . By -saturation of and compactness, we can find some in such that , which implies that for some , i.e., forks over . But this contradicts the assumption on .

Finally for this subsection, we prove Theorem 1.1. For NIP groups, definable amenability is characterized by the existence of a type with a bounded orbit, proving Petrykowski’s conjecture for NIP theories (see Reference New12, Conjecture 0.1). In fact, existence of a measure with a bounded orbit is sufficient.

Theorem 3.12.

Let be NIP, let , and let be a definable group. Then the following are equivalent:

(1)

is definably amenable;

(2)

for some

(3)

some measure has a bounded -orbit.

Proof.

(1) (2): If is definably amenable, then it has a strongly -generic type by Fact 3.3 and such a type is -invariant. In particular its orbit has size at most .

(2) (3): This is obvious.

(3) (1): Assume that , with strong limit and is -saturated. Let be a model with , let be an -saturated submodel of of size (exists as is a strong limit cardinal), and let be a strict Morley sequence in contained in (exists by -saturation of and Fact 2.4(2)). In particular is an -saturated extension of for all .

Let . It is enough to show that for some , the measure does not fork over for any , as then any type in the support of extends to a global type strongly -generic over by Lemma 3.11, and we can conclude by Fact 3.3.

Assume not. Then for each we have some and some such that but forks over .

As the orbit of is bounded, by throwing away some ’s we may assume that there is some such that for all , in particular . By the pigeonhole principle and the assumption on we may assume also that there are some and such that and for all , and that the sequence is indiscernible (i.e., the ’s occupy the same place in the enumeration of , for all , and the sequence is indiscernible by construction). Applying Fact 2.11 to the measure we conclude that is consistent. But as is a strict Morley sequence, this contradicts the assumption that divides over for all , in view of Fact 2.4(3).

Remark 3.13.
(1)

In the special case of types in -minimal expansions of real closed fields, this was proved in Reference CP12, Corollary 4.12.

(2)

Theorem 3.12 also shows that the issues with absoluteness of the existence of a bounded orbit considered in Reference New12 do not arise when one restricts to NIP groups.

3.2. Measures in definably amenable groups

3.2.1. Construction

Again, we are assuming throughout this section that is an NIP group. We generalize the connection between -invariant measures and strongly -generic types from Fact 3.3 to -generic types in definably amenable groups.

First we generalize Proposition 3.8 to measures.

Proposition 3.14.

Let be definably amenable, and let be a Keisler measure on . The following are equivalent:

(1)

The measure is -generic, that is implies is -generic for all .

(2)

All types in the support are -generic.

(3)

The measure is -invariant.

(4)

The orbit of is bounded.

Proof.

The equivalence of (1) and (2) is clear by compactness, (1) implies (3) is immediate by Lemma 3.6, and (3) implies (4) as the size of the orbit of a -invariant measure is bounded by .

: Assume that we have some -dividing with . As in the proof of Proposition 3.4 (3) (2), we can find an arbitrarily long indiscernible sequence with such that is -inconsistent, for some fixed .

In view of Fact 2.11 for any fixed there can be only finitely many such that . But . This implies that for all but finitely many , which then implies that the orbit of is unbounded.

In Reference HP11, Proposition 5.6 it is shown that one can lift the Haar measure on to a global -invariant measure on all definable subsets of an NIP group using a strongly -generic type. We point out that in a definably amenable NIP group, an -generic type works just as well. For this we need a local version of the argument used there.

Fix a small model , and let be the set of formulas of the form or , for .

Proposition 3.15.

Let be definably amenable, and let be a maximal finitely consistent set of formulas in . Then is -generic if and only if is -invariant for every .

Proof.

Notice that is also a set of formulas in . Assume that is not -invariant. Then for some and . Hence and (by Fact 3.3(2)). Then is not -generic by Lemma 3.6, and so is not -generic, a contradiction.

Conversely, assume that some formula implied by is not -generic. Let contain the parameters of . Then there is some indiscernible over such that is -inconsistent. Then , but for some . So is not -invariant and, thus, also not -invariant.

Definition 3.16.

Let be definably amenable, and let be -generic. Keeping in mind that (as well as all its translates) is -invariant (by Proposition 3.8), we define a measure on by

where is the normalized Haar measure on the compact group and .

We have to check that this definition makes sense; that is, that the set we take the measure of is indeed measurable. Let be a small model over which is defined. Let be the restriction of to formulas from (as defined above). By Proposition 3.15, is -invariant. It follows that extends to some complete -invariant type (by Remark 2.6). Then we can use Borel definability of invariant types (applied to the family of all translates of ), exactly as in Reference HP11, Proposition 5.6 to conclude.

Remark 3.17.
(1)

The measure that we just constructed is clearly -invariant and -strongly invariant (that is, for ). Besides, for any , .

(2)

We have . Indeed, if and arbitrary, then , which by the definition of implies that for some .

Question 3.18.

Footnote1 Let be an NIP group. Are the following two properties equivalent?

1

We have claimed an affirmative answer in an earlier version of this article, however a mistake in our argument was pointed out by the referees.

(1)

is definably amenable.

(2)

admits a global -generic type (equivalently, the family of all non--generic subsets of is an ideal).

3.2.2. Approximation lemmas

Throughout this section, is a definably amenable NIP group. Given a -invariant type and a formula , let .

Note that and , where is the image of in .

Lemma 3.19.

For a fixed formula , let be the family of all where varies over and varies over all -generic types on . Then has finite VC-dimension.

Proof.

Let be shattered by . Then for any there is some which cuts out that subset. Take representatives of the ’s. Let , then we have if and only if . Hence the VC-dimension of is at most that of , so it is finite by NIP.

Replacing the formula by , we may assume that any translate of an instance of is again an instance of . Note also that then for any parameters we have

for some . Using this and applying Lemma 3.19 to , we get the following corollary.

Corollary 3.20.

For any , the family

has finite VC-dimension.

We would now like to apply the VC-theorem to . This requires verifying an additional technical hypothesis (assumptions (2) and (3) in Fact 2.1), which we are only able to show for certain (sufficiently representative) subfamilies of .

Fix and let be a set of global -generic types. Let

Lemma 3.21.

If is countable and is countable, then satisfies all of the assumptions of Fact 2.1 with respect to the measure .

Proof.

First of all, the family of sets has finite VC-dimension by Corollary 3.20 and the obvious inclusion .

Next, (1) is satisfied by the assumption that consists of -generic types and an argument as in the discussion after Definition 3.16 (using countability of the language).

For a set of global -generic types, let

For (2) and (3) we need to show that and are measurable for all . Note that and . Since is countable, it is enough to show that for a fixed -generic type the functions and are measurable.

Let . By -invariance of on both the left and the right, we have

and

Then it is enough to show that for a fixed , the set

is measurable. But we can write as the projection of where is the intersection of for and for . As group multiplication is continuous and is Borel, those sets are Borel as well. Hence is analytic. Now is a Polish space (as is countable, by Remark 2.15), and analytic subsets of Polish spaces are universally measurable (see, e.g., Reference Kec95, Theorem 29(7)). In particular they are measurable with respect to the complete Haar measure .

The next lemma will allow us to reduce to a countable sublanguage.

Lemma 3.22.

Let be a sublanguage of , let be the -reduct of , let be an -definable group definably amenable (in the sense of and let be a formula from . Let be a global -type which is -generic, and let .

(1)

In the sense of , the group is definably amenable NIP and is an -generic type.

(2)

Let be the connected component computed in , and let be the -invariant measure on -definable (resp., -definable) subsets of given by Definition 3.16 in (resp., in . Then .

Proof.

(1) The first assertion is clear. Similarly, it is easy to see that if is -dividing in , then it is -dividing in (by extracting an -indiscernible sequence from an indiscernible sequence). Then is -generic by Fact 3.4 applied in .

(2) Let and , then by definition and , where is the Haar measure on and is the Haar measure on . The map is a surjective group homomorphism, and it is continuous with respect to the logic topology. Note that for any we have , so . Let be the push-forward measure, it is an invariant measure on . But by the uniqueness of the Haar measure, it follows that , and so , i.e., as wanted.

Proposition 3.23.

For any , and a countable set of -generic types there are some such that for any and we have .

Proof.

First assume that the language is countable. Using Lemma 3.21, we can apply the VC-theorem (Fact 2.1) to the family and find some such that for any we have . Let be some representative of , for . Let and be arbitrary. Recall that and that , where . Then and we have .

Now let be an arbitrary language, let be an arbitrary countable sublanguage such that and is -definable, and let be the corresponding reduct. Let , by Lemma 3.22 it is a countable set of -generic types in the sense of . Applying the countable case with respect to inside , we find some such that for any and we have . Let be arbitrary, and take . On the one hand, the right-hand side is equal to . On the other hand, as and is -generic, by Lemma 3.22 the left-hand side is equal to , as wanted.

Proposition 3.24.

Let be an -generic type, and assume that . Then is -generic and .

Proof.

First of all, is -generic because the orbit of consists of -generic types and the set of -generic types is closed.

Take a formula and , and let be as given by Proposition 3.23 for . Then we have . As , there is some such that for each we have . But we also have , which together with implies . As and were arbitrary, we conclude.

Proposition 3.25.

Let be an -generic type. Then for any definable set , if , then there is a finite union of translates of which covers the support (so in particular it has -measure .

Proof.

As (Remark 3.17), any type weakly random for is -generic and satisfies by Proposition 3.24. Hence , so some translate of must be in . It follows that the closed compact set can be covered by translates of , so by finitely many of them.

Lemma 3.26.

Let be -invariant. Then for any and , there are some -generic such that

for any .

Proof.

As before, we may assume that every translate of an instance of is an instance of . Fix .

By Fact 2.9 there are some such that for all . It follows by -invariance of and the assumption on that for any and , .

By Proposition 3.14, all of the ’s are -generic. By Proposition 3.23 with , for every there are some such that for any , .

So let be arbitrary, and choose the corresponding for it. By the previous remarks we have

Thus .

Corollary 3.27.

Let be a -invariant measure, and assume that for some -generic . Then .

Proof.

Let and let be arbitrary. By Lemma 3.26 we can find some -generic such that . But as , it follows by Proposition 3.24 that for all , so .

3.3. Weak genericity and almost periodic types

Now we return to the notions of genericity for definable subsets of definable groups and add to the picture another one motivated by topological dynamics, due to Newelski.

We will be using the standard terminology from topological dynamics: Given a group , a -flow is a compact space equipped with an action of such that every , is a homeomorphism of . We will usually write a -flow as a pair . A set is said to be a subflow if is closed and -invariant. The flows relevant to us are and for a small model .

Definition 3.28 (Reference New09Reference Poi87).
(1)

A formula is (left-) generic if there are some finitely many such that .

(2)

A formula is (left-) weakly generic if there is formula which is not generic but such that is generic.

(3)

A (partial) type is (weakly) generic if it only contains (weakly) generic formulas.

(4)

A type is called almost periodic if it belongs to a minimal flow in (i.e., a minimal -invariant closed set), equivalently if for any we have .

Fact 3.29 (Reference New09, Section 1).

The following hold, in an arbitrary theory:

(1)

The formula is weakly generic if and only if for some finite , is not generic.

(2)

The set of nonweakly generic formulas forms a -invariant ideal. In particular, there are always global weakly generic types by compactness.

(3)

The set of all weakly generic types is exactly the closure of the set of all almost periodic types in .

(4)

Every generic type is weakly generic. Moreover, if there is a global generic type, then every weakly generic type is generic, and the set of generic types is the unique minimal flow in .

(5)

A type is almost periodic if and only if for every , the set is covered by finitely many left translates of .

We connect these definitions to the notions of genericity from the previous sections. As before, we always assume that is NIP.

Proposition 3.30.

Let be definably amenable, and let be a weakly generic formula. Then it is -generic.

Proof.

We adapt the argument from Reference NP06, Lemma 1.8. As is weakly generic, let be nongeneric, and let be a finite set such that . We may assume that and that is defined over . Assume that is not -generic over . The set of formulas which are not -generic is -invariant, and moreover it is an ideal by Corollary 3.5. Thus is not -generic, which implies that there is some such that divides over . That is, there is an -indiscernible sequence such that .

As , we also have for every . Thus . But this means that is generic, a contradiction.

Proposition 3.31.

Assume that is definably amenable.

(1)

If is almost periodic, then it is -generic and .

(2)

Minimal flows in are exactly the sets of the form for some -generic .

(3)

If are almost periodic and then .

Proof.

(1) An almost periodic type contains only weakly generic formulas and hence is -generic by Proposition 3.30. As (see Remark 3.17), it follows by minimality that .

(2) For an -generic , the set is a subflow by -invariance of . If and , then and by Proposition 3.25 there are finitely many translates of which cover , so in particular they cover . Thus is almost periodic (by the usual characterization of almost periodic types from Fact 3.29(5)).

(3) This is clear.

In particular, for any -generic type there is some almost periodic type with . However, the following question remains open.⁠Footnote2

2

While this paper was under review, a negative answer was obtained in Reference PY16.

Question 3.32.

Is every -generic type almost periodic? Equivalently, does always hold?

Now toward the converse.

Proposition 3.33.

Let be definably amenable. Assume that does not -divide. Then there are some global almost periodic types such that for any there is some such that holds.

Proof.

Let be as given by Fact 2.2 for the VC-family . We claim that satisfies the -property for some . If not, then by compactness we can find an infinite indiscernible sequence in such that is -inconsistent, and so -divides.

By Fact 2.2 and compactness it follows that there are some which satisfy

for every , for some , we have .

Now consider the action of on with the product topology, and let

It is a subflow, and besides every satisfies . (It is clear for translates of ; if for some we have , then since is an open subset of with respect to the product topology containing , it follows that belongs to it for some , which is impossible.) Let be a minimal subflow of , and notice that the projection of on any coordinate is a minimal subflow of . Thus, taking , it follows that is almost periodic for every , and every translate of belongs to one of the .

Corollary 3.34.

Let be definably amenable. If is -generic, then for some global -generic type .

Proof.

Let be some global almost periodic types given by Proposition 3.33, which are also -generic by Proposition 3.31. Let . As and each of the ’s is measurable, it follows that for some . But then .

Summarizing, we have demonstrated that all notions of genericity that we have considered coincide in definable amenable NIP groups.

Theorem 3.35.

Let be definably amenable NIP. Let be a definable subset of . Then the following are equivalent:

(1)

is -generic;

(2)

is not -dividing;

(3)

is weakly-generic;

(4)

for some -invariant measure

(5)

for some global -generic type .

Proof.

(1) and (2) are equivalent by Proposition 3.4, (1) implies (3) by Proposition 3.33, and (3) implies (1) by Proposition 3.30. Finally, (1) implies (5) by Corollary 3.34, that (5) implies (4) is obvious, and (4) implies (1) by Lemma 3.14.

3.4. Unique ergodicity

We now characterize the case when admits a unique -invariant measure. Following standard terminology in topological dynamics, we call such a uniquely ergodic (indeed, it will follow from the next section in which this condition is equivalent to having a unique regular ergodic measure).

Recall that a -invariant measure is called generic if for any definable set , implies that is generic. It follows that any is generic.

Theorem 3.36.

A definably amenable NIP group is uniquely ergodic if and only if it admits a generic type (in which case it has a unique minimal flow—the support of the unique measure).

Proof.

If admits a generic type , then for any type , belongs to the closure (if then for some , so for some ). In particular, for an arbitrary -generic type we have (by Proposition 3.24). By Lemma 3.26, this implies that any invariant measure is equal to , hence there is a unique invariant measure.

Conversely, assume that admits a unique -invariant measure . We claim that is generic. Assume not, and let be a definable set of positive -measure, and assume that is not generic. Then for any , the union is not generic. Hence its complement is weakly generic. By Theorem 3.35 we conclude that the partial type is -generic and hence extends to a complete -generic type . The measure associated to gives measure 0, so , which contradicts unique ergodicity.

Remark 3.37.

In particular, in a uniquely ergodic group every -generic type is almost periodic and generic.

Recall from Reference HP11 that an NIP group is fsg if it admits a global type such that for some small model , all translates of are finitely satisfiable over . It is proved that an fsg group admits a unique invariant measure and that this measure is generic. So the previous proposition was known in this special case. We now give an example (pointed out to us by Hrushovski) of a uniquely ergodic group which is not fsg.

Remark 3.38.

Let be a model of ACVF and consider the additive group. By C-minimality, the partial type concentrating on the complement of all balls is a complete type and is -invariant. There can be no other -invariant measure since any nontrivial ball in -divides, hence cannot have positive measure for any -invariant measure. Finally, the group is not fsg since is not finitely satisfiable.

4. Regular ergodic measures

In this section, we are going to characterize regular ergodic measures on for a definably amenable NIP group , but first we recall some general notions and facts from functional analysis and ergodic theory (see, e.g., Reference Wal82). As we are going to deal with more delicate measure-theoretic issues here, we will be specific about our measures being regular or not. The reader should keep in mind that all the results in the previous sections only apply to regular measures on .

The set of all regular (Borel, probability) measures on can be naturally viewed as a subset of , the dual space of the topological vector space of continuous functions on , with the weak topology of pointwise convergence (i.e., if for all ). It is easy to check that this topology coincides with the logic topology on the space of measures (Remark 2.8). This space carries a natural structure of a real topological vector space containing a compact convex set of -invariant measures.

We will need the following version of a “converse” to the Krein–Milman theorem (see, e.g., Reference Jer54, Theorem 1, and we refer to, e.g., Reference Sim11, Chapter 8 for a discussion of convexity in topological spaces).

Fact 4.1.

Let be a real, locally convex Hausdorff topological vector space. Let be a compact convex subset of , and let be a subset of . Then the following are equivalent:

(1)

, the closed convex hull of .

(2)

The closure of includes all extreme points of .

Now we recall the definition of an ergodic measure.

Fact 4.2 (Reference Phe01, Proposition 12.4).

Let be a group acting on a topological space with a Borel map for each , and let be a -invariant Borel probability measure on . Then the following are equivalent:

(1)

The measure is an extreme point of the convex set of -invariant measures on .

(2)

For every Borel set such that for all , we have that either or .

A -invariant measure is ergodic if it satisfies any of the equivalent conditions above. Under many natural conditions on and the two notions above are equivalent to the following property of : for every -invariant Borel set , either or . However this is not the case in general.

Proposition 4.3.

The map from the (closed) set of global -generic types to the (closed) set of global -invariant measures on is continuous.

Proof.

Fix and , and let be the set of all global -generic with . It is enough to show that is closed. Let belong to the closure of , in particular is -generic. Let be some countable language such that is -definable and , and let .

Now let be some countable model of over which is defined, and let . Let , i.e., the restriction of to all formulas of the form , , and let . By Lemma 3.22, and all elements of are -generic in the sense of . By Lemma 3.15 applied in we know that and all elements of are -invariant. Working in , let be the space of all global -types invariant over . It follows from the assumption that (i.e., the closure of in the sense of the topology on ).

By Fact 2.7 we know that is a limit of a countable sequence of types from . Each of is -generic in , so in as well (easy to verify using equivalence to -dividing both in and ), and it extends to some global -generic -type by Corollary 3.5.

Now work in , and let be arbitrary. By Proposition 3.23 with , there are some such that for all , as well as . As for any , , and the same for , it follows that for all large enough, we have . But this implies that for any , , and so and .

Corollary 4.4.
(1)

The set is closed in the set of all -invariant measures.

(2)

Given a -invariant measure , the set of -generic types for which is a subflow.

Proof.

This follows from Proposition 4.3.

Theorem 4.5.

Let be definably amenable. Then regular ergodic measures on are exactly the measures of the form for some -generic .

Proof.

Fix a global -generic type , and assume that is not an extreme point. Then there is some and some -invariant measures such that . First, it is easy to verify using regularity of that both and are regular. Second, it follows that . By Corollary 3.27 which we may apply as are regular, it follows that , a contradiction.

Now for the converse, let be an arbitrary regular -invariant measure which is an extreme point, and let . Let be the closed convex hull of . By Lemma 3.26, is a limit of the averages of measures from , so and it is still an extreme point of . Then we actually have (by Fact 4.1, as (1) is automatically satisfied for , then (2) holds as well). But by Corollary 4.4(1).

Corollary 4.6.

The set of all regular ergodic measures in is closed.

Let denote the closed -invariant set of all -generic types in . By Proposition 3.8 we have a well-defined action of on (not necessarily continuous or even measurable). If is an arbitrary regular -invariant measure, then by Proposition 3.14, and we can naturally view as a -invariant measure on Borel subsets of .

Question 4.7.

Consider the action . Is it measurable? It is easy to see that is continuous for a fixed and measurable for a fixed . In many situations this is sufficient for joint measurability of the map, but our case does not seem to be covered by any result in the literature.

5. Generic compact domination and the Ellis group conjecture

5.1. Baire-generic compact domination

Let be a definably amenable NIP group, and let be a small model of . Let be a global type strongly -generic over . Let be the canonical projection. It naturally lifts to a continuous map . Fix a formula , and we define .

Proposition 5.1.

The set is a constructible subset of (namely, a boolean combination of closed sets).

Proof.

Note that with .

As explained in section 2.2, we have for some , where

Note that are type definable (over and the parameters of ). Define

These are also type-definable sets. Let . We check that . Note:

(1)

is -invariant (because is);

(2)

all of are -invariant (by definition);

(3)

.

First, if , say , then there is such that and . As , also , and so (by (2) and (3)). Hence , and by (1) also . So .

Assume that , and let be maximal for which there is such that . Then for a corresponding , we still have by (1) and (2). In particular, . As , necessarily . This means that there is some such that . As is still in by (1), it follows that for some , but by the definition of the ’s this is only possible if , contradicting the choice of . Thus .

Now, we have since and are all -invariant. As , are closed, we conclude that is constructible.

Let , and we define

Remark 5.2.

Let be an arbitrary topological space, and let be a constructible set. Then the boundary has empty interior.

Proof.

This is easily verified as is a boolean combination of closed sets, for any sets , and has empty interior if is either closed or open.

Theorem 5.3 (Baire-generic compact domination).

The set is closed and has empty interior. In particular it is meager.

Proof.

We have and , are closed subsets of , hence is closed.

We may assume that concentrates on , as replacing by for some does not change , and thus does not change .

Let be given, and let be an arbitrary open subset of containing . As the map is continuous, the set is an open subset of . By the definition of , there must exist such that and , . As , it follows that there are some such that and . But then, as concentrates on , and (where is as defined before Proposition 5.1). As was an arbitrary neigborhood of , it follows that , hence . By Proposition 5.1, is constructible. Hence has empty interior by Remark 5.2, and so has empty interior as well.

5.2. Connected components in an expansion by externally definable sets

Given a small model of , an externally definable subset of is an intersection of an -definable subset of with . One defines an expansion in a language by adding a new predicate symbol for every externally definable subset of , for all . Recall that a global type is finitely satisfiable in if lies in the topological closure of , where is identified with its image in under the map sending to the type . There is a canonical bijection (even homeomorphism) between and the subspace of types in finitely satisfiable in . Recall also that a coheir of a type is a type over a larger model which extends and is finitely satisfiable in .

Let . Note that automatically any quantifier-free -type over is definable (using -formulas). The following is a fundamental theorem of Shelah Reference She09 (see also Reference CS13 for a refined version).

Fact 5.4.

Let be NIP, and let be a model of . Then eliminates quantifiers. It follows that is NIP and that all (-) types over are definable.

Assume now that is an -definable group, and let be a monster model for such that is a monster for . In general there will be many new -definable subsets and subgroups of which are not -definable. In Reference CPS14 it is demonstrated however that many properties of definable groups are preserved when passing to .

Fact 5.5.

Let be NIP, and let be a small model of . Let be an -definable group.

(1)

If is definably amenable in the sense of , then it is definably amenable in the sense of as well.

(2)

The group computed in coincides with computed in .

In particular this implies that is the same group when computed in or in . Note also that the logic topology on computed in coincides with the logic topology computed in : any open set in the sense of is also open in the sense of , and both are compact Hausdorff topologies; therefore, they must coincide.

Remark 5.6.

In view of Remark 2.15, if is countable, then is still a Polish space with respect to the -induced logic topology.

5.3. Ellis group conjecture

We recall the setting of definable topological dynamics and enveloping semigroups (originally from Reference New09, Section 4, but we are following the notation from Reference CPS14).

Let be a small model of a theory , and assume that all types over are definable. Then acts on by homeomorphisms, and the identity element has a dense orbit. The set admits a natural semigroup structure  extending the group operation on and which is continuous in the first coordinate: for , is where realizes and realizes the unique coheir of over . This semigroup is precisely the enveloping Ellis semigroup of (see, e.g., Reference Gla07a). In particular left ideals of are precisely the closed -invariant subflows of , there is a minimal subflow and there is an idempotent . Moreover, is a subgroup of the semigroup whose isomorphism type does not depend on the choice of and . It is called the Ellis group (attached to the data). The quotient map from to factors through the tautological map from to , and we let denote the resulting map from . It is a surjective semigroup homomorphism, and for any minimal subflow of and , the restriction of to is a surjective group homomorphism.

Now, let be NIP, and let be an arbitrary model. Then we consider , an expansion of by naming all externally definable subsets of for all , in a new language extending . Then is still NIP, and all -types over are definable (by Fact 5.4), so the construction from the previous paragraph applies to . Let be a monster model for , so that is a monster model for . By Fact 5.5, if is definably amenable in the sense of , then it remains definably amenable in the sense of , and (the first one is computed in with respect to -definable subgroups, while the second one is computed in with respect to -definable subgroups). Newelski asked in Reference New09 if the Ellis group was equal to for some nice classes of groups. Gismatullin, Penazzi, and Pillay Reference GPP15 show that this is not always the case for NIP groups ( is a counterexample). The following modified conjecture was then suggested by Pillay (see Reference CPS14):

Ellis group conjecture.

Suppose is a definably amenable NIP group. Then the restriction of to is an isomorphism, for some/any minimal subflow of and idempotent (i.e., is injective).

Theorem 5.7.

The Ellis group conjecture is true; i.e., is an isomorphism.

Proof.

Fix notations as above. Throughout this proof, we work in . Let be strongly -generic over . Let , and let . Note that is a subflow of : it is closed as a continuous image of a compact set into a Hausdorff space, and it is -invariant as is -invariant. Let be a minimal subflow of . It has to be of the form for some . So replacing by (which is still strongly -generic over ) we may assume that is minimal.

Let be an idempotent. We will show that if and (i.e., they determine the same coset of ), then there is some such that . By the general theory of Ellis semigroups (see, e.g., Reference Gla07a, Proposition 2.5(5)) this will imply that , as wanted.

Let be the filter of comeager subsets of , and let be some ultrafilter extending it. Let be some global types extending , respectively. For each , let be a type in with . Let . Note that .

Let be a larger monster of . Let be such that for . For each let be the unique coheir of over , and let . Finally, let , the unique coheir of over , and let realize .

Claim 1.

for .

This follows by left continuity of the semigroup operation, but we give the details. Let be arbitrary, and let be such that , where is some small model over which is defined. Then we have

The second equivalence is by -invariance of , and the fourth one is by -invariance of .

Claim 2.

.

Assume not. Say there exists some such that , , so (according to the choice of and the definition of the semigroup operation on ). We may assume that both and concentrate on . By Claim 1 we have

As is meager by Theorem 5.3, we have , and so there is some such that .

For an arbitrary open set containing , we can choose such that and holds. Indeed, let , which is open by continuity of . Then there is an -definable set such that and . By finite satisfiability of , take satisfying . As and is closed by Theorem 5.3, we find such an with .

Note that as concentrate on , and that . It follows that a contradiction.

Corollary 5.8.

In a definably amenable NIP group, the Ellis group of the dynamical system is independent of the model .

6. Further remarks

6.1. Left vs. right actions

Until now, we have only considered the action of the group on itself by left-translations. One could also let act on the right and define analogous notions of right--generic, right-invariant measure, etc. In a stable group, a type is left-generic if and only if it is right-generic, so we obtain nothing new. However, in general, left and right notions may differ.

We start with an example of a left-invariant measure which is not right-invariant.

Example 6.1.

Let , where the two-element group acts on by multiplication. Consider as a group defined in a model of RCF with universe and multiplication defined by . Let be the type whose restriction to is the type at and which implies . Define similarly . Then is left-invariant, but not right-invariant.

However, some things can be said.

Lemma 6.2.

Let be definably amenable. Then there is always a measure on which is both left- and right-invariant.

Proof.

Let be a left-invariant measure on which is also invariant over some small model (always exists in a definably amenable NIP group, e.g., by Reference HP11, Lemma 5.8).

Let be defined by for every definable set , where . Then is also a measure, -invariant (as for any automorphism ) and right invariant (as for any ).

For any , we define . That is, for any definable set and a model containing and such that is -definable, we have , where for every , for some/any (well-defined by -invariance of , see section 2.3). Then is an -invariant measure, and given any and such that and are -definable, for any and we have

(1)

, by left invariance of .

(2)

, and

as by right invariance.

Hence is both left- and right-invariant.

Proposition 6.3.

Let be definably amenable, and let . If is left-generic, then it is right--generic.

Proof.

By the previous lemma, let be a left- and right-invariant measure on . Then as is left-generic, we must have . But as is also right-invariant, this implies that is right--generic (by the right-hand side counterpart of Proposition 3.14).

As the following example shows, no other implication holds.

Example 6.4.

Let be a saturated real closed field, and let with the canonical action, seen as a definable group in . For let be the angular region defined by . Finally, let .

Note that any two translates of intersect. Hence any two right translates of intersect: Let . Then ; hence is nonempty and in fact has surjective projection on . This shows that is right--generic.

On the other hand, multiplying on the left has the effect of turning it: . If is infinitesimal, then there are infinitely many pairwise disjoint left-translates of , hence is not left--generic. If however is not infinitesimal, then we can cover by finitely many -conjugates of , and hence cover by finitely many left-translates of .

We conclude that if is infinitesimal, then is right--generic but not left--generic, and if is not infinitesimal, then is left-generic but not right-generic.

6.2. Actions on definable homogeneous spaces

While the theory above was developed for the action of a definably amenable group on , we remark that (with obvious rephrasements) it works just as well for a definably amenable group acting on for a definable homogeneous -space (i.e., is a definable set, the graph of the action map is definable, and the action is transitive). We show that given a definable homogeneous space for a definably amenable group , every -invariant measure on pushes forward to a -invariant measure on and, conversely, any -invariant measure on lifts to a -invariant measure on , possibly nonuniquely.

Lemma 6.5.

Let be a boolean algebra, and let be an ideal such that is contained in the zero-ideal of , a measure on .

Let be the collection of all sets for which there is some such that . Then is a boolean algebra with . Moreover, extends to a global measure on such that all sets from have -measure .

Proof.

It can be checked straightforwardly that is a boolean algebra containing and . Now for , let , where is some set in with .

(1)

is well-defined. If we have some with , then , so . But by assumption this implies that , so .

(2)

is a measure on extending . Given , let be such that . Then , as wanted.

Now extends to a global measure by compactness; see, e.g., Reference Sim15a, Lemma 7.3.

Proposition 6.6.

Let be a definable homogeneous -space, and let be an arbitrary point in .

(1)

Let be a measure on . For every definable subset of , let . Then is a measure on . Moreover, if is -invariant, then is -invariant as well. If is also right-invariant, then does not depend on the choice of .

(2)

Assume moreover that is definably amenable NIP. Let be a -invariant measure on . Then there is some (possibly nonunique) -invariant measure on such that the procedure from induces .

Proof.

(1) It is clearly a measure as , and if are disjoint subsets of , then are disjoint subsets of . If is -invariant, then for any , we have .

Finally, assume that is also right-invariant. Let and be arbitrary. Then by transitivity of the action there is some such that . We have , as wanted.

(2) Now let be a -invariant measure on , and fix . Let be the family of subsets of of the form , where is a definable subset of . For , define . The following can be easily verified using that is a -invariant measure:

Claim.

The family is a boolean algebra closed under -translates, and is a -invariant measure on .

Next, let be the collection of all non--generic definable subsets of . We know by Corollary 3.5 that it is an ideal. As in Proposition 3.14, is contained in the zero-ideal of . Then, applying Lemma 6.5, we obtain a global measure on extending and such that all types in its support are -generic. Note that is -invariant: for any and there are some such that for any , (by Fact 2.9), and each is -invariant (by Proposition 3.8). Consider the map . It is well-defined and -measurable (using an argument as in the proof of Lemma 3.21). Finally, we define . It is easy to check that is a -invariant measure on (and that the procedure from (1) applied to returns ).

Acknowledgments

We are grateful to the referees for pointing out a mistake in the previous version of the article (see Question 3.18) and for other numerous suggestions on improving the presentation.

Table of Contents

  1. Abstract
  2. 1. Introduction
    1. Theorem 1.1.
    2. Theorem 1.2.
    3. Theorem 1.3.
    4. Theorem 1.4.
    5. Theorem 1.5 (Ellis group conjecture).
  3. 2. Preliminaries
    1. 2.1. Combinatorics of VC-families
    2. Fact 2.1.
    3. Fact 2.2 (Mat04).
    4. 2.2. Forking in NIP theories
    5. Definition 2.3 (CK12).
    6. Fact 2.4 (CK12).
    7. Fact 2.5 (See, e.g., HP11, Proposition).
    8. Fact 2.7 (Sim15b).
    9. 2.3. Keisler measures
    10. Fact 2.9.
    11. Corollary 2.10.
    12. Fact 2.11.
    13. 2.4. Model-theoretic connected components
    14. Fact 2.12 (See, e.g., Sim15a, Chapter 8 and references therein).
    15. Fact 2.14 (Pil04).
  4. 3. Generic sets and measures
    1. 3.1. -dividing, bounded orbits, and definable amenability
    2. Definition 3.1.
    3. Definition 3.2.
    4. Fact 3.3.
    5. Proposition 3.4.
    6. Corollary 3.5.
    7. Lemma 3.6.
    8. Definition 3.7.
    9. Proposition 3.8.
    10. Proposition 3.9.
    11. Example 3.10.
    12. Lemma 3.11.
    13. Theorem 3.12.
    14. 3.2. Measures in definably amenable groups
    15. Proposition 3.14.
    16. Proposition 3.15.
    17. Definition 3.16.
    18. Question 3.18.
    19. Lemma 3.19.
    20. Corollary 3.20.
    21. Lemma 3.21.
    22. Lemma 3.22.
    23. Proposition 3.23.
    24. Proposition 3.24.
    25. Proposition 3.25.
    26. Lemma 3.26.
    27. Corollary 3.27.
    28. 3.3. Weak genericity and almost periodic types
    29. Definition 3.28 (New09Poi87).
    30. Fact 3.29 (New09, Section 1).
    31. Proposition 3.30.
    32. Proposition 3.31.
    33. Question 3.32.
    34. Proposition 3.33.
    35. Corollary 3.34.
    36. Theorem 3.35.
    37. 3.4. Unique ergodicity
    38. Theorem 3.36.
  5. 4. Regular ergodic measures
    1. Fact 4.1.
    2. Fact 4.2 (Phe01, Proposition 12.4).
    3. Proposition 4.3.
    4. Corollary 4.4.
    5. Theorem 4.5.
    6. Corollary 4.6.
    7. Question 4.7.
  6. 5. Generic compact domination and the Ellis group conjecture
    1. 5.1. Baire-generic compact domination
    2. Proposition 5.1.
    3. Theorem 5.3 (Baire-generic compact domination).
    4. 5.2. Connected components in an expansion by externally definable sets
    5. Fact 5.4.
    6. Fact 5.5.
    7. 5.3. Ellis group conjecture
    8. Theorem 5.7.
    9. Claim 1.
    10. Claim 2.
    11. Corollary 5.8.
  7. 6. Further remarks
    1. 6.1. Left vs. right actions
    2. Example 6.1.
    3. Lemma 6.2.
    4. Proposition 6.3.
    5. Example 6.4.
    6. 6.2. Actions on definable homogeneous spaces
    7. Lemma 6.5.
    8. Proposition 6.6.
    9. Claim.
  8. Acknowledgments

Mathematical Fragments

Theorem 1.1.

Let be a definable NIP group with a sufficiently saturated model. Then the following are equivalent:

(1)

is definably amenable (i.e., admits a -invariant measure on its definable subsets).

(2)

The action of on admits a small orbit.

Theorem 1.2.

Let be a definably amenable NIP group. Then the following are equivalent for a definable set

(1)

does not -divide (i.e., there is no infinite sequence of elements of and natural number such that any sets in have empty intersection, see Definition 3.2

(2)

is weakly generic (i.e., there is some nongeneric such that is generic, see Definition 3.28

(3)

for some -invariant measure

(4)

is -generic (meaning that for any small model over which is defined, no -translate of forks over see Definition 3.2

Moreover, for a global type the following are equivalent:

(1)

is -generic (i.e., every formula in is -generic

(2)

has a small -orbit;

(3)

Fact 2.1.

For any and there is satisfying the following.

Let be a probability space, and let be a family of subsets of of VC-dimension such that:

(1)

every set from is measurable;

(2)

for each , the function given by

is measurable;

(3)

for each , the function

is measurable.

Then there is some tuple such that for any we have .

Fact 2.2 (Reference Mat04).

Let be a family of subsets of some set . Assume that has finite VC-dimension. Then there is some such that for every , there is an integer such that for every finite subfamily , if has the -property, then there is an -point set intersecting all members of .

Fact 2.4 (Reference CK12).

Assume that is NIP, and let be an extension base.

(1)

A formula forks over if and only if it divides over , i.e., the set of formulas dividing over forms an ideal.

(2)

Every admits a global extension strictly nonforking over .

(3)

Assume that forks (equivalently, divides) over , and let in be an infinite strict Morley sequence in . Then is inconsistent.

Remark 2.6.

In particular, in view of Fact 2.4, if is a partial type that does not divide over (e.g., if is -invariant), then it extends to a global -invariant type.

Fact 2.7 (Reference Sim15b).

Let be a countable model, and let be NIP. For any set and , if (i.e., in the topological closure of ), then is the limit of a countable sequence of elements of .

Remark 2.8.

Let denote the set of measures over in variable ; it is naturally equipped with a compact topology as a closed subset of with the product topology. Every type over can be identified with the -measure concentrating on it; thus is identified with a closed subset of .

Fact 2.9.

Let be NIP. Let a measure over , let be a finite set of -formulas, and let be arbitrary. Then there are some types such that for every and , we have

Furthermore, we may assume that , the support of , for all .

Fact 2.11.

Let be a Keisler measure, let be a formula, and let be an indiscernible sequence. Assume that for some we have for every . Then the partial type is consistent.

Remark 2.15.

If is countable, then is a Polish space with respect to the logic topology. Indeed, there is a countable model such that every closed set is a projection of a partial type over , and is a countable basis of the topology.

Definition 3.1.

The group is definably amenable if it admits a global Keisler measure on definable subsets of which is invariant under (left-) translation by elements of .

Definition 3.2.
(1)

Let be a subset of defined over some model . We say that (left-) -divides if there is an -indiscernible sequence such that is inconsistent.

(2)

The formula is (left-) -generic over if no translate of forks over . We say that is -generic if it is -generic over some small . A (partial) type is -generic if every formula implied by it is -generic.

(3)

A global type is called (left-) strongly f-generic over if no -translate of forks over . A global type is strongly -generic if it is strongly -generic over some small model .

Fact 3.3.
(1)

If admits a strongly -generic type over some small model , then it admits a strongly -generic type over any model .

(2)

If is strongly -generic, then for any small model ).

(3)

The group admits a -invariant measure if and only if there is a global strongly -generic type in .

Proposition 3.4.

Let be a definably amenable group, and let . Let also be strongly -generic, -invariant and take . Then the following are equivalent:

(1)

is -generic over ;

(2)

does not -divide;

(3)

does not fork over .

Corollary 3.5.

Let be definably amenable. The family of nonf-generic formulas (equivalently, -dividing formulas) forms an ideal. In particular, every partial -generic type extends to a global one.

Lemma 3.6.

Let be definably amenable, let be a formula, and let . Then is not -generic (and hence it -divides by Proposition 3.4).

Proposition 3.8.

Let be definably amenable. For , the following are equivalent:

(1)

is -generic;

(2)

is -invariant (and

(3)

has a bounded orbit.

Proposition 3.9.

Let be definably amenable. A type is strongly -generic if and only if it is -generic and -invariant over some small model .

Lemma 3.11.

Let be -saturated, and let be such that does not fork over for every . Then extends to a global type strongly -generic over .

Theorem 3.12.

Let be NIP, let , and let be a definable group. Then the following are equivalent:

(1)

is definably amenable;

(2)

for some

(3)

some measure has a bounded -orbit.

Proposition 3.14.

Let be definably amenable, and let be a Keisler measure on . The following are equivalent:

(1)

The measure is -generic, that is implies is -generic for all .

(2)

All types in the support are -generic.

(3)

The measure is -invariant.

(4)

The orbit of is bounded.

Proposition 3.15.

Let be definably amenable, and let be a maximal finitely consistent set of formulas in . Then is -generic if and only if is -invariant for every .

Definition 3.16.

Let be definably amenable, and let be -generic. Keeping in mind that (as well as all its translates) is -invariant (by Proposition 3.8), we define a measure on by

where is the normalized Haar measure on the compact group and .

Remark 3.17.
(1)

The measure that we just constructed is clearly -invariant and -strongly invariant (that is, for ). Besides, for any , .

(2)

We have . Indeed, if and arbitrary, then , which by the definition of implies that for some .

Question 3.18.

Footnote1 Let be an NIP group. Are the following two properties equivalent?

1

We have claimed an affirmative answer in an earlier version of this article, however a mistake in our argument was pointed out by the referees.

(1)

is definably amenable.

(2)

admits a global -generic type (equivalently, the family of all non--generic subsets of is an ideal).

Lemma 3.19.

For a fixed formula , let be the family of all where varies over and varies over all -generic types on . Then has finite VC-dimension.

Corollary 3.20.

For any , the family

has finite VC-dimension.

Lemma 3.21.

If is countable and is countable, then satisfies all of the assumptions of Fact 2.1 with respect to the measure .

Lemma 3.22.

Let be a sublanguage of , let be the -reduct of , let be an -definable group definably amenable (in the sense of and let be a formula from . Let be a global -type which is -generic, and let .

(1)

In the sense of , the group is definably amenable NIP and is an -generic type.

(2)

Let be the connected component computed in , and let be the -invariant measure on -definable (resp., -definable) subsets of given by Definition 3.16 in (resp., in . Then .

Proposition 3.23.

For any , and a countable set of -generic types there are some such that for any and we have .

Proposition 3.24.

Let be an -generic type, and assume that . Then is -generic and .

Proposition 3.25.

Let be an -generic type. Then for any definable set , if , then there is a finite union of translates of which covers the support (so in particular it has -measure .

Lemma 3.26.

Let be -invariant. Then for any and , there are some -generic such that

for any .

Corollary 3.27.

Let be a -invariant measure, and assume that for some -generic . Then .

Definition 3.28 (Reference New09Reference Poi87).
(1)

A formula is (left-) generic if there are some finitely many such that .

(2)

A formula is (left-) weakly generic if there is formula which is not generic but such that is generic.

(3)

A (partial) type is (weakly) generic if it only contains (weakly) generic formulas.

(4)

A type is called almost periodic if it belongs to a minimal flow in (i.e., a minimal -invariant closed set), equivalently if for any we have .

Fact 3.29 (Reference New09, Section 1).

The following hold, in an arbitrary theory:

(1)

The formula is weakly generic if and only if for some finite , is not generic.

(2)

The set of nonweakly generic formulas forms a -invariant ideal. In particular, there are always global weakly generic types by compactness.

(3)

The set of all weakly generic types is exactly the closure of the set of all almost periodic types in .

(4)

Every generic type is weakly generic. Moreover, if there is a global generic type, then every weakly generic type is generic, and the set of generic types is the unique minimal flow in .

(5)

A type is almost periodic if and only if for every , the set is covered by finitely many left translates of .

Proposition 3.30.

Let be definably amenable, and let be a weakly generic formula. Then it is -generic.

Proposition 3.31.

Assume that is definably amenable.

(1)

If is almost periodic, then it is -generic and .

(2)

Minimal flows in are exactly the sets of the form for some -generic .

(3)

If are almost periodic and then .

Proposition 3.33.

Let be definably amenable. Assume that does not -divide. Then there are some global almost periodic types such that for any there is some such that holds.

Corollary 3.34.

Let be definably amenable. If is -generic, then for some global -generic type .

Theorem 3.35.

Let be definably amenable NIP. Let be a definable subset of . Then the following are equivalent:

(1)

is -generic;

(2)

is not -dividing;

(3)

is weakly-generic;

(4)

for some -invariant measure

(5)

for some global -generic type .

Theorem 3.36.

A definably amenable NIP group is uniquely ergodic if and only if it admits a generic type (in which case it has a unique minimal flow—the support of the unique measure).

Fact 4.1.

Let be a real, locally convex Hausdorff topological vector space. Let be a compact convex subset of , and let be a subset of . Then the following are equivalent:

(1)

, the closed convex hull of .

(2)

The closure of includes all extreme points of .

Proposition 4.3.

The map from the (closed) set of global -generic types to the (closed) set of global -invariant measures on is continuous.

Corollary 4.4.
(1)

The set is closed in the set of all -invariant measures.

(2)

Given a -invariant measure , the set of -generic types for which is a subflow.

Theorem 4.5.

Let be definably amenable. Then regular ergodic measures on are exactly the measures of the form for some -generic .

Proposition 5.1.

The set is a constructible subset of (namely, a boolean combination of closed sets).

Remark 5.2.

Let be an arbitrary topological space, and let be a constructible set. Then the boundary has empty interior.

Theorem 5.3 (Baire-generic compact domination).

The set is closed and has empty interior. In particular it is meager.

Fact 5.4.

Let be NIP, and let be a model of . Then eliminates quantifiers. It follows that is NIP and that all (-) types over are definable.

Fact 5.5.

Let be NIP, and let be a small model of . Let be an -definable group.

(1)

If is definably amenable in the sense of , then it is definably amenable in the sense of as well.

(2)

The group computed in coincides with computed in .

Lemma 6.5.

Let be a boolean algebra, and let be an ideal such that is contained in the zero-ideal of , a measure on .

Let be the collection of all sets for which there is some such that . Then is a boolean algebra with . Moreover, extends to a global measure on such that all sets from have -measure .

References

[BGT12]
Emmanuel Breuillard, Ben Green, and Terence Tao, The structure of approximate groups, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221. MR3090256, Show rawAMSref\bib{breuillard2012}{article}{ author={Breuillard, Emmanuel}, author={Green, Ben}, author={Tao, Terence}, title={The structure of approximate groups}, journal={Publ. Math. Inst. Hautes \'Etudes Sci.}, volume={116}, date={2012}, pages={115--221}, issn={0073-8301}, review={\MR {3090256}}, } Close amsref.
[CK12]
Artem Chernikov and Itay Kaplan, Forking and dividing in theories, J. Symbolic Logic 77 (2012), no. 1, 1–20. MR2951626, Show rawAMSref\bib{CheKap}{article}{ author={Chernikov, Artem}, author={Kaplan, Itay}, title={Forking and dividing in ${\rm NTP}_2$ theories}, journal={J. Symbolic Logic}, volume={77}, date={2012}, number={1}, pages={1--20}, issn={0022-4812}, review={\MR {2951626}}, } Close amsref.
[CPS14]
Artem Chernikov, Anand Pillay, and Pierre Simon, External definability and groups in NIP theories, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 213–240. MR3245144, Show rawAMSref\bib{ChePilSim}{article}{ author={Chernikov, Artem}, author={Pillay, Anand}, author={Simon, Pierre}, title={External definability and groups in NIP theories}, journal={J. Lond. Math. Soc. (2)}, volume={90}, date={2014}, number={1}, pages={213--240}, issn={0024-6107}, review={\MR {3245144}}, } Close amsref.
[CP12]
Annalisa Conversano and Anand Pillay, Connected components of definable groups and -minimality I, Adv. Math. 231 (2012), no. 2, 605–623. MR2955185, Show rawAMSref\bib{AnnalisaAnand}{article}{ author={Conversano, Annalisa}, author={Pillay, Anand}, title={Connected components of definable groups and $o$-minimality I}, journal={Adv. Math.}, volume={231}, date={2012}, number={2}, pages={605--623}, issn={0001-8708}, review={\MR {2955185}}, } Close amsref.
[CS13]
Artem Chernikov and Pierre Simon, Externally definable sets and dependent pairs, Israel J. Math. 194 (2013), no. 1, 409–425. MR3047077, Show rawAMSref\bib{ExtDefI}{article}{ author={Chernikov, Artem}, author={Simon, Pierre}, title={Externally definable sets and dependent pairs}, journal={Israel J. Math.}, volume={194}, date={2013}, number={1}, pages={409--425}, issn={0021-2172}, review={\MR {3047077}}, } Close amsref.
[Gla07a]
Eli Glasner, Enveloping semigroups in topological dynamics, Topology Appl. 154 (2007), no. 11, 2344–2363. MR2328017, Show rawAMSref\bib{Glasner1}{article}{ author={Glasner, Eli}, title={Enveloping semigroups in topological dynamics}, journal={Topology Appl.}, volume={154}, date={2007}, number={11}, pages={2344--2363}, issn={0166-8641}, review={\MR {2328017}}, } Close amsref.
[Gla07b]
Eli Glasner, The structure of tame minimal dynamical systems, Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1819–1837. MR2371597, Show rawAMSref\bib{glasner2007structure}{article}{ author={Glasner, Eli}, title={The structure of tame minimal dynamical systems}, journal={Ergodic Theory Dynam. Systems}, volume={27}, date={2007}, number={6}, pages={1819--1837}, issn={0143-3857}, review={\MR {2371597}}, } Close amsref.
[GPP15]
Jakub Gismatullin, Davide Penazzi, and Anand Pillay, Some model theory of , Fund. Math. 229 (2015), no. 2, 117–128. MR3315377, Show rawAMSref\bib{gismatullin2012some}{article}{ author={Gismatullin, Jakub}, author={Penazzi, Davide}, author={Pillay, Anand}, title={Some model theory of $\rm {SL}(2,\mathbb {R})$}, journal={Fund. Math.}, volume={229}, date={2015}, number={2}, pages={117--128}, issn={0016-2736}, review={\MR {3315377}}, } Close amsref.
[Hal50]
Paul R. Halmos, Measure theory, Vol. 2, D. Van Nostrand Company, Inc., New York, NY, 1950. MR0033869, Show rawAMSref\bib{halmos1950measure}{book}{ author={Halmos, Paul R.}, title={Measure theory}, volume={2}, publisher={D. Van Nostrand Company, Inc., New York, NY}, date={1950}, pages={xi+304}, review={\MR {0033869}}, } Close amsref.
[Hru12]
Ehud Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), no. 1, 189–243. MR2833482, Show rawAMSref\bib{hr_appx}{article}{ author={Hrushovski, Ehud}, title={Stable group theory and approximate subgroups}, journal={J. Amer. Math. Soc.}, volume={25}, date={2012}, number={1}, pages={189--243}, issn={0894-0347}, review={\MR {2833482}}, } Close amsref.
[Hru01]
Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115. MR1854232, Show rawAMSref\bib{hrushovski2001manin}{article}{ author={Hrushovski, Ehud}, title={The Manin-Mumford conjecture and the model theory of difference fields}, journal={Ann. Pure Appl. Logic}, volume={112}, date={2001}, number={1}, pages={43--115}, issn={0168-0072}, review={\MR {1854232}}, } Close amsref.
[Hru96]
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR1333294, Show rawAMSref\bib{MordellLang}{article}{ author={Hrushovski, Ehud}, title={The Mordell-Lang conjecture for function fields}, journal={J. Amer. Math. Soc.}, volume={9}, date={1996}, number={3}, pages={667--690}, issn={0894-0347}, review={\MR {1333294}}, } Close amsref.
[Hru90]
Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117–138. MR1081816, Show rawAMSref\bib{unidimensional}{article}{ author={Hrushovski, Ehud}, title={Unidimensional theories are superstable}, journal={Ann. Pure Appl. Logic}, volume={50}, date={1990}, number={2}, pages={117--138}, issn={0168-0072}, review={\MR {1081816}}, } Close amsref.
[HP11]
Ehud Hrushovski and Anand Pillay, On NIP and invariant measures, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1005–1061. MR2800483, Show rawAMSref\bib{NIP2}{article}{ author={Hrushovski, Ehud}, author={Pillay, Anand}, title={On NIP and invariant measures}, journal={J. Eur. Math. Soc. (JEMS)}, volume={13}, date={2011}, number={4}, pages={1005--1061}, issn={1435-9855}, review={\MR {2800483}}, } Close amsref.
[HPP08]
Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR2373360, Show rawAMSref\bib{NIP1}{article}{ author={Hrushovski, Ehud}, author={Peterzil, Ya'acov}, author={Pillay, Anand}, title={Groups, measures, and the NIP}, journal={J. Amer. Math. Soc.}, volume={21}, date={2008}, number={2}, pages={563--596}, issn={0894-0347}, review={\MR {2373360}}, } Close amsref.
[HPS13]
Ehud Hrushovski, Anand Pillay, and Pierre Simon, Generically stable and smooth measures in NIP theories, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2341–2366. MR3020101, Show rawAMSref\bib{HruPilSimMeas}{article}{ author={Hrushovski, Ehud}, author={Pillay, Anand}, author={Simon, Pierre}, title={Generically stable and smooth measures in NIP theories}, journal={Trans. Amer. Math. Soc.}, volume={365}, date={2013}, number={5}, pages={2341--2366}, issn={0002-9947}, review={\MR {3020101}}, } Close amsref.
[Jer54]
Meyer Jerison, A property of extreme points of compact convex sets, Proc. Amer. Math. Soc. 5 (1954), 782–783. MR0065021, Show rawAMSref\bib{Jerison1954}{article}{ author={Jerison, Meyer}, title={A property of extreme points of compact convex sets}, journal={Proc. Amer. Math. Soc.}, volume={5}, date={1954}, pages={782--783}, issn={0002-9939}, review={\MR {0065021}}, } Close amsref.
[Kec95]
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR1321597, Show rawAMSref\bib{Kechris1995}{book}{ author={Kechris, Alexander S.}, title={Classical descriptive set theory}, series={Graduate Texts in Mathematics}, volume={156}, publisher={Springer-Verlag, New York}, date={1995}, pages={xviii+402}, isbn={0-387-94374-9}, review={\MR {1321597}}, } Close amsref.
[KL07]
David Kerr and Hanfeng Li, Independence in topological and -dynamics, Math. Ann. 338 (2007), no. 4, 869–926. MR2317754, Show rawAMSref\bib{kerr2007independence}{article}{ author={Kerr, David}, author={Li, Hanfeng}, title={Independence in topological and $C^*$-dynamics}, journal={Math. Ann.}, volume={338}, date={2007}, number={4}, pages={869--926}, issn={0025-5831}, review={\MR {2317754}}, } Close amsref.
[Las92]
Michael C. Laskowski, Vapnik-Chervonenkis classes of definable sets, J. London Math. Soc. (2) 45 (1992), no. 2, 377–384. MR1171563, Show rawAMSref\bib{LasVC}{article}{ author={Laskowski, Michael C.}, title={Vapnik-Chervonenkis classes of definable sets}, journal={J. London Math. Soc. (2)}, volume={45}, date={1992}, number={2}, pages={377--384}, issn={0024-6107}, review={\MR {1171563}}, } Close amsref.
[Mat04]
Jiří Matoušek, Bounded VC-dimension implies a fractional Helly theorem, Discrete Comput. Geom. 31 (2004), no. 2, 251–255. MR2060639, Show rawAMSref\bib{Matousek}{article}{ author={Matou\v sek, Ji\v r\'\i }, title={Bounded VC-dimension implies a fractional Helly theorem}, journal={Discrete Comput. Geom.}, volume={31}, date={2004}, number={2}, pages={251--255}, issn={0179-5376}, review={\MR {2060639}}, } Close amsref.
[MS14]
Alice Medvedev and Thomas Scanlon, Invariant varieties for polynomial dynamical systems, Ann. of Math. (2) 179 (2014), no. 1, 81–177. MR3126567, Show rawAMSref\bib{medvedev-scanlon}{article}{ author={Medvedev, Alice}, author={Scanlon, Thomas}, title={Invariant varieties for polynomial dynamical systems}, journal={Ann. of Math. (2)}, volume={179}, date={2014}, number={1}, pages={81--177}, issn={0003-486X}, review={\MR {3126567}}, } Close amsref.
[New12]
Ludomir Newelski, Bounded orbits and measures on a group, Israel J. Math. 187 (2012), 209–229. MR2891705, Show rawAMSref\bib{New3}{article}{ author={Newelski, Ludomir}, title={Bounded orbits and measures on a group}, journal={Israel J. Math.}, volume={187}, date={2012}, pages={209--229}, issn={0021-2172}, review={\MR {2891705}}, } Close amsref.
[New09]
Ludomir Newelski, Topological dynamics of definable group actions, J. Symbolic Logic 74 (2009), no. 1, 50–72. MR2499420, Show rawAMSref\bib{New4}{article}{ author={Newelski, Ludomir}, title={Topological dynamics of definable group actions}, journal={J. Symbolic Logic}, volume={74}, date={2009}, number={1}, pages={50--72}, issn={0022-4812}, review={\MR {2499420}}, } Close amsref.
[NP06]
Ludomir Newelski and Marcin Petrykowski, Weak generic types and coverings of groups. I, Fund. Math. 191 (2006), no. 3, 201–225. MR2278623, Show rawAMSref\bib{NewPetr}{article}{ author={Newelski, Ludomir}, author={Petrykowski, Marcin}, title={Weak generic types and coverings of groups. I}, journal={Fund. Math.}, volume={191}, date={2006}, number={3}, pages={201--225}, issn={0016-2736}, review={\MR {2278623}}, } Close amsref.
[Phe01]
Robert R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR1835574, Show rawAMSref\bib{phelps2001lectures}{book}{ author={Phelps, Robert R.}, title={Lectures on Choquet's theorem}, series={Lecture Notes in Mathematics}, volume={1757}, edition={2}, publisher={Springer-Verlag, Berlin}, date={2001}, pages={viii+124}, isbn={3-540-41834-2}, review={\MR {1835574}}, } Close amsref.
[Pil04]
Anand Pillay, Type-definability, compact Lie groups, and o-minimality, J. Math. Log. 4 (2004), no. 2, 147–162. MR2114965, Show rawAMSref\bib{PillayLogicTop}{article}{ author={Pillay, Anand}, title={Type-definability, compact Lie groups, and o-minimality}, journal={J. Math. Log.}, volume={4}, date={2004}, number={2}, pages={147--162}, issn={0219-0613}, review={\MR {2114965}}, } Close amsref.
[Poi01]
Bruno Poizat, Stable groups, Mathematical Surveys and Monographs, vol. 87, American Mathematical Society, Providence, RI, 2001. Translated from the 1987 French original by Moses Gabriel Klein. MR1827833, Show rawAMSref\bib{PoizatStableGroups}{book}{ author={Poizat, Bruno}, title={Stable groups}, series={Mathematical Surveys and Monographs}, volume={87}, note={Translated from the 1987 French original by Moses Gabriel Klein}, publisher={American Mathematical Society, Providence, RI}, date={2001}, pages={xiv+129}, isbn={0-8218-2685-9}, review={\MR {1827833}}, } Close amsref.
[Poi87]
Bruno Poizat, Groupes stables (French), Nur al-Mantiq wal-Marifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987. Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR902156, Show rawAMSref\bib{poizat1987groupes}{book}{ author={Poizat, Bruno}, title={Groupes stables}, language={French}, series={Nur al-Mantiq wal-Marifah [Light of Logic and Knowledge]}, volume={2}, note={Une tentative de conciliation entre la g\'eom\'etrie alg\'ebrique et la logique math\'ematique. [An attempt at reconciling algebraic geometry and mathematical logic]}, publisher={Bruno Poizat, Lyon}, date={1987}, pages={vi+218}, isbn={2-9500919-1-1}, review={\MR {902156}}, } Close amsref.
[PY16]
Anand Pillay and Ningyuan Yao, On minimal flows, definably amenable groups, and -minimality, Adv. Math. 290 (2016), 483–502. MR3451930, Show rawAMSref\bib{pillay2016minimal}{article}{ author={Pillay, Anand}, author={Yao, Ningyuan}, title={On minimal flows, definably amenable groups, and $o$-minimality}, journal={Adv. Math.}, volume={290}, date={2016}, pages={483--502}, issn={0001-8708}, review={\MR {3451930}}, } Close amsref.
[Sel13]
Z. Sela, Diophantine geometry over groups VIII: Stability, Ann. of Math. (2) 177 (2013), no. 3, 787–868. MR3034289, Show rawAMSref\bib{Sela-stable}{article}{ author={Sela, Z.}, title={Diophantine geometry over groups VIII: Stability}, journal={Ann. of Math. (2)}, volume={177}, date={2013}, number={3}, pages={787--868}, issn={0003-486X}, review={\MR {3034289}}, } Close amsref.
[She09]
Saharon Shelah, Dependent first order theories, continued, Israel J. Math. 173 (2009), 1–60. MR2570659, Show rawAMSref\bib{shelah2009dependent}{article}{ author={Shelah, Saharon}, title={Dependent first order theories, continued}, journal={Israel J. Math.}, volume={173}, date={2009}, pages={1--60}, issn={0021-2172}, review={\MR {2570659}}, } Close amsref.
[She08]
Saharon Shelah, Minimal bounded index subgroup for dependent theories, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1087–1091. MR2361885, Show rawAMSref\bib{shelah2008minimal}{article}{ author={Shelah, Saharon}, title={Minimal bounded index subgroup for dependent theories}, journal={Proc. Amer. Math. Soc.}, volume={136}, date={2008}, number={3}, pages={1087--1091}, issn={0002-9939}, review={\MR {2361885}}, } Close amsref.
[She71]
Saharon Shelah, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Ann. Math. Logic 3 (1971), no. 3, 271–362. MR0317926, Show rawAMSref\bib{Sh10}{article}{ author={Shelah, Saharon}, title={Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory}, journal={Ann. Math. Logic}, volume={3}, date={1971}, number={3}, pages={271--362}, issn={0168-0072}, review={\MR {0317926}}, } Close amsref.
[Sim15a]
Pierre Simon, A guide to NIP theories, Lecture Notes in Logic, vol. 44, Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015. MR3560428, Show rawAMSref\bib{SimBook}{book}{ author={Simon, Pierre}, title={A guide to NIP theories}, series={Lecture Notes in Logic}, volume={44}, publisher={Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge}, date={2015}, pages={vii+156}, isbn={978-1-107-05775-3}, review={\MR {3560428}}, } Close amsref.
[Sim15b]
Pierre Simon, Rosenthal compacta and NIP formulas, Fund. Math. 231 (2015), no. 1, 81–92. MR3361236, Show rawAMSref\bib{Simon2014rosenthal}{article}{ author={Simon, Pierre}, title={Rosenthal compacta and NIP formulas}, journal={Fund. Math.}, volume={231}, date={2015}, number={1}, pages={81--92}, issn={0016-2736}, review={\MR {3361236}}, } Close amsref.
[Sim11]
Barry Simon, Convexity, Cambridge Tracts in Mathematics, vol. 187, Cambridge University Press, Cambridge, 2011. An analytic viewpoint. MR2814377, Show rawAMSref\bib{simon2011convexity}{book}{ author={Simon, Barry}, title={Convexity}, series={Cambridge Tracts in Mathematics}, volume={187}, note={An analytic viewpoint}, publisher={Cambridge University Press, Cambridge}, date={2011}, pages={x+345}, isbn={978-1-107-00731-4}, review={\MR {2814377}}, } Close amsref.
[Sta17]
Sergei Starchenko, NIP, Keisler measures and combinatorics, Astérisque 390 (2017), Exp. No. 1114, 303–334. Séminaire Bourbaki. Vol. 2015/2016. Exposés 1104–1119. MR3666030, Show rawAMSref\bib{starchenko2016nip}{article}{ author={Starchenko, Sergei}, title={NIP, Keisler measures and combinatorics}, note={S\'eminaire Bourbaki. Vol. 2015/2016. Expos\'es 1104--1119}, journal={Ast\'erisque}, number={390}, date={2017}, pages={Exp. No. 1114, 303--334}, issn={0303-1179}, isbn={978-2-85629-855-8}, review={\MR {3666030}}, } Close amsref.
[VČ71]
V. N. Vapnik and A. Ja. Červonenkis, The uniform convergence of frequencies of the appearance of events to their probabilities (Russian, with English summary), Teor. Verojatnost. i Primenen. 16 (1971), 264–279. MR0288823, Show rawAMSref\bib{vapnik1971uniform}{article}{ author={Vapnik, V. N.}, author={\v Cervonenkis, A. Ja.}, title={The uniform convergence of frequencies of the appearance of events to their probabilities}, language={Russian, with English summary}, journal={Teor. Verojatnost. i Primenen.}, volume={16}, date={1971}, pages={264--279}, issn={0040-361x}, review={\MR {0288823}}, } Close amsref.
[Wag00]
Frank O. Wagner, Simple theories, Mathematics and its Applications, vol. 503, Kluwer Academic Publishers, Dordrecht, 2000. MR1747713, Show rawAMSref\bib{WagnerBook}{book}{ author={Wagner, Frank O.}, title={Simple theories}, series={Mathematics and its Applications}, volume={503}, publisher={Kluwer Academic Publishers, Dordrecht}, date={2000}, pages={xii+260}, isbn={0-7923-6221-7}, review={\MR {1747713}}, } Close amsref.
[Wal82]
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York–Berlin, 1982. MR648108, Show rawAMSref\bib{walters2000introduction}{book}{ author={Walters, Peter}, title={An introduction to ergodic theory}, series={Graduate Texts in Mathematics}, volume={79}, publisher={Springer-Verlag, New York--Berlin}, date={1982}, pages={ix+250}, isbn={0-387-90599-5}, review={\MR {648108}}, } Close amsref.
[Zil93]
Boris Zilber, Uncountably categorical theories, Translations of Mathematical Monographs, vol. 117, American Mathematical Society, Providence, RI, 1993. Translated from the Russian by D. Louvish. MR1206477, Show rawAMSref\bib{zilberuncountably}{book}{ author={Zilber, Boris}, title={Uncountably categorical theories}, series={Translations of Mathematical Monographs}, volume={117}, note={Translated from the Russian by D. Louvish}, publisher={American Mathematical Society, Providence, RI}, date={1993}, pages={vi+122}, isbn={0-8218-4586-1}, review={\MR {1206477}}, } Close amsref.

Article Information

MSC 2010
Primary: 03C45 (Classification theory, stability and related concepts), 37B05 (Transformations and group actions with special properties), 03C60 (Model-theoretic algebra)
Secondary: 03C64 (Model theory of ordered structures; o-minimality), 22F10 (Measurable group actions), 28D15 (General groups of measure-preserving transformations)
Author Information
Artem Chernikov
IMJ-PRG, Université Paris Diderot, Paris 7, L’Equipe de Logique Mathématique, UFR de Mathématiques case 7012, 75205 Paris Cedex 13, France
chernikov@math.ucla.edu
MathSciNet
Pierre Simon
Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
simon@math.univ-lyon1.fr
MathSciNet
Additional Notes

The research leading to this paper has been partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291111 and by ValCoMo (ANR-13-BS01-0006).

The first author was partially supported by the Fondation Sciences Mathematiques de Paris (ANR-10-LABX-0098), by the NSF (grants DMS-1600796 and DMS-1651321), and by the Sloan Foundation.

The second author was partially supported by the NSF (grant DMS-1665491) and by the Sloan Foundation.

Journal Information
Journal of the American Mathematical Society, Volume 31, Issue 3, ISSN 1088-6834, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , , and published on .
Copyright Information
Copyright 2018 American Mathematical Society
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