• bounded invariant equivalence relations
• broom lemma
• definably amenable groups
• dependent theories
• Ellis Group Conjecture
• Ellis semigroup
• enveloping semigroup
• f-generic types
• forking
• honest definitions
• invariant types
• Keisler measures
• model theory
• NIP theories
• topological dynamics

DEFINABLE GROUPS, NIP THEORIES, AND THE ELLIS GROUP CONJECTURE

A \emph{definable group} $G$ is a group which is definable in a first-order structure. Despite the name, it is not a single group, but a family of groups given by interpreting the defining formulas in \emph{elementary extensions} of the structure defining the group. For instance, algebraic groups are definable in the complex field using first-order formulas. These include matrix groups and abelian varieties such as elliptic curves. Among groups which are definable with first-order formulas in the real field there are $\operatorname{GL}(n, \mathbb R)$, $\operatorname{SO}(n, \mathbb R)$, and other Lie groups.

The two families of examples above are, in a sense, orthogonal. The field $\mathbb C$ falls into the class of \emph{stable} structures, which are, in a nutshell, the ones that do not define an order relation on an infinite set. Stable theories have been a central and fruitful topic in the model theory of the past decades (e.g. [1,11]), and there is a huge literature on stable groups (for instance [3,13]). Unfortunately, since stability is destroyed by the presence of a total infinite order, the field structure on $\mathbb R$ lives outside this realm, and more generally \emph{o-minimal} structures, another important class in which it is possible to provide a framework for \emph{tame geometry} (see [14]), are not stable. Model theorists have therefore tried to generalize methods from stability theory to broader contexts. One robust, simultaneous generalization of both stability and o-minimality is found in the class of \emph{dependent}, or \textsc{nip} theories. \textsc{Nip} structures can be roughly described as the ones that do not code a membership relation on an infinite set; this viewpoint is intimately connected to \textsc{vc}-dimension, a fundamental tool of statistical learning theory. This thesis explores a problem, which we are now going to outline, concerning the relation between two groups that can be attached to any group definable in a \textsc{nip} structure.



In \textsc{nip} theories, to every definable group is associated a concrete compact Hausdorff topological group called $G/G^{00}$. As an example, it can be proven that if $G$ is a definably compact group definable over $\emptyset$ in a real closed field, for instance $\operatorname{SO}(3, M)$ for $M\succ \mathbb R$ an hyperreal field, then $G/G^{00}$ is exactly $G(\mathbb R)$, and the projection to $G/G^{00}$ behaves like a ``standard part'' map. If $G$ is not compact then this may not be true, as in the case of $\operatorname{SL}(n, M)$ where $G/G^{00}$ is trivial. In general (see [2]), for a group which is definable in an o-minimal structure, $G/G^{00}$ is a real Lie group. As a stable example, if $G$ is the additive group in the structure of the integers with sum (but without product), then $G/G^{00}$ is isomorphic to $\hat{\mathbb Z}=\varprojlim \mathbb Z/n\mathbb Z$. All these isomorphisms preserve the topology, i.e. are isomorphisms of topological groups. This canonical quotient is the first protagonist of the problem studied in the thesis. In order to introduce the second one, some preliminary explanations are needed.

An important concept in the study of stable groups is the one of a \emph{generic type}. Trying to find a well-behaved analogue in the unstable context, Newelski noticed that a certain notion, namely that of a \emph{weak generic type}, is well understood when bringing topological dynamics into the picture\footnot{Briefly, in the dynamical context ``generic'' becomes ``syndetic'', and ``weak generic'' corresponds to ``piecewise syndetic''.}. In topological dynamics one is often interested in $G$-flows, actions of a group $G$ on compact Hausdorff spaces by homeomorphisms; soon one turns the attention to the ones that have a dense orbit ($G$-ambits) and to the ones in which all orbits are dense (minimal flows). A very special $G$-flow is the universal $G$-ambit $\beta G$ of ultrafilters on $G$: every $G$-ambit can be seen as a quotient of $\beta G$, and its minimal subflows enjoy a similar universal property. A ``tame'' counterpart of $\beta G$ is the space $S_G(M)$ of types over a model $M$ concentrating on $G$, i.e. the ultrafilters on definable subsets of $G(M)$, and one could develop a theory of \emph{tame topological dynamics} ([7,12]) and hope for $S_G(M)$ to be universal with respect to \emph{definable} $G(M)$-flows. Now, one important tool in the study of a $G$-flow $X$ is its \emph{enveloping semigroup} $E(X)$; it turns out that $\beta G\cong E(\beta G)$ and this equips the former with a semigroup structure. Once some technical obstacles are overcome, this construction can be carried out for $S_G(M)$ too, or at least for a certain bigger type space called $S_G^{\textnormal{ext}}(M)$.


Applying the theory of enveloping semigroups to $E(\beta G)\cong\beta G$ produces a certain family of sub-semigroups that are indeed groups, and furthermore all in the same isomorphism class: this is the \emph{ideal group}, or \emph{Ellis group} associated to the flow. Modulo the complications mentioned above, an Ellis group can also be associated to $S_G(M)$. Even if this may depend on $M$, a comparison with $G/G^{00}$ can be made, and indeed the latter is always a quotient of the former, the projection $\pi$ being the restriction of a certain natural map $S_G(M)\to G/G^{00}$. Since in stable groups a similar situation arises replacing the Ellis group with the subspace of generic types of $S_G(M)$, and in that case the relevant map is injective, the next question is: is this $\pi$ an isomorphism?



Even in tame context, this need not be the case: it was shown in [8] that the Ellis group of $\operatorname{SL}(2,\mathbb R)$ is the group with two elements, but its $G/G^{00}$ is trivial. A property that is \emph{not} satisfied by $\operatorname{SL}(2, \mathbb R)$ is amenability: there is no finitely additive, left-translation-invariant probability measure defined on $\mathscr P(\operatorname{SL}(2, \mathbb R))$. Another group lacking amenability is $\operatorname{SO}(3, \mathbb R)$; this is essentially the Banach-Tarski paradox. The reasons behind the non-amenability of these two groups are, however, different. If one searches for a left-translation-invariant (finitely additive) measure defined not on the whole power-set, but only on the Boolean algebra of \emph{definable} subsets of $\operatorname{SO}(3, \mathbb R)$, then such a measure \emph{does} exist, and we say that $\operatorname{SO}(3, \mathbb R)$ is \emph{definably amenable}. A similar thing happens with free groups on at least two generators. This is due to the fact that the non-measurable sets arising from the Banach-Tarski paradox are very complicated, and certainly not definable in the first-order structure of $\mathbb R$, and so this kind of obstructions to amenability disappear when we only want a measure on an algebra of ``simple'' sets. On the contrary, $\operatorname{SL}(2,\mathbb R)$ is not even definably amenable, thus being more inherently pathological under this point of view. In [4] Pillay then proposed the \emph{Ellis Group Conjecture}. Several special cases were proven in the same paper and, thereafter, the conjecture was proven true in [5] by Chernikov and Simon, hence we state it as a Theorem.
\begin{theorem}[5, Theorem 5.6]
If $G$ is a definably amenable \textsc{nip} group, the restriction of the natural map $S_G^{\textnormal{ext}}(M)\to G/G^{00}$ to any ideal group of $G$ is an isomorphism.
\end{theorem}

Remarkably, the model-theoretic techniques involved in stating, approaching, and proving the conjecture are anything but peculiar to this particular problem, and the main focus of this thesis is on the development and understanding of said techniques. This is reflected in the fact that we will deal with Ellis semigroups only in the first chapter and in the closing section. We start in Chapter 1 by studying enveloping semigroups, first in the classical context ([6]) and then in the definable one, without any kind of tameness assumption ([9, 10]). In Chapter 2 we introduce some techniques, still without assuming anything on the underlying theory beyond being first-order complete. In Chapter 3 we introduce dependent theories, see how the previously introduced tools behave in this context, and explore some constructions that heavily exploit the \textsc{nip} hypothesis. In Chapter 4 we bring in the last ingredient, i.e. definable amenability, see that under our hypotheses it is preserved when passing to Shelah's expansion, characterize it in terms of f-generic types, and conclude by studying the proof of the Ellis Group Conjecture.

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