We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is n
ot definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.
For $G$ an algebraic group definable over a model of $operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connec
ted and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $widehat{S}$ onto a definable group which is internal to the value group. We show that, in case $S$ is defined over a complete valued field $K$ with value group a subgroup of $mathbb{R}$, this map descends to an $S(K)$-equivariant strong deformation retraction of the Berkovich analytification $S^{mathrm{an}}$ of $S$ onto a piecewise linear group, namely onto the skeleton of $S^{mathrm{an}}$. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of $S$. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group $G$ is a directed union of $infty$-definable subgroups which all stabilize a generically stable Keisler measure on $G$.
We work in a first-order setting where structures are spread out over a metric space, with quantification allowed only over bounded subsets. Assuming a doubling property for the metric space, we define a canonical {em core} $mathcal{J}$ associated to
such a theory, a locally compact structure that embeds into the type space over any model. The automorphism group of $mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact group $mathcal{G}$. The automorphism groups of models of the theory are related with $mathcal{G}$, not in general via a homomorphism, but by a {em quasi-homomorphism}, respecting multiplication up to a certain canonical compact error set. This fundamental structure is applied to describe the nature of approximate subgroups. Specifically we obtain a full classification of (properly) approximate lattices of $SL_n({mathbb{R}})$ or $SL_n({mathbb{Q}}_p)$.
We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variabl
es. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupi{n}ski, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $omega$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupi{n}ski, A. Pillay] . In the special case when amenability is witnessed by $emptyset$-definable global Keisler measures (which is for example the case for amenable $omega$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.
We identify a canonical structure J associated to any first-order theory, the {it space of definability patterns}. It generalizes the imaginary algebraic closure in a stable theory, and the hyperimaginary bounded closure in simple theories. J admits
a compact topology, not necessarily Hausdorff, but the Hausdorff part can already be bigger than the Kim-Pillay space. Using it, we obtain simple proofs of a number of results previously obtained using topological dynamics, but working one power set level lower. The Lascar neighbour relation is represented by a canonical relation on the compact Hausdorff part J; the general Lascar group can be read off this compact structure. This gives concrete form to results of Krupinski, Newelski, Pillay, Rzepecki and Simon, who used topological dynamics applied to large models to show the existence of compact groups mapping onto the Lascar group. In an appendix, we show that a construction analogous to the above but using infinitary patterns recovers the Ellis group of cite{kns}, and use this to sharpen the cardinality bound for their Ellis group from $beth_5$ to $beth_3$, showing the latter is optimal. There is also a close connection to another school of topological dynamics, set theory and model theory, centered around the Kechris-Pestov-Todorv cevic correspondence. We define the Ramsey property for a first order theory, and show - as a simple application of the construction applied to an auxiliary theory - that any theory admits a canonical minimal Ramsey expansion. This was envisaged and proved for certain Fraisse classes, first by Kechris-Pestov-Todorv cevic for expansions by orderings, then by Melleray, Nguyen Van The, Tsankov and Zucker for more general expansions.
Motivated by Emmanuel Kowalskis exponential sums over definable sets in finite fields, we generalize Axs theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weils Riemann hypothesis for
curves over finite fields is taken by the `Weil bound on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions $Ff_p^n to Cc$ obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $Psi_p(xi)$ at certain algebraic functions $xi$ of the variables, where $Psi(n mod p) = exp(2 pi i n/p)$ is the standard additive character.
We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, domination implies is
olation which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $pin S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.
We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version
of the stabilizer theorem, and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain ``weak Bohr compactification introduced in [24]. In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a ``definable-topological context, confirming the main conjectures from [24]. We also introduce $bigvee$-definable group topologies on a given $emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. Thirdly, we give an example of a $emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $mathbb{F}_2 times mathbb{Z}$ in a suitable language (where $mathbb{F}_2$ is the free group in 2-generators) for which the $bigvee$-definable group $H:=langle X rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) ``model exists for each approximate subgroup does not work in general (they proved in [29] that it works for definably amenable approximate subgroups).
We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignmen
ts are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate.
We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (ove
r a sort $Gamma$) if every type over a sufficiently rich base structure can be viewed as part of a $Gamma$-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from $Gamma$, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.
