We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context.
As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.
We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.
Poincaré series of $p$-adic, definable equivalence relations have been studied in various cases since Igusa's and Denef's work related to counting solutions of polynomial equations modulo $p^n$ for prime $p$. General semi-algebraic equivalence relations on local fields have been studied uniformly in $p$ recently in \cite{16}. Here we generalize the rationality result of \cite{16} to the analytic case, unifomly in $p$, building further on the appendix of \cite{16} and on \cite{13b}, \cite{03}. In particular, the results hold for large positive characteristic local fields. We also introduce rati…
We express integrals of definable functions over definable sets uniformly for non-Archimedean local fields, extending results of Pas. We apply this to Chevalley groups, in particular proving that zeta functions counting conjugacy classes in congruence quotients of such groups depend only on the size of the residue field, for sufficiently large residue characteristic. In particular, the number of conjugacy classes in a congruence quotient depends only on the size of the residue field. The same holds for zeta functions counting dimensions of Hecke modules of intertwining operators associated to …
Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence F and a parameter R subseteq H(kappa) such that functions f,g:kappa --> 2 are equivalent iff for some h:kappa --> 2, the structure (H(kappa),in,R,f,g,h) satisfies F, where in, R, f, g, and h are interpretations of the symbols appearing in F. All the values mu, 1 leq mu leq kappa^+ or mu=2^kappa, are possible numbers of equivalence classes for such a Sigma_1^1-equivalence relation. Additionally, the possibilities are close…
Consider an o-minimal structure on the real field. Let $M$ be a definable $C^r$ manifold, where $r$ is a nonnegative integer. We first demonstrate an equivalence of the category of definable $C^r$ vector bundles over $M$ with the category of finitely generated projective modules over the ring $C_{\text{df}}^r(M)$. Here, the notation $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$. We also show an equivalence of the category of definable $C^r$ bilinear spaces over $M$ with the category of bilinear spaces over the ring $C_{\text{df}}^r(M)$. The main theorems of this pap…
We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they …
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.
The paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These materials allow us to compare the motivic measure of Cluckers-Loeser and the one of Denef-Loeser in certain classes of definable subassignments.
In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable cove…
In \cite{HPP}, Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that $T$ has the canonical base property in a strong form, " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than \cite{HPP}, and prove its {\em equivalence} with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that $1$-based groups are rigid.
We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …
The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via…
Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{R}^n$ is definable in $\mathcal{R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-d…
We describe all binary simple homogeneous structures M in terms of 0-definable equivalence relations on M, which "coordinatize" M and control dividing, and extension properties that respect these equivalence relations.
We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before. As a by-product, we show some analogous results in purely topological context (without direct use of model theory).
The correspondence between definable connected groupoids in a theory $T$ and internal generalised imaginary sorts of $T$, established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaboration on Hrushovki's original co…
If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \in ω$, then define $$[X]^n_E = \{(x_0, ..., x_{n - 1}) \in {}^nX : (\forall i,j)(i \neq j \Rightarrow \neg(x_i \ E \ x_j))\}.$$ For $n \in ω$, a set $X$ has the $n$-Jónsson property if and only if for every function $f : [X]^n_= \rightarrow X$, there exists some $Y \subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=] \neq X$. A set $X$ has the Jónsson property if and only for every function $f : (\bigcup_{n \in ω}[X]^n_=) \rightarrow X$, there exists some $Y \subseteq X$ with $X$ and $Y$ in bijection so that $f[\bigcup_{…
Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $E$ be a $\mathbfΣ^1_1$ equivalence relation coded in $\mathrm{HOD}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\mathrm{HOD} \models E$ is unpinned. $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$ proves $E$-class section uniformization when $E$ is a $\mathbfΣ^1_1$ equivalence relation on $\mathbb{R}$ which is pinned in every transitive model of $\mathsf{ZFC}$ containing the real which codes $E$: Suppose $R$ is a relation on $\mathbb{R}$ such that each section $R_…
For a finite lattice $Λ$, $Λ$-ultrametric spaces have, among other reasons, appeared as a means of constructing structures with lattices of equivalence relations embedding $Λ$. This makes use of an isomorphism of categories between $Λ$-ultrametric spaces and structures equipped with certain families of equivalence relations. We extend this isomorphism to the case of infinite lattices. We also pose questions about representing a given finite lattice as the lattice of $\emptyset$-definable equivalence relations of structures with model-theoretic symmetry properties.
This paper is the 10th in a sequence on the structure of sets of solutions to systems of equations over groups, projections of such sets (Diophantine sets), and the structure of definable sets over few classes of groups. In the 10th paper we answer affirmatively a question of R. L. Vaught on the elementary equivalence of free products of pairs of elementarily equivalent groups, and obtain a generalization of Tarski's problem on the elementary equivalence of non-abelian free groups. Finally, we prove that free products of stable groups is stable, generalizing a previous result on the stability …
This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class---those equinumerous elements of the equivalence class with equinumerous complements---can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved m…
We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (conc…
Let $Γ$ and $Δ$ be sufficiently distinct countable groups. We show that there is an orbit equivalence relation $E$, induced by an action of the Polish wreath product group $Γ\wrΓ$, so that $E$ is generically $F$-ergodic for any orbit equivalence relation $F$ induced by an action of $Δ\wrΔ$. More generally, we establish generic ergodicity between $Γ$-jumps and the iterated $Δ$-jumps, answering a question of Clemens and Coskey. The proofs follow a translation between Borel homomorphisms and definable pins.
We examine the degree structure $\mathbf{ER}$ of equivalence relations on $ω$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\mathbf{ER}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf{d}\oplus \mathbf…
Ordinal analysis induces a partition of $Σ^1_1$-definable and $Π^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $Π^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $Σ^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.
This paper considers "definable cardinalities" arising from Polish group actions. The first part of the paper answers a question of Becker-Kechris by showing that under suitable determinacy assumptions in ZF+DC, every action by a Polish group on a metric space is either as complicated as the Vitali equivalence relation or no more complicated than the equality relation on bounded subsets of \aleph_1. The same argument shows that if hereditarily wellorderabel sets can be used as complete invariants, then elements in HC will do. The second half gives a new proof of Howard Becker's recent theorem …
Let $K \subset {\mathbb R}^n$ be a compact definable set in an o-minimal structure over $\mathbb R$, e.g., a semi-algebraic or a subanalytic set. A definable family $\{ S_δ|\> 0< δ\in {\mathbb R} \}$ of compact subsets of $K$, is called a monotone family if $S_δ\subset S_η$ for all sufficiently small $δ> η>0$. The main result of the paper is that when $\dim K \le 2$ there exists a definable triangulation of $K$ such that for each (open) simplex $Λ$ of the triangulation and each small enough $δ>0$, the intersection $S_δ\cap Λ$ is equivalent to one of the five standard families in the standard s…
We show that induction over $Δ(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.
Let $\mathcal M=(M,<,...)$ be a linearly ordered first-order structure and $T$ its complete theory. We investigate conditions for $T$ that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin's work, we label three conditions expressing properties of types of $T$ and/or automorphisms of models of $T$. We prove several results which indicate the "geometric" simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, …
Let $X = \left\{P \in [0,1]^{\bf N} : \left(\forall ν\in {\bf N} \right) \left(P \left(\{ν\} \right) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} P \left(\{ν\} \right) = 1 \right\} $ be the Polish space of probability measures on ${\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \in X$, let $Γ_{P} = \left\{ξ\in L^{1}({\bf N}, P) : \left(\forall ν\in {\bf N} \right) \left(ξ(ν) > 0 \right) \wedge \sum\limits_{ν= 0}^{\infty} ξ(ν) P \left(\{ν\} \right) = 1 \right\} $ and let $Φ_{P} : Γ_{P} \ni ξ\mapsto Φ_{P}(ξ) \in X$ be defined by the relation $\l…
