We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T_2, the equivalence relation called ``the equality of countable sets of the reals'', is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with G_δclasses, some ideals, and is closed under the Fubini product over $Fin.$ Both results and their corollaries extend some earlier irreducibility theorems by Hjorth and Kechris.
We study the class of Borel equivalence relations under continuous reducibility. In particular , we characterize when a Borel equivalence relation with countable equivalence classes is $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$). We characterize when all the equivalence classes of such a relation are $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewi…
We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by…
We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations ${\mathbb F}_2$ and ${\mathbb E}_1$.…
Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting …
In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbfΔ_1^1$ classes and every $σ$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbfΔ_1^1$ subsets is proper, there exists some $I^+$ $\mathbfΔ_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbfΔ_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with…
For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$. We show that $\mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $\…
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechr…
Given an analytic equivalence relation, we tend to wonder whether it is Borel. When it is non Borel, there is always the hope it will be Borel on a "large" set -- nonmeager or of positive measure. That has led Kanovei, Sabok and Zapletal to ask whether every proper $σ$ ideal satisfies the following property: given $E$ an analytic equivalence relation with Borel classes, there exists a set $B$ which is Borel and $I$-positive such that $E\restriction_{B}$ is Borel. We propose a related problem -- does every proper $σ$ ideal satisfy: given $A$ an analytic subset of the plane with Borel sections, …
We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence relation induced by a Borel action of a nonarchimedean Polish group.
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.
This paper addresses the notion of time change equivalence for Borel multidimensional flows. We show that all free flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.
We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998). For example, we show that the equivalence relation $\cong^\ast_{ω+1,0}$ is strictly below $\cong^\ast_{ω+1,<ω}$ in Borel reducibility. By results of Hjorth-Kechris-Louveau, $\cong^\ast_{ω+1,<ω}$ provides invariants for $Σ^0_{ω+1}$ equivalence relations induced by actions of $S_\infty$, while $\cong^\ast_{ω+1,0}$ provides invariants for $Σ^0_{ω+1}$…
This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to $μ$ then $E^\mathbb{N}$ is strongly ergodic with respect to $μ^\mathbb{N}$. We answer questions of Clemens and Coskey regarding their recently defined $Γ$-jump operations, in particular showing that the $\mathbb{Z}^2$-jump of $E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$. We study a notion of equivalence relations which can be classified by infinite sequences…
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning …
Let $F_{ω_1}$ be the countable admissible ordinal equivalence relation defined on ${}^ω2$ by $x \ F_{ω_1} \ y$ if and only if $ω_1^x = ω_1^y$. It will be shown that $F_{ω_1}$ is classifiable by countable structures and must be classified by structures of high Scott rank. If $E$ and $F$ are equivalence relations, then $E$ is almost Borel reducible to $F$ if and only if there is a Borel reduction of $E$ to $F$, except possibly on countably many $E$-classes. Let $E_{ω_1}$ denote the equivalence of order types of reals coding well-orderings. It will be shown that in the constructible universe $L$ …
We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$, i.e., groups such that every diagonal action $α\times β$ of $G$ induces a Borel orbit equivalence relation, provided that the actions $α$, $β$ of $…
We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.
We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group $Γ$ we introduce the $Γ$-jump. We study the elementary properties of the $Γ$-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups $Γ$, the $Γ$-jump is \emph{proper} in the sense that for any Borel equivalence relation $E$ the $Γ$-jump of $E$ is strictly higher than $E$ in the Borel reducibility hierarchy. On the other hand there are examples of groups $Γ$ for which the $Γ$-jump is not proper. To establish…
Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces
我们将有关可分巴拿赫空间之间同构关系复杂性的已知结果改进为Borel可约性，并且我们使用经典空间$c_0$、$\ell_p$和$L_p$、$1 \leq p <2$实现了这一点。更准确地说，我们表明，关系$E_{K_σ}$是Borel可约同构和补充双嵌入之间的$c_0$或$\ell_p, 1 \leq p <2$的子空间。我们证明了关系$E_{K_σ} \otimes =^+$是Borel可约的同构，补双嵌入，和Lipschitz等价的子空间之间的$L_p, 1 \leq p <2$。
