Borel Reducibility Research Articles

Analytic equivalence relations and bi-embeddability

AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

On the complexity of the uniform homeomorphism relation between separable Banach spaces

We investigate the uniform homeomorphism relation between separable Banach spaces and the related relation of local equivalence. We completely characterize the descriptive complexity of local equivalence in the Borel reducibility hierarchy. This also provides a lower bound for the complexity of the uniform homeomorphism.

Infinite Time Decidable Equivalence Relation Theory

We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely Δ 2 1 reductions, and hence to the infinite time computable reductions.

The effective theory of Borel equivalence relations

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P ( ω ) , the power set of ω , and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P ( ω ) . In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is that of the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [5,16]) establishing for any recursive ordinal α the existence of Π 1 0 singletons whose α -jumps are Turing incomparable.

Comparing Borel Reducibility and Depth of an ω-Stable Theory

In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories ( T α ) α < ω 1 with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility.

Borel Reductibility and Classification of von Neumann Algebras

AbstractWe announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.

Borel reducibility as an additive property of domains

We prove that under certain requirements, if E and F are Borel equivalence relations, [FORMULA], is a countable union of Borel sets, and E ↾ X n , is Borel reducible to F for all n, then E ↾ X is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains. Bibliography: 19 titles.

Isomorphism of Homogeneous Structures

We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation for certain classes of homogeneous and ultrahomogeneous metric spaces.

BASIS THEOREMS FOR NON-POTENTIALLY CLOSED SETS AND GRAPHS OF UNCOUNTABLE BOREL CHROMATIC NUMBER

We show that there is an antichain basis for neither (1) the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor (2) the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

Borel complexity of isomorphism between quotient Boolean algebras

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

PROPERTY τ AND COUNTABLE BOREL EQUIVALENCE RELATIONS

We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2(ℤ[1/p1, … , 1/pt]), where {p1, …, pt} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines ℚp∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

Coloring linear orders with Rado's partial order

AbstractLet ⪯R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order (a standard example of a wqo which is not a bqo). We show that ⪯R has fairly high complexity with respect to Borel reducibility (e.g. if P is a Borel preorder, then P ≤B ⪯R), although its exact classification remains open. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

THE CLASSIFICATION PROBLEM FOR p-LOCAL TORSION-FREE ABELIAN GROUPS OF RANK TWO

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

Analytic partial orders and oriented graphs

We prove that there is no maximum element, under Borel reducibility, in the class of analytic partial orders and in the class of analytic oriented graphs. We also provide a natural jump operator for these two classes.

A confinal family of equivalence relations and Borel ideals generating them

An increasing θ1-sequence of Borel equivalence relations on a Polish space that is cofinal (in the sense of Borel reducibility) in the family of all Borel equivalence relations is defined as a development of Rosendal’s construction. It is proved that equivalence relations from this sequence are generated by explicitly defined Borel ideals.

Cofinal families of Borel equivalence relations and quasiorders

AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to beKσcomplete.

Bi-Borel reducibility of essentially countable Borel equivalence relations

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

Complete analytic equivalence relations

We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.

Rigidity theorems for actions of product groups and countable Borel equivalence relations

This Memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here E is said to be Borel reducible to F if there is a Borel function f with xEy if and only if f(x)Ff(y). Moreover, E is orbit equivalent to F if the respective measure spaces equipped with the extra structure provided by the equivalence relations are almost everywhere isomorphic. We consider product groups acting ergodically and by measure preserving transformations on standard Borel probability spaces. In general terms, the basic parts of the monograph show that if the groups involved have a suitable notion of “boundary (we make this precise with the definition of near hyperbolic), then one orbit equivalence relation can only be Borel reduced to another if there is some kind of algebraic resemblance between the product groups and coupling of the action. This also has consequence for orbit equivalence. In the case that the original equivalence relations do not have non-trivial almost invariant sets, the techniques lead to relative ergodicity results. An equivalence relation E is said to be relatively ergodic to F if any f with xEy⇒ f(x)Ff(y) has [f(x)]F constant almost everywhere. This underlying collection of lemmas and structural theorems is employed in a number of different ways. One of the most pressing concerns was to give completely self-contained proofs of results which had previously only been obtained using Zimmer's superrigidity theory. We present elementary proofs that there are incomparable countable Borel equivalence relations (Adams-Kechris), inclusion does not imply reducibility (Adams), and (n + 1)E is not necessarily reducible to nE (Thomas). In the later parts of the paper we give applications of the theory to specific cases of product groups. In particular, we catalog the actions of products of the free group and obtain additional rigidity theorems and relative ergodicity results in this context. There is a rather long series of appendices, whose primary goal is to give the reader a comprehensive account of the basic techniques. But included here are also some new results. For instance, we show that the Furstenberg-Zimmer lemma on cocycles from amenable groups fails with respect to Baire category, and use this to answer a question of Weiss. We also present a different proof that F_2 has the Haagerup approximation property.

Superrigidity and countable Borel equivalence relations

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of SL 3( Z) on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.