Basis Problem for Turbulent Actions II: c ₀ -Equalities

Abstract

This note is a contribution to the study of reducibility between Borel equivalence relations (see [14,11,9], or § 1). If E and F are Borel equivalence relations on Polish spaces X and Y, then we say that E is Borel reducible to F (in symbols,for all x; y P X. The early success in the study of the Borelreducibility was marked by the discovery of two dichotomy theorems by Silver [25] and Harrington, Kechris and Louveau [6]. For example, the latter result (socalled Glimm±Effros dichotomy, since it extends earlier theorems by J. Glimm and E. Effros) says that either a Borel equivalence relation E is smooth, that is, Borelreducible to the equality relation on some separable, completely metrizable (or Polish) space, or the Vitali equivalence relation, E 0 (see Example 3.2), is continuously reducible to E. Since E 0 is not smooth, this result can be interpreted as saying that E 0 is the minimal non-smooth Borel equivalence relation.Therefore the Glimm±Effros dichotomy gives one answer to the questioǹ When is an equivalence relation simply classi®able?' Another criterion for the simplicity of a Borel equivalence relation is that of being effectively reducible to the isomorphism relation of countable models. An equivalence relation E on a Polish space X is effectively reducible to the isomorphism relation of countable models if there is a Borel-measurable mapping f : X 3 2 N < N such that f x and f y code isomorphic countable structures if and only if x E y. This is a natural extension of smoothness which occurs in classical classi®cation results, like Cantor±Bendixson analysis, Ulm invariants, spectral theorems, . . . (see [9, § 0]).The ®rst example of a Borel equivalence relation which is not effectively reducible to the isomorphism relation of countable models was given by H. Friedman [5]. Later, Hjorth isolated the dynamical property of Polish group actions called turbulence (see [9,14]), and proved that in the realm of orbit equivalence relations arising from continuous Polish group actions,`effectively reducible to the isomorphism relation of countable models' is essentially equivalent with`induced by a turbulent action' [9, Theorem 0.14]. It is natural to ask whether a dichotomy along the lines of the Glimm±Effros dichotomy can be proved in the context of turbulence, namely whether there is the minimum turbulent orbit equivalence relation. The answer is known to be negative, since there are turbulent orbit equivalence relations E and F such that no turbulent orbit equivalence relation is Borel-reducible to both E and F (see [8]). We say that such E and F are incompatible. This still leaves a possibility that there is a ®nite list 2000 Mathematics Subject Classi®cation: 03E15.

Recent developments in the theory of Borel reducibility

“…Theorem 5.2 (Farah; see [6]). There is no finite or even countably in- [7]). There is a Polish group G and a turbulent Polish G-space X which is above no minimal turbulent orbit equivalence relation; that is to say, for each Polish group H 0 and turbulent Polish…”

Borel cardinalities below $c_0$

“…Thus we define: Proof. By Theorem 1.5, this is immediate once we know ∼ LV is a c 0 -equality, which is noted in [Far01] (and also easy to check directly).…”

Characterization of ‐like and c0‐like equivalence relations

“…The study of c 0 ‐like equivalence relations was commenced by Farah . Definition Let be a sequence of pseudo‐metric spaces.…”

“…Another condition was introduced by Farah for investigating c 0 ‐equalities. It is easy to check that (*) holds iff for arbitrary , there exist such that and does not converge to 0.…”