The Classification of Hypersmooth Borel Equivalence Relations

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel automorphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism. An equivalence relation E on X is Borel reducible to an equivalence relation F on Y if there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). We write then E ≤ F . If E ≤ F and F ≤ E we say that E,F are Borel bi-reducible, in symbols E ≈∗ F . When E ≈∗ F the quotient spaces X/E, Y/F have the same “effective” or “definable” cardinality. We say that E,F are Borel isomorphic if there exists a Borel bijection f : X → Y with xEy ⇔ f(x)Ff(y). Below we denote by E0, Et the equivalence relations on the Cantor space 2 N given by: xE0y ⇔ ∃n∀m ≥ n(xm = ym), xEty ⇔ ∃n∃k∀m(xn+m = yk+m). We denote by ∆X the equality relation on X , and finally we call E smooth if E ≤ ∆2N . This just means that elements of X can be classified up to E-equivalence by concrete invariants which are members of some Polish space. It is shown now in [DJK] that up to Borel bi-reducibility there is exactly one non-smooth hyperfinite Borel E, namely E0, and up to Borel isomorphism there are exactly countably many non-smooth hyperfinite aperiodic (i.e., having no finite equivalence classes) Borel E, namely Et, E0 ×∆n (1 ≤ n ≤ א0), E0 ×∆2N (where ∆n = ∆X , with card(X) = n, if 1 ≤ n ≤ א0). In this paper we investigate and classify the class of Borel equivalence relations which are the “continuous” analogs of the hyperfinite ones. We call a Borel equivalence relation E hypersmooth if it can be written as E = ⋃ nEn, where E0 ⊆ E1 ⊆ · · · is an increasing sequence of smooth Borel equivalence relations. These have been also studied (in a measure theoretic context) in the Russian literature under the name tame equivalence relations. They include many interesting examples such as: The increasing union of a sequence of closed or even Gδ equivalence relations (like for example the coset equivalence relation of a Polish group modulo a subgroup, which is the increasing union of a sequence of closed subgroups), the hyperfinite equivalence relations, the “tail” equivalence relations