Recent developments in the theory of Borel reducibility

Abstract

Let E 0 be the Vitali equivalence relation and E 3 the product of countably many copies of E 0 . Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E 3 , either E is reducible to E 0 or else E 3 is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible to a countable Borel equivalence relation or else E 3 is reducible to E.We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.[21]of as a subgroup of R N in the usual way.24 G. Hjorth and A. S. KechrisDefinition 2.5. A Borel equivalence relation E on X is said to be countable if every equivalence class is countable. It is then said to be treeable if there is a symmetric Borel relation R ⊆ X×X which has no cycles and whose connected components form the equivalence classes of E; in other words, we may, in a Borel manner, place the structure of a tree on each [x] E .An equivalence relation E is said to be smooth or concretely classifiable if it is Borel reducible to R.It is a non-trivial fact that in the ≤ B ordering there is a maximal countable Borel equivalence relation, E ∞ , and a maximal countable treeable Borel equivalence relation, E T∞ . For the sake of definiteness, we give each one an instantiation, but the reader should see [4] or [13] for a more detailed analysis.

Non-archimedean abelian Polish groups and their actions

“…We now use a theorem of Hjorth and Kechris from [11], Theorem 7.3, to get a sequence of Polish spaces (Z m ) m∈N + , with a Borel action of H m on Z m for each m ∈ N + , such that…”

Comparing Classes of Finite Structures

“…In descriptive set theory, there is a body of work using a notion of "Borel embedding" to compare the classification problems for various classes of structures (fields, graphs, groups, etc.) [7], [3], [11], [12], [13]. In this work, each class consists of structures with the same countable universe, say ω, and with the same language, usually finite.…”

A weak dichotomy below E1×E3

“…4 Then the pair a, H a δ (ϕ) obviously still belongs to D n and H a δ (H a δ (ϕ)) = ϕ . Coming back to a triple of y, ξ, η ∈ R N such that ξ E 3 η , let δ = δ ξ η .…”