[Paper Review] Hjorth Analysis of General Polish Group Actions

Ohad Drucker|arXiv (Cornell University)|Dec 20, 2015

Advanced Topology and Set Theory5 references3 citations

TL;DR

This paper establishes a boundedness principle for Hjorth's rank in general Polish group actions, proving that the orbit equivalence relation is Borel if and only if Hjorth ranks are uniformly bounded. It confirms a conjecture of Hjorth by showing that for every limit ordinal α, the set of elements with Hjorth rank less than α is Borel, and proves Nadel's theorem for Hjorth rank: δ(x) ≤ ω₁^ck(x).

ABSTRACT

Hjorth has introduced a Scott analysis for general Polish group actions, and has asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank - namely, the orbit equivalence relation is Borel if and only if Hjorth ranks are bounded. We present the principles of Hjorth analysis and Hjorth rank, and answer Hjorth's question positively. From that we get a positive answer to a conjecture due to Hjorth - for every limit ordinal $α$ , the set of elements whose orbit is of complexity less than $α$ is a Borel set. We then show Nadel's theorem for Hjorth rank - the rank of $x$ is no more than $ω_{1}^{ck(x)}$.

Motivation & Objective

• To resolve Hjorth's open question on whether the boundedness principle for Hjorth rank holds analogously to Scott rank.
• To prove that for every limit ordinal α, the set {x : δ(x) < α} is Borel, confirming a conjecture by Hjorth.
• To establish Nadel's theorem for Hjorth rank, showing that δ(x) ≤ ω₁^ck(x) for all x in a Polish G-space.
• To analyze the Borel complexity of rank comparisons, such as {(x,y) : δ(x) ≤ δ(y)} and {(x,y) : δ(x) < δ(y)}.
• To reprove the Becker-Kechris theorem using the new Hjorth analysis framework.

Proposed method

• Constructs a decreasing sequence of Borel, G-invariant equivalence relations ≡_α using a non-symmetric, transitive relation ≤_α between pairs (x,U) with x ∈ X and U ⊆ G open.
• Defines Hjorth rank δ(x) as the least ordinal α such that x ≡_α+ω y implies orbit equivalence for all y.
• Uses Vaught transforms and Borel determinacy to show that sets of the form U·x are Borel and have complexity close to that of the orbit G·x.
• Applies a back-and-forth argument to prove Scott's isomorphism theorem for Hjorth analysis: x ≡_{δ(x)+ω} y ⇒ x and y are orbit equivalent.
• Employs recursive ordinal analysis to bound δ(x) via ω₁^ck(x), the least non-recursive ordinal relative to x_G, the G-orbit information of x.
• Translates forcing-based arguments into Vaught transform terminology to maintain accessibility without requiring forcing knowledge.

Experimental results

Research questions

• RQ1 Does the boundedness principle for Hjorth rank hold: is E_G^X Borel if and only if Hjorth ranks are uniformly bounded?
• RQ2 Is the set {x : δ(x) < α} Borel for every limit ordinal α?
• RQ3 Does Nadel's theorem extend to Hjorth rank: is δ(x) ≤ ω₁^ck(x) for all x?
• RQ4 What is the Borel complexity of the relation {(x,y) : δ(x) ≤ δ(y)}?
• RQ5 Can the Becker-Kechris theorem for the logic action be reproven using Hjorth analysis?

Key findings

• The boundedness principle holds: E_G^X is Borel if and only if there exists α < ω₁ such that δ(x) ≤ α for all x ∈ X.
• For every limit ordinal α, the set {x : δ(x) < α} is Borel, confirming a conjecture of Hjorth.
• Nadel's theorem is proven for Hjorth rank: δ(x) ≤ ω₁^ck(x) for all x ∈ X.
• The relation {(x,y) : δ(x) ≤ δ(y)} is analytic, and {(x,y) : δ(x) < δ(y)} is Π₁¹, which is optimal.
• In a counterexample to the Vaught conjecture, there exists a non-meager orbit, and the union of all G_δ orbits is comeager.
• At least one G_δ orbit in a counterexample to Vaught's conjecture is not F_σ, implying a structural complexity in the orbit space.