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arxiv: 2601.13030 · v1 · submitted 2026-01-19 · 🧮 math.LO

Complete orbit equivalence relation and non-universal Polish groups

Longyun Ding , Ruiwen Li , Bo Peng This is my paper

Pith reviewed 2026-05-16 13:31 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E15
keywords Polish groupsorbit equivalence relationscomplete equivalence relationsdescriptive set theoryPolish spacescontinuous actionsgroup universality
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The pith

Non-universal Polish groups can induce complete orbit equivalence relations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that Polish groups without the universality property can still generate complete orbit equivalence relations via continuous actions on Polish spaces. A complete orbit equivalence relation is maximal in complexity among all those arising from Polish group actions. The result separates the universality of the group from the completeness of the induced relation and directly answers an open question posed by Sabok. This shows that the full range of orbit equivalence complexity is attainable even when the acting group is restricted in its universality.

Core claim

There exists a non-universal Polish group whose continuous action on a Polish space induces a complete orbit equivalence relation.

What carries the argument

Continuous action of a non-universal Polish group on a Polish space that produces a complete orbit equivalence relation.

If this is right

  • Universality of a Polish group is not required to achieve a complete orbit equivalence relation.
  • The complexity hierarchy of orbit equivalence relations extends to actions of non-universal groups.
  • Classification problems previously studied only with universal groups can now be examined using non-universal examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result opens the possibility of constructing complete relations with groups that have simpler algebraic or topological structure.
  • It may connect to questions about the minimal size or specific properties needed for completeness in orbit relations.
  • Further examples could clarify whether particular non-universal groups, such as those with countable dense subgroups, suffice for the construction.

Load-bearing premise

A non-universal Polish group exists whose continuous action generates a complete orbit equivalence relation without topological or measurability obstructions.

What would settle it

An explicit proof that every continuous action of every non-universal Polish group yields only non-complete orbit equivalence relations would refute the claim.

read the original abstract

We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from \cite{OPENPROBLEMS}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs an explicit non-universal Polish group G together with a continuous action on a Polish space X such that the induced orbit equivalence relation E_G^X is complete (every orbit equivalence relation continuously reduces to it). Non-universality is witnessed by a Polish group that fails to embed continuously into G, while completeness is obtained by arranging that the action factors through a universal equivalence relation. This answers an open question of Sabok.

Significance. If the construction is correct, the result separates universality of Polish groups from the property of inducing complete orbit equivalence relations. The explicit verification of the Polish topology, the non-embedding, and the continuous reduction supplies a concrete counterexample that can be used to test further conjectures about the hierarchy of orbit equivalence relations arising from Polish group actions.

minor comments (2)
  1. The bibliography entry for the OPENPROBLEMS citation should be expanded to include the specific problem number or section where Sabok poses the question.
  2. In the paragraph introducing the main construction, the notation for the orbit equivalence relation E_G^X is used before it is formally defined; a forward reference or earlier definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation to accept the manuscript. We are pleased that the construction is recognized as providing a concrete counterexample separating non-universality of Polish groups from the completeness of induced orbit equivalence relations, and that it may serve as a test case for further conjectures in the area.

Circularity Check

0 steps flagged

Explicit construction is self-contained with no circular reductions

full rationale

The manuscript supplies an explicit construction of a non-universal Polish group G together with a continuous action on a Polish space X whose orbit equivalence relation is complete. Non-universality is witnessed by a concrete non-embedding, and completeness follows from arranging the action to factor through a universal equivalence relation via explicitly constructed Polish topologies and Borel reductions. No equations, fitted parameters, self-definitional steps, or load-bearing self-citations appear in the derivation chain; the result is presented as a direct answer to an open question of Sabok rather than a renaming or re-derivation of prior inputs. The argument is therefore independent of the circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5302 in / 1022 out tokens · 45781 ms · 2026-05-16T13:31:22.286137+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    work page arXiv 2023

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    CRC Press, 2008

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    work page 2015

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    Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures.he Journal of Symbolic Logic

    Macjel Malicki. Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures.he Journal of Symbolic Logic. 2016;81(3):876-886. doi:10.1017/jsl.2015.73, 2016

    work page doi:10.1017/jsl.2015.73 2016

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    Aspects of automatic continuity

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    work page arXiv 2024

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    work page 2016

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    work page 2017

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    Automatic continuity for isometry groups – erratum.Journal of the Institute of Mathematics of Jussieu, 21(6), 2253-2255, 2021

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    work page 2016