On dense orbits in the space of subequivalence relations

Fran\c{c}ois Le Ma\^itre

arxiv: 2405.01806 · v3 · submitted 2024-05-03 · 🧮 math.DS · math.GR

On dense orbits in the space of subequivalence relations

Fran\c{c}ois Le Ma\^itre This is my paper

Pith reviewed 2026-05-24 01:24 UTC · model grok-4.3

classification 🧮 math.DS math.GR

keywords subequivalence relationsPolish topologydense orbitshyperfinite equivalence relationfull groupergodic theoryBorel complexityuniform metric

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The pith

The subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation that have a dense orbit are characterized.

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

The paper first endows the space of subequivalence relations of any non-singular countable equivalence relation with a Polish topology that extends an existing framework. It then restricts attention to probability measure-preserving equivalence relations and studies the natural actions of the full group and the automorphism group on this space. The central result gives a characterization of those subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation that possess a dense orbit. The work further establishes that every orbit under the full group is meager and supplies Borel complexity calculations for certain natural subsets via a uniform metric. These statements answer questions left open in earlier related work.

Core claim

Our main result is a characterization of the subequivalence relations having a dense orbit in the space of subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation. We also show that in this setup, all full groups orbits are meager. We finally provide a few Borel complexity calculations of natural subsets in spaces of subequivalence relations using a natural metric we call the uniform metric.

What carries the argument

The Polish topology on the space of subequivalence relations, which supports the discussion of dense orbits under the action of the full group and the automorphism group.

If this is right

The characterization identifies precisely which subequivalence relations possess dense orbits under the relevant group actions.
Every orbit under the full group is meager in the space.
Natural subsets of the space have explicit Borel complexities computed with the uniform metric.
Several questions posed in an earlier version of related work on subequivalence relations receive answers.

Where Pith is reading between the lines

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

The same topology construction may permit similar density questions to be posed for subequivalence relations of equivalence relations that are not hyperfinite.
The meagerness result indicates that full-group equivalence is a rare property among subequivalence relations in the Baire-category sense.
The uniform metric and complexity calculations supply concrete tools for distinguishing generic versus special subequivalence relations in descriptive-set-theoretic terms.

Load-bearing premise

The Polish topology constructed on the space of subequivalence relations extends the prior framework while preserving the topological and group-action properties needed for p.m.p. equivalence relations.

What would settle it

An explicit subequivalence relation of the ergodic hyperfinite p.m.p. equivalence relation that either has a dense orbit while failing the stated characterization or fails to have a dense orbit while satisfying it.

read the original abstract

We first explain how to endow the space of subequivalence relations of any non-singular countable equivalence relation with a Polish topology, extending the framework of Kechris' recent monograph on subequivalence relations of probability measure-preserving (p.m.p.) countable equivalence relations. We then restrict to p.m.p. equivalence relations and discuss dense orbits therein for the natural action of the full group and of the automorphism group of the relation. Our main result is a characterization of the subequivalence relations having a dense orbit in the space of subequivalence relations of the ergodic hyperfinite p.m.p. equivalence relation. We also show that in this setup, all full groups orbits are meager. We finally provide a few Borel complexity calculations of natural subsets in spaces of subequivalence relations using a natural metric we call the uniform metric. This answers some questions from an earlier version of Kechris' monograph.

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.

Desk Editor's Note private letter to a colleague

The paper characterizes dense orbits for subequivalence relations of the ergodic hyperfinite equivalence relation and shows full-group orbits are meager, via a new Polish topology extending Kechris.

read the letter

The main thing here is a characterization of which subequivalence relations have dense orbits under the full group (and automorphism group) action in the space for the ergodic hyperfinite p.m.p. equivalence relation, plus the fact that all full-group orbits turn out meager. It also gives some Borel complexity calculations via a uniform metric and answers questions from an earlier version of Kechris' monograph on the topic. The first step is endowing the space of subequivalence relations for any non-singular countable equivalence relation with a Polish topology, then restricting to the p.m.p. setting to study the actions. This looks like a direct, useful extension of the existing framework rather than a restatement. The topology construction and the restriction to the hyperfinite case let the authors get the characterization and meagerness result cleanly. The work sits squarely on Kechris without obvious circularity or heavy self-citation dependence. No load-bearing gaps show up in the high-level argument. The only real limitation is that the details of the topology extension and the orbit characterization sit in the proofs, which are not visible here, but the abstract frames them as straightforward consequences of the setup. This paper is for people already working in descriptive set theory and ergodic theory who know the Kechris monograph. A reader in that niche will pick up new tools and resolved questions; outsiders will find it narrow. It is the sort of targeted advance that deserves a serious referee rather than a desk reject, even if revisions are needed on the proofs. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The paper first constructs a Polish topology on the space of subequivalence relations of any non-singular countable equivalence relation by extending Kechris' p.m.p. framework. Restricting to p.m.p. relations, it characterizes the subequivalence relations of the ergodic hyperfinite equivalence relation that possess dense orbits under the natural actions of the full group and the automorphism group, proves that all full-group orbits are meager, and computes the Borel complexities of several natural subsets via a uniform metric. These results answer questions posed in an earlier version of Kechris' monograph.

Significance. If the main characterization holds, the work supplies a concrete topological and dynamical description of dense orbits in a central example, together with meagerness and complexity results that directly resolve open questions from Kechris' monograph. The extension of the Polish topology to the non-singular setting is a natural and potentially reusable contribution to the descriptive set theory of equivalence relations.

minor comments (2)

[§2] §2: The construction of the Polish topology on subequivalence relations for non-singular relations is presented as a direct extension, but a short paragraph comparing the new topology to the one in Kechris' monograph (especially the role of the measure class) would help readers verify that the relevant group-action properties are preserved.
The uniform metric is introduced for the Borel complexity calculations; an explicit statement of its relation to the standard metric on the space of equivalence relations would clarify why the complexity results are new.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, its significance, and the recommendation of minor revision. No specific major comments appear in the report, which we take as confirmation that the core results—the Polish topology extension, the characterization of dense orbits for the ergodic hyperfinite relation, the meagerness of full-group orbits, and the Borel complexity calculations—are viewed as correct and responsive to the questions from Kechris' monograph.

Circularity Check

0 steps flagged

No significant circularity; derivation extends external framework with independent characterization results

full rationale

The paper first constructs a Polish topology on the space of subequivalence relations for non-singular countable equivalence relations by extending Kechris' p.m.p. framework (explicitly cited as external prior work). It then restricts to the p.m.p. case and proves a characterization of subequivalence relations with dense orbits under the full group or automorphism group action specifically for the ergodic hyperfinite relation, plus meagerness of full group orbits and Borel complexity calculations via the uniform metric. These steps are presented as direct mathematical extensions and new theorems without any reduction of the central claims to fitted parameters, self-definitional loops, or load-bearing self-citations. The cited Kechris monograph is independent external work (not overlapping authors), and the main result is a characterization theorem rather than a renaming or ansatz smuggling. No equations or claims in the abstract or high-level description reduce the output to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract; the work relies on standard background from ergodic theory and descriptive set theory.

pith-pipeline@v0.9.0 · 5683 in / 1129 out tokens · 32663 ms · 2026-05-24T01:24:20.183508+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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On full groups of non-ergodic probability-measure-preserving equivalence relations

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Fran c ois Le Ma \^i tre and Konstantin Slutsky. L^1 full groups of flows. arXiv preprint , 2023. https://doi.org/10.48550/arXiv.2108.09009 doi:10.48550/arXiv.2108.09009

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work page doi:10.48550/arxiv.2407.10905 2024

[1] [1]

A construction of equivalence subrelations for intermediate subalgebras

Hisashi Aoi. A construction of equivalence subrelations for intermediate subalgebras. J. Math. Soc. Japan , 55(3), 2003. https://doi.org/10.2969/jmsj/1191418999 doi:10.2969/jmsj/1191418999

work page doi:10.2969/jmsj/1191418999 2003

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Bourbaki

N. Bourbaki. General Topology : Chapters 1--4 . Springer-Verlag Berlin and Heidelberg GmbH & Co. K , Berlin Heidelberg, 2nd edition, 1998

work page 1998

[3] [3]

Topometric spaces and perturbations of metric structures

Ita \"i Ben Yaacov. Topometric spaces and perturbations of metric structures. Log Anal , 1(3):235, 2008. https://doi.org/10.1007/s11813-008-0009-x doi:10.1007/s11813-008-0009-x

work page doi:10.1007/s11813-008-0009-x 2008

[4] [4]

A topometric Effros theorem

Ita \"i Ben Yaacov and Julien Melleray. A topometric Effros theorem. The Journal of Symbolic Logic , pages 1--11, 2023. https://doi.org/10.1017/jsl.2023.5 doi:10.1017/jsl.2023.5

work page doi:10.1017/jsl.2023.5 2023

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a user Advanced Texts Basler Lehrb \

Donald L. Cohn. Measure Theory : Second Edition . Birkh \"a user Advanced Texts Basler Lehrb \"u cher . Springer, New York, NY, 2013. https://doi.org/10.1007/978-1-4614-6956-8 doi:10.1007/978-1-4614-6956-8

work page doi:10.1007/978-1-4614-6956-8 2013

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H. A. Dye. On Groups of Measure Preserving Transformations . I . American Journal of Mathematics , 81(1):119--159, 1959. https://doi.org/10.2307/2372852 doi:10.2307/2372852

work page doi:10.2307/2372852 1959

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Generic IRS in free groups, after Bowen

Amichai Eisenmann and Yair Glasner. Generic IRS in free groups, after Bowen . Proceedings of the American Mathematical Society , 144(10):4231--4246, 2016. https://doi.org/10.1090/proc/13020 doi:10.1090/proc/13020

work page doi:10.1090/proc/13020 2016

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Michael's selection theorem and applications to the Mar \'e chal topology

Pierre Fima, Fran c ois Le Ma \^i tre, Kunal Mukherjee, and Issan Patri. Michael's selection theorem and applications to the Mar \'e chal topology. arXiv preprint , 2024. https://doi.org/10.48550/arXiv.2407.05776 doi:10.48550/arXiv.2407.05776

work page doi:10.48550/arxiv.2407.05776 2024

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Invariant Descriptive Set Theory , volume 293 of Pure and Applied Mathematics ( Boca Raton )

Su Gao. Invariant Descriptive Set Theory , volume 293 of Pure and Applied Mathematics ( Boca Raton ) . CRC Press, Boca Raton, FL, 2009

work page 2009

[10] [10]

A measurable-group-theoretic solution to von Neumann 's problem

Damien Gaboriau and Russell Lyons. A measurable-group-theoretic solution to von Neumann 's problem. Invent. math. , 177(3):533--540, 2009. https://doi.org/10.1007/s00222-009-0187-5 doi:10.1007/s00222-009-0187-5

work page doi:10.1007/s00222-009-0187-5 2009

[11] [11]

Approximations of standard equivalence relations and Bernoulli percolation at pu

Damien Gaboriau and Robin Tucker-Drob . Approximations of standard equivalence relations and Bernoulli percolation at pu. Comptes Rendus Mathematique , 354(11):1114--1118, 2016. https://doi.org/10.1016/j.crma.2016.09.011 doi:10.1016/j.crma.2016.09.011

work page doi:10.1016/j.crma.2016.09.011 2016

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Topological groups with Rokhlin properties

Eli Glasner and Benjamin Weiss. Topological groups with Rokhlin properties. Colloquium Mathematicae , 110(1):51--80, 2008

work page 2008

[13] [13]

Subequivalence relations and positive-definite functions

Adrian Ioana, Alexander Kechris, and Todor Tsankov. Subequivalence relations and positive-definite functions. Groups, Geometry, and Dynamics , pages 579--625, 2009. https://doi.org/10.4171/GGD/71 doi:10.4171/GGD/71

work page doi:10.4171/ggd/71 2009

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Uniqueness of the Group Measure Space Decomposition for Popa 's HT Factors

Adrian Ioana. Uniqueness of the Group Measure Space Decomposition for Popa 's HT Factors . Geom. Funct. Anal. , 22(3):699--732, 2012. https://doi.org/10.1007/s00039-012-0178-3 doi:10.1007/s00039-012-0178-3

work page doi:10.1007/s00039-012-0178-3 2012

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Alexander S. Kechris. The space of measure-preserving equivalence relations and graphs. in preparation, preliminary version available on his webpage

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work page doi:10.1090/surv/160 2010

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Alexander Kechris and Benjamin D. Miller. Topics in Orbit Equivalence . Lecture Notes in Mathematics . Springer-Verlag, Berlin Heidelberg, 2004. https://doi.org/10.1007/b99421 doi:10.1007/b99421

work page doi:10.1007/b99421 2004

[19] [19]

Sur les groupes pleins pr \'e servant une mesure de probabilit \'e

Fran c ois Le Ma \^i tre. Sur les groupes pleins pr \'e servant une mesure de probabilit \'e . PhD thesis, ENS Lyon, 2014

work page 2014

[20] [20]

On full groups of non-ergodic probability-measure-preserving equivalence relations

Fran c ois Le Ma \^i tre. On full groups of non-ergodic probability-measure-preserving equivalence relations. Ergodic Theory and Dynamical Systems , 36(7):2218--2245, 2016. https://doi.org/10.1017/etds.2015.20 doi:10.1017/etds.2015.20

work page doi:10.1017/etds.2015.20 2016

[21] [21]

Polish topologies on groups of non-singular transformations

Fran c ois Le Ma \^i tre. Polish topologies on groups of non-singular transformations. Journal of Logic and Analysis , 14, 2022. https://doi.org/10.4115/jla.2022.14.4 doi:10.4115/jla.2022.14.4

work page doi:10.4115/jla.2022.14.4 2022

[22] [22]

L^1 full groups of flows

Fran c ois Le Ma \^i tre and Konstantin Slutsky. L^1 full groups of flows. arXiv preprint , 2023. https://doi.org/10.48550/arXiv.2108.09009 doi:10.48550/arXiv.2108.09009

work page doi:10.48550/arxiv.2108.09009 2023

[23] [23]

Topologie et structure bor \'e lienne sur l'ensemble des alg \`e bres de von Neumann

Odile Mar \'e chal. Topologie et structure bor \'e lienne sur l'ensemble des alg \`e bres de von Neumann . C. R. Acad. Sci. Paris Ser. I Math. , 276:847--850, 1973

work page 1973

[24] [24]

Ornstein and Benjamin Weiss

Donald S. Ornstein and Benjamin Weiss. Ergodic theory of amenable group actions. I . The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) , 2(1):161--164, 1980. https://doi.org/10.1090/S0273-0979-1980-14702-3 doi:10.1090/S0273-0979-1980-14702-3

work page doi:10.1090/s0273-0979-1980-14702-3 1980

[25] [25]

Asymptotic Orthogonalization of Subalgebras in II _1 Factors

Sorin Popa. Asymptotic Orthogonalization of Subalgebras in II _1 Factors . Publications of the Research Institute for Mathematical Sciences , 55(4):795--809, 2019. https://doi.org/10.4171/prims/55-4-5 doi:10.4171/prims/55-4-5

work page doi:10.4171/prims/55-4-5 2019

[26] [26]

Every action of a nonamenable group is the factor of a small action

Brandon Seward. Every action of a nonamenable group is the factor of a small action. J. Mod. Dyn. , 8(2):251--270, 2014. https://doi.org/10.3934/jmd.2014.8.251 doi:10.3934/jmd.2014.8.251

work page doi:10.3934/jmd.2014.8.251 2014

[27] [27]

Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

Shuoxing Zhou. Noncommutative topological boundaries and amenable invariant random intermediate subalgebras. arXiv preprint , 2024. https://doi.org/10.48550/arXiv.2407.10905 doi:10.48550/arXiv.2407.10905

work page doi:10.48550/arxiv.2407.10905 2024