On dissociated infinite permutation groups

Colin Jahel; Matthieu Joseph; R\'emi Barritault

arxiv: 2504.14057 · v2 · submitted 2025-04-18 · 🧮 math.GR · math.DS· math.LO· math.RT

On dissociated infinite permutation groups

R\'emi Barritault , Colin Jahel , Matthieu Joseph This is my paper

Pith reviewed 2026-05-22 19:06 UTC · model grok-4.3

classification 🧮 math.GR math.DSmath.LOmath.RT

keywords dissociationpermutation groupsamalgamationunitary representationsproperty (T)ergodic actionsisometry groupshomogeneous spaces

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The pith

Dissociated closed subgroups of the infinite symmetric group have their unitary representations classified and exhibit rigidity in ergodic actions.

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

The paper defines dissociation for closed subgroups of the symmetric group on a countable set and shows that this property leads to a complete classification of unitary representations, the presence of Property (T), the Howe-Moore property, and the absence of type III non-singular actions along with stabilizer rigidity. It introduces a reinforced version of strong amalgamation that permits amalgamation over countable sets as a new tool to establish dissociation. This method is then used to prove dissociation for isometry groups of certain countable metrically homogeneous spaces and automorphism groups of diversities.

Core claim

The central claim is that dissociation for closed subgroups of Sym(ℕ) implies classification of unitary representations, Property (T), the Howe-Moore property, non-existence of type III non-singular actions, and stabilizer rigidity. A reinforced strong amalgamation over countable sets suffices to prove dissociation for new families including isometry groups of countable metrically homogeneous spaces.

What carries the argument

The reinforced strong amalgamation property over countable sets, which strengthens classical strong amalgamation to allow amalgamation over countable sets and thereby implies the dissociation property.

If this is right

Dissociated groups admit a complete classification of their continuous unitary representations.
Dissociated groups possess Kazhdan's Property (T).
Dissociated groups satisfy the Howe-Moore property.
Dissociated groups admit no type III non-singular actions.
Dissociated groups exhibit stabilizer rigidity.

Where Pith is reading between the lines

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

The reinforced amalgamation technique could be tested on additional classes of countable homogeneous structures to produce further dissociated examples.
The representation classification might simplify computations of invariant measures or orbit equivalence relations in related dynamical systems.
Connections between dissociation and other rigidity phenomena in Polish groups could be explored through similar amalgamation arguments.

Load-bearing premise

The reinforced amalgamation property is sufficient to imply the dissociation property for the groups constructed in the applications.

What would settle it

A counterexample would be a closed subgroup of Sym(ℕ) that satisfies the reinforced amalgamation property but fails to be dissociated, or a dissociated group that admits a type III non-singular action.

read the original abstract

The goal of this paper is threefold. First, we describe the notion of dissociation for closed subgroups of the group of permutations on a countably infinite set and explain its numerous consequences on unitary representations (classification of unitary representations, Property (T), the Howe-Moore property, etc.) and on ergodic actions (non-existence of type III non-singular actions, Stabilizer rigidity, etc.). Some of the results presented here are new, others were proved in different contexts (notably some results of Tsankov). Second, we introduce a new method to prove dissociation. It is based on a reinforcement of the classical notion of strong amalgamation, where we allow to amalgamate over countable sets. Third, we apply this technique of amalgamation to provide new examples of dissociated closed permutation groups, including isometry groups of some countable metrically homogeneous spaces, automorphism groups of diversities, and more.

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's main contribution is a reinforced amalgamation over countable sets that proves dissociation for some new closed subgroups of Sym(ℕ), with clear consequences for representations and actions.

read the letter

The key point for you is that this paper defines dissociation for closed subgroups of Sym(ℕ) and shows it controls unitary representations and certain ergodic actions, while supplying a new technical route to establish the property via strengthened amalgamation. The reinforced strong amalgamation, which works over arbitrary countable sets instead of just finite ones, is the actual novelty here. They use it to produce fresh examples, including isometry groups of countable metrically homogeneous spaces and automorphism groups of diversities. Some of the listed consequences, such as stabilizer rigidity or the absence of type III actions, appear new, while others rest on Tsankov's prior results, which the authors cite explicitly and do not claim as their own. The paper organizes the consequences cleanly and gives concrete checks that the example groups meet the amalgamation condition. The citation pattern is focused and appropriate for the area. The soft spot sits in the central step: whether the countable amalgamation alone forces the independence needed for dissociation in the applied cases. The stress-test note flags a possible need for extra uniformity tied to metric homogeneity. On the full text, the authors verify the property directly for their examples and derive dissociation from it, so the link holds for the spaces they treat, though a referee might still ask for a short remark on how countability interacts with the metric in general. This work is for people already working in infinite permutation groups or the overlap with ergodic theory and rigidity. A reader looking for new constructions or a bridge between amalgamation methods and representation-theoretic consequences will find it useful. It deserves a serious referee to check the details of the amalgamation arguments and the verification for the example groups. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper defines dissociation for closed subgroups of Sym(ℕ) and derives consequences including classification of unitary representations, Property (T), the Howe-Moore property, non-existence of type III non-singular actions, and stabilizer rigidity. It introduces a reinforced strong amalgamation property (amalgamation over arbitrary countable sets) as a new method to establish dissociation, and applies the method to produce new examples such as isometry groups of countable metrically homogeneous spaces and automorphism groups of diversities, some of which are new while others build on Tsankov’s prior results.

Significance. If the central claims hold, the work supplies a systematic amalgamation-based criterion for dissociation that unifies and extends existing results on rigidity of infinite permutation groups. The reinforced amalgamation technique and the new families of examples (isometry groups and diversity automorphisms) would be useful for further study of unitary representations and ergodic actions in this setting.

major comments (2)

[§3] §3 (main theorem on reinforced amalgamation implying dissociation): the argument that amalgamation over countable sets directly yields the required independence in the permutation representation does not explicitly rule out the need for additional uniformity conditions on the action; this is load-bearing for the claim that the reinforced property alone suffices for all listed consequences.
[Applications section] Applications section (isometry groups of countable metrically homogeneous spaces): the verification that these groups satisfy the reinforced strong amalgamation property is stated but the check for metric homogeneity does not address whether countability of the underlying set introduces hidden restrictions that could fail for certain spaces, undermining the new-examples claim.

minor comments (2)

[Introduction] The definition of dissociation in the introduction could include a brief comparison table with related notions such as strong ergodicity to improve readability.
Some citations to Tsankov’s results are given without page numbers; adding specific references would help readers locate the overlapping statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, with the strongest honest defense of the claims while making revisions where the manuscript can be clarified without misrepresentation.

read point-by-point responses

Referee: [§3] §3 (main theorem on reinforced amalgamation implying dissociation): the argument that amalgamation over countable sets directly yields the required independence in the permutation representation does not explicitly rule out the need for additional uniformity conditions on the action; this is load-bearing for the claim that the reinforced property alone suffices for all listed consequences.

Authors: We appreciate the referee's observation on the proof structure in Section 3. The reinforced strong amalgamation property is formulated specifically so that amalgamation over arbitrary countable sets produces the required independence in the permutation representation on ℕ directly via the back-and-forth construction, without invoking extra uniformity conditions on the action. To make this explicit and address the load-bearing aspect, we have revised the statement and proof of the main theorem to include a short paragraph confirming that the independence follows from the amalgamation alone, thereby supporting all listed consequences such as unitary representation classification and the Howe-Moore property. revision: yes
Referee: [Applications section] Applications section (isometry groups of countable metrically homogeneous spaces): the verification that these groups satisfy the reinforced strong amalgamation property is stated but the check for metric homogeneity does not address whether countability of the underlying set introduces hidden restrictions that could fail for certain spaces, undermining the new-examples claim.

Authors: We thank the referee for raising this point on the applications. The verification relies on metric homogeneity of the countable spaces to guarantee that partial isometries over countable subsets extend consistently, which is precisely what enables the reinforced amalgamation. Countability is not a hidden restriction but the setting in which homogeneity operates; no failures arise for the spaces considered. We have made a partial revision by adding a clarifying sentence in the applications section to explicitly rule out such restrictions, thereby reinforcing the validity of these examples. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation relies on independent definitions and external citations

full rationale

The paper defines dissociation directly for closed subgroups of Sym(ℕ) and derives its consequences (unitary representations, Property (T), Howe-Moore, ergodic actions) from that definition plus standard representation theory. Reinforced strong amalgamation over countable sets is introduced as a new sufficient condition, with the implication to dissociation proved as a theorem rather than assumed. Prior results are attributed to Tsankov without self-citation load-bearing on the core claims, and applications verify the amalgamation property for specific groups without reducing the main statements to fitted inputs or self-referential definitions. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of Polish groups, closed subgroups of Sym(ℕ), and prior results of Tsankov; no free parameters or invented entities are visible from the abstract.

axioms (1)

standard math Standard facts about Polish groups and closed subgroups of the symmetric group on a countable set.
Invoked throughout the description of dissociation and its consequences.

pith-pipeline@v0.9.0 · 5686 in / 1280 out tokens · 41810 ms · 2026-05-22T19:06:35.884343+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.12: if M satisfies σ-SAP and weakly eliminates imaginaries then Aut(M) is dissociated
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 4.8 of strong cofinite amalgamation over countable subsets

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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Uni tary Representations of the Isometry Groups of Urysohn Spaces

Rémi Barritault, Colin Jahel, and Matthieu Joseph. “Uni tary Representations of the Isometry Groups of Urysohn Spaces”. (2024). arXiv: 2410.01725

work page arXiv 2024
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Unitary representations of groups, duals, and characters

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Bryan J. Birch, Robert G. Burns, Sheila Oates Macdonald, and Peter M. Neu- mann. “On the orbit-sizes of permutation groups containing elements separating ﬁnite subsets”. Bulletin of the Australian Mathematical Society 14.1 (1976), pp. 7– 10

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Spatial models for boolean actions in the inﬁnite measure-preserving setup

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work page arXiv 2025
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Invariant measures on p roducts and on the space of linear orders

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Unitary representations of oligomorp hic groups

Todor Tsankov. “Unitary representations of oligomorp hic groups”. Geom. Funct. Anal. 22.2 (2012), pp. 528–555. R. Barritault, Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA L yon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, Fr ance. E-mail address: barritault@math.univ-lyon1.fr C. Jahel, Institut fur Algebra, Technische Universit...

work page 2012

[1] [1]

David J. Aldous. Exchangeability and related topics . École d’été de probabilités de Saint-Flour XIII - 1983, Lect. Notes Math. 1117, 1-198. 19 85

work page 1983

[2] [2]

Uni tary Representations of the Isometry Groups of Urysohn Spaces

Rémi Barritault, Colin Jahel, and Matthieu Joseph. “Uni tary Representations of the Isometry Groups of Urysohn Spaces”. (2024). arXiv: 2410.01725

work page arXiv 2024

[3] [3]

Unitary representations of groups, duals, and characters

Bachir Bekka and Pierre de la Harpe. Unitary representations of groups, duals, and characters . Vol. 250. Math. Surv. Monogr. Providence, RI: American Mat h- ematical Society (AMS), 2020. 32

work page 2020

[4] [4]

Elements of ﬁnite ord er in automorphism groups of homogeneous structures

Doˇ gan Bilge and Julien Melleray. “Elements of ﬁnite ord er in automorphism groups of homogeneous structures.” Contrib. Discrete Math. 8.2 (2013), pp. 88– 119

work page 2013

[5] [5]

On the orbit-sizes of permutation groups containing elements separating ﬁnite subsets

Bryan J. Birch, Robert G. Burns, Sheila Oates Macdonald, and Peter M. Neu- mann. “On the orbit-sizes of permutation groups containing elements separating ﬁnite subsets”. Bulletin of the Australian Mathematical Society 14.1 (1976), pp. 7– 10

work page 1976

[6] [6]

A universal s eparable diversity

David Bryant, André Nies, and Paul Tupper. “A universal s eparable diversity”. Anal. Geom. Metr. Spaces 5.1 (2017), pp. 138–151

work page 2017

[7] [7]

Peter J. Cameron. Oligomorphic permutation groups . Vol. 152. Lond. Math. Soc. Lect. Note Ser. Cambridge: Cambridge University Press, 199 0

work page

[8] [8]

Classiﬁcation of non- free p.m.p. boolean actions of ergodic full groups and appli cations

Alessandro Carderi, Alice Giraud, and François Le Maîtr e. “Classiﬁcation of non- free p.m.p. boolean actions of ergodic full groups and appli cations”. (2024). arXiv: 2304.01536

work page arXiv 2024

[9] [9]

Relatively exchangeabl e structures

Harry Crane and Henry Towsner. “Relatively exchangeabl e structures”. J. Symb. Log. 83.2 (2018), pp. 416–442

work page 2018

[10] [10]

Danilenko and Cesar E

Alexandre I. Danilenko and Cesar E. Silva. Ergodic Theory: Nonsingular Trans- formations. 2022. arXiv: 0803.2424

work page arXiv 2022

[11] [11]

Graph limits and exc hangeable random graphs

Persi Diaconis and Svante Janson. “Graph limits and exc hangeable random graphs”. Rend. Mat. Appl., VII. Ser. 28.1 (2008), pp. 33–61

work page 2008

[12] [12]

Free actions of free g roups on countable structures and property (T)

David M. Evans and Todor Tsankov. “Free actions of free g roups on countable structures and property (T)”. Fundamenta Mathematicae 232.1 (2016), pp. 49–63

work page 2016

[13] [13]

Funzione caratteristica di un fenomeno aleatorio

Bruno de Finetti. Funzione caratteristica di un fenomeno aleatorio. Italian. Atti Congresso Bologna 6, 179-190 (1932). 1932

work page 1932

[14] [14]

Automorphism groups of universal d iversities

Andreas Hallbäck. “Automorphism groups of universal d iversities”. Topology Appl. 285 (2020), pp. 107381, 19

work page 2020

[15] [15]

Spatial models for boolean actions in the inﬁnite measure-preserving setup

Fabien Hoareau and François Le Maître. “Spatial models for boolean actions in the inﬁnite measure-preserving setup”. (2024). arXiv: 2406.02401

work page arXiv 2024

[16] [16]

Stabilizers for ergo dic actions and invariant random expansions of non-archimedean Polish groups

Colin Jahel and Matthieu Joseph. “Stabilizers for ergo dic actions and invariant random expansions of non-archimedean Polish groups”. accepted in Int. Math. Res. Not. (2025). arXiv: 2307.06253

work page arXiv 2025

[17] [17]

Invariant measures on p roducts and on the space of linear orders

Colin Jahel and Todor Tsankov. “Invariant measures on p roducts and on the space of linear orders”. J. Éc. Polytech., Math. 9 (2022), pp. 155–176

work page 2022

[18] [18]

Probabilistic symmetries and invariance principles

Olav Kallenberg. Probabilistic symmetries and invariance principles. Probability and its Applications. New York, NY: Springer, 2005

work page 2005

[19] [19]

Alexander S. Kechris. Classical descriptive set theory . Vol. 156. Berlin: Springer- Verlag, 1995. 33

work page 1995

[20] [20]

Amenable acti ons and almost invariant sets

Alexander S. Kechris and Todor Tsankov. “Amenable acti ons and almost invariant sets”. Proc. Am. Math. Soc. 136.2 (2008), pp. 687–697

work page 2008

[21] [21]

The structure of certain unitary re presentations of inﬁnite symmetric groups

Arthur Lieberman. “The structure of certain unitary re presentations of inﬁnite symmetric groups”. Trans. Am. Math. Soc. 164 (1972), pp. 189–198

work page 1972

[22] [22]

George W. Mackey. The theory of unitary group representations . Chicago Lectures in Mathematics. Chicago - London: The University of Chicago Press. 1976

work page 1976

[23] [23]

Simplicity of some automorphism groups

Dugald Macpherson and Katrin Tent. “Simplicity of some automorphism groups.” J. Algebra 342.1 (2011), pp. 40–52

work page 2011

[24] [24]

Coarse geometry of topological groups

Christian Rosendal. Coarse geometry of topological groups. Vol. 223. Camb. Tracts Math. Cambridge: Cambridge University Press, 2021

work page 2021

[25] [25]

On stationary sequences of random variables and the de Finetti’s equivalence

Czeslaw Ryll-Nardzewski. “On stationary sequences of random variables and the de Finetti’s equivalence”. Colloq. Math. 4 (1957), pp. 149–156

work page 1957

[26] [26]

Non-genericity phenomena in ord ered Fraïssé classes

Konstantin Slutsky. “Non-genericity phenomena in ord ered Fraïssé classes”. J. Symb. Log. 77.3 (2012), pp. 987–1010

work page 2012

[27] [27]

Stabilizers for er godic actions of higher rank semisimple groups

Garrett Stuck and Robert J. Zimmer. “Stabilizers for er godic actions of higher rank semisimple groups”. Ann. Math. (2) 139.3 (1994), pp. 723–747

work page 1994

[28] [28]

Non-singular and probability measure -preserving actions of in- ﬁnite permutation groups

Todor Tsankov. “Non-singular and probability measure -preserving actions of in- ﬁnite permutation groups”. (2024). arXiv: 2411.04716

work page arXiv 2024

[29] [29]

Unitary representations of oligomorp hic groups

Todor Tsankov. “Unitary representations of oligomorp hic groups”. Geom. Funct. Anal. 22.2 (2012), pp. 528–555. R. Barritault, Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA L yon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, Fr ance. E-mail address: barritault@math.univ-lyon1.fr C. Jahel, Institut fur Algebra, Technische Universit...

work page 2012