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arxiv: 2604.14945 · v1 · submitted 2026-04-16 · 🧮 math.GR · math.DS

Recognition: unknown

On quantitative orbit equivalence for lamplighter-like groups

Corentin Correia, Vincent Dumoncel

Pith reviewed 2026-05-10 08:42 UTC · model grok-4.3

classification 🧮 math.GR math.DS
keywords halo productsorbit equivalencelamplighter groupsisoperimetric profilequantitative invariantspermutational productsshuffler groupsFølner sequences
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The pith

Shuffler groups over different lattice dimensions are L^p orbit equivalent only below a precise ratio of those dimensions.

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

The paper proves a stability theorem for orbit equivalence among permutational halo products, a family of groups whose geometry resembles lamplighters. It introduces a notion of orbit equivalence of pairs that extends beyond ordinary halo products. By linking this stability to the asymptotics of isoperimetric profiles computed earlier, the authors establish that most of these constructions achieve quantitative optimality. A key example states that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p is less than k over k plus ℓ. The work also constructs the required couplings explicitly via Følner tiling sequences.

Core claim

We prove a stability result for orbit equivalence of permutational halo products, going beyond the framework of standard halo products, using a new notion of orbit equivalence of pairs. Combined with our asymptotics of isoperimetric profiles obtained in an earlier article, we prove that most of these constructions are quantitatively optimal. For instance, we show that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p < k/(k+ℓ). We finally build orbit equivalence couplings using the notion of Følner tiling sequences.

What carries the argument

Stability of orbit equivalence of pairs for permutational halo products, controlled by their isoperimetric profiles.

If this is right

  • Most permutational halo products attain quantitative optimality under the new stability result.
  • Shuffler groups over Z to different powers become L^p orbit equivalent exactly below the ratio of their dimensions.
  • Følner tiling sequences yield concrete orbit equivalence couplings for these groups.
  • The isoperimetric profile links quantitative orbit equivalence to large-scale geometry for lamplighter-like constructions.

Where Pith is reading between the lines

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

  • The same stability technique may distinguish finer invariants among other wreath products or amenable groups with similar geometry.
  • Isoperimetric profiles could serve as a systematic tool to measure how much non-quasi-isometric groups differ in their orbit equivalence relations.
  • The method opens a route toward classifying all L^p orbit equivalences within the broader class of halo products.

Load-bearing premise

The isoperimetric profile functions as a reliable quasi-isometry invariant that governs quantitative orbit equivalence for these amenable groups.

What would settle it

An explicit computation or construction showing two permutational halo products that are L^p orbit equivalent for some p larger than the ratio predicted by their isoperimetric profiles.

read the original abstract

We focus on halo products, a class of groups introduced by Genevois and Tessera, and whose geometry mimics lamplighters. Famous examples are lampshufflers. Motivated by their work on the classifications up to quasi-isometry of these groups, we initiate a more quantitative study of their geometry. Indeed, it follows from the work of Delabie, Koivisto, Le Ma\^itre and Tessera that quantitative orbit equivalence between amenable groups is closely related to their large scale geometry, such a connection being justified by the use, in their main results, of a well-known quasi-isometry invariant: the isoperimetric profile. Inspired by their work on quantitative orbit equivalence between lamplighters, we prove a stability result for orbit equivalence of permutational halo products, going beyond the framework of standard halo products, using a new notion of orbit equivalence of pairs. Combined with our asymptotics of isoperimetric profiles obtained in an earlier article, we prove that most of these constructions are quantitatively optimal. For instance, we show that $\mathsf{Shuffler}(\mathbb{Z}^{k+\ell})$ and $\mathsf{Shuffler}(\mathbb{Z}^{k})$ are $\mathrm{L}^p$ orbit equivalent if and only if $p<\frac{k}{k+\ell}$, thus quantifying how much the geometries of these non-quasi-isometric groups differ. We finally build orbit equivalence couplings using the notion of F{\o}lner tiling sequences.

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

2 major / 2 minor

Summary. The paper proves a stability theorem for quantitative orbit equivalence between permutational halo products (extending standard halo products) via a new notion of orbit equivalence of pairs. Combined with isoperimetric profile asymptotics from the authors' prior work, it establishes quantitative optimality results, including that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p < k/(k+ℓ). The manuscript also constructs orbit equivalence couplings using Følner tiling sequences.

Significance. If the central claims hold, the work provides a quantitative refinement of orbit equivalence distinctions for amenable groups with lamplighter-like geometry, offering explicit thresholds that measure geometric differences between non-quasi-isometric examples. The extension to permutational cases and the constructive couplings via Følner sequences strengthen the framework initiated by Delabie-Koivisto-Le Maître-Tessera, with potential implications for classifying such groups up to quantitative invariants.

major comments (2)
  1. [the paragraph combining the stability result with prior asymptotics (near the end of the introduction and in the main OE] The optimality claims, including the iff statement that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent precisely when p < k/(k+ℓ), depend on the isoperimetric asymptotics from the authors' earlier article applying unchanged to the permutational halo products. The manuscript does not explicitly verify or cite the conditions under which those asymptotics remain valid for the generalized (permutational) actions and generators considered here.
  2. [the section defining orbit equivalence of pairs and the stability theorem] The justification that the isoperimetric profile serves as a reliable invariant for quantitative orbit equivalence under the new notion of orbit equivalence of pairs relies on the external reference to Delabie, Koivisto, Le Maître and Tessera; the manuscript should detail the precise applicability conditions and error controls when extending from standard halo products to the permutational setting.
minor comments (2)
  1. [Abstract] The abstract contains LaTeX artifacts (e.g., Le Ma^itre, F{ø}lner) that should be rendered consistently in the main text; ensure all group notations like Shuffler(Z^{k+ℓ}) are defined upon first use.
  2. [the section on Følner tiling sequences] The final construction of orbit equivalence couplings via Følner tiling sequences would benefit from a brief remark on how the quantitative parameters (e.g., L^p norms) are controlled in these explicit couplings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and have revised the manuscript to provide the requested explicit verifications, applicability conditions, and error controls.

read point-by-point responses

  1. Referee: The paragraph combining the stability result with prior asymptotics (near the end of the introduction and in the main OE] The optimality claims, including the iff statement that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent precisely when p < k/(k+ℓ), depend on the isoperimetric asymptotics from the authors' earlier article applying unchanged to the permutational halo products. The manuscript does not explicitly verify or cite the conditions under which those asymptotics remain valid for the generalized (permutational) actions and generators considered here.

    Authors: We agree that the connection to the prior isoperimetric asymptotics would benefit from explicit verification in the present setting. In the revised manuscript we have inserted a dedicated paragraph immediately following the statement of the optimality claims (both in the introduction and in the relevant section of the main text) that recalls the precise hypotheses of the earlier article and verifies that they hold for the permutational halo products under consideration. The verification relies on the fact that the permutational actions are by automorphisms of bounded degree that preserve the same family of Følner sequences used in the earlier work; consequently the asymptotic estimates and error terms carry over verbatim. We also add a citation to the specific theorems of the prior paper that are invoked. revision: yes

  2. Referee: The justification that the isoperimetric profile serves as a reliable invariant for quantitative orbit equivalence under the new notion of orbit equivalence of pairs relies on the external reference to Delabie, Koivisto, Le Maître and Tessera; the manuscript should detail the precise applicability conditions and error controls when extending from standard halo products to the permutational setting.

    Authors: We have expanded the section introducing orbit equivalence of pairs (now containing an additional lemma) to spell out the precise conditions from Delabie–Koivisto–Le Maître–Tessera that are needed for the isoperimetric profile to descend to a quantitative invariant. The new lemma checks that our permutational actions satisfy the required measurability and invariance properties, and it derives explicit distortion bounds on the isoperimetric profile in terms of the parameters of the orbit-equivalence coupling. These bounds are obtained directly from the Følner tiling sequences constructed later in the paper, thereby making the error controls self-contained within the present work rather than relying solely on the external reference. revision: yes

Circularity Check

1 steps flagged

Quantitative optimality relies on self-cited isoperimetric asymptotics from prior work

specific steps
  1. self citation load bearing [Abstract]
    "Combined with our asymptotics of isoperimetric profiles obtained in an earlier article, we prove that most of these constructions are quantitatively optimal. For instance, we show that Shuffler(Z^{k+ℓ}) and Shuffler(Z^k) are L^p orbit equivalent if and only if p < k/(k+ℓ)"

    The iff direction establishing non-equivalence (hence quantitative optimality) for p at or above the threshold is justified solely by invoking the isoperimetric asymptotics from the authors' own prior paper; the present work does not re-derive or verify these asymptotics for the permutational case, making the optimality conclusion dependent on the self-citation.

full rationale

The paper's novel contribution is a stability theorem for orbit equivalence of pairs extending to permutational halo products. The quantitative optimality claims, including the specific L^p threshold iff statement, are obtained by combining this theorem with isoperimetric profile asymptotics computed in the authors' earlier article. This constitutes self-citation load-bearing for the optimality direction but leaves the stability result itself independent and non-circular. No self-definitional reductions, fitted predictions, or ansatz smuggling appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on an external theorem linking quantitative orbit equivalence to isoperimetric profiles and on the authors' own prior asymptotics; it introduces one new conceptual entity (orbit equivalence of pairs) without independent evidence outside the paper.

axioms (1)
  • domain assumption Quantitative orbit equivalence between amenable groups is closely related to their large-scale geometry through the isoperimetric profile (Delabie, Koivisto, Le Maître, Tessera).
    Invoked to justify using isoperimetric profiles as the bridge between geometry and orbit equivalence.
invented entities (1)
  • orbit equivalence of pairs no independent evidence
    purpose: To formulate and prove a stability result that extends beyond standard halo products to permutational versions.
    New notion introduced in the paper; no independent evidence or falsifiable prediction is supplied in the abstract.

pith-pipeline@v0.9.0 · 5565 in / 1524 out tokens · 59556 ms · 2026-05-10T08:42:37.337762+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Isoperimetric profiles of lamplighter-like groups

    [Aus16] Tim Austin. “Integrable measure equivalence for groups of polynomial growth”. In:Groups, Geometry and Dynamics10.1 (2016), pp. 117–154.doi:10.4171/ GGD/345. [BFS13] Uri Bader, Alex Furman, and Roman Sauer. “Integrable measure equivalence and rigidity of hyperbolic lattices”. In:Inventiones mathematicae194.2 (2013), pp. 313–379.doi:10.1007/s00222-0...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00222-012-0445-9 2016

  2. [2]

    Isopérimétrie pour les groupes et les variétés

    [CSC93] Thierry Coulhon and Laurent Saloff-Coste. “Isopérimétrie pour les groupes et les variétés”. In:Revista Matemática Iberoamericana9.2 (1993), pp. 293–314. [DLIT25] Thiebout Delabie, Claudio Llosa Isenrich, and Romain Tessera.L p measure equivalence of nilpotent groups. 2025.doi:10.48550/ARXIV.2505.17865. [DKLMT22] Thiebout Delabie, Juhani Koivisto, ...

    work page doi:10.48550/arxiv.2505.17865 1993

  3. [3]

    On isoperimetric profiles of finitely generated groups

    [Ers03] Anna Erschler. “On isoperimetric profiles of finitely generated groups”. In:Ge- ometriae Dedicata100.1 (2003), pp. 157–171.doi:10.1023/A:1025849602376. [EZ21] Anna Erschler and Tianyi Zheng. “Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s propertyHF D”. In:Annales de l’Institut Fourier 70.4 (2021), pp. 1363–1402.doi:10....

    work page doi:10.1023/a:1025849602376 2003

  4. [4]

    Transitivity degrees of countable groups and acylindrical hyperbolicity

    London Math. Soc. Lecture Note Ser. Cambridge: Cam- bridge Univ. Press, 1993, pp. 1–295. [HO16] Michael Hull and Denis Osin. “Transitivity degrees of countable groups and acylindrical hyperbolicity”. In:Israel Journal of Mathematics216.1 (2016), pp. 307–353.doi:10.1007/s11856-016-1411-9. [OW80] Donald S. Ornstein and Benjamin Weiss. “Ergodic theory of ame...

    work page doi:10.1007/s11856-016-1411-9 1993