Obstructions for quantitative measure equivalence between locally compact groups

Corentin Correia; Juan Paucar

arxiv: 2510.25321 · v2 · submitted 2025-10-29 · 🧮 math.GR · math.DS

Obstructions for quantitative measure equivalence between locally compact groups

Corentin Correia , Juan Paucar This is my paper

Pith reviewed 2026-05-18 03:42 UTC · model grok-4.3

classification 🧮 math.GR math.DS

keywords measure equivalencelocally compact groupscocycle integrabilityunimodular groupscompactly generated groupsquantitative obstructions

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The pith

Unimodular compactly generated locally compact groups have explicit upper bounds on cocycle integrability in measure equivalence, and the threshold cannot be achieved.

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

The paper extends prior bounds on the integrability of cocycles arising from measure equivalence couplings, which had been established for finitely generated groups, to the setting of unimodular compactly generated locally compact groups. It supplies explicit upper bounds that any such coupling must obey. The work further shows that the specific integrability threshold appearing in these bounds is never attained. A reader would care because these limits give concrete quantitative obstructions to how closely two groups can resemble each other under measure equivalence.

Core claim

In any measure equivalence coupling between two unimodular compactly generated locally compact groups, the associated cocycles satisfy explicit upper bounds on integrability, and the integrability threshold described in the statements cannot be achieved.

What carries the argument

The cocycles associated to a measure equivalence coupling and the quantitative integrability bounds they must obey.

Load-bearing premise

The groups are unimodular and compactly generated, and the measure equivalence coupling exists with the usual measurability and invariance properties.

What would settle it

An explicit construction of a measure equivalence coupling between two unimodular compactly generated locally compact groups whose cocycle exceeds the stated integrability upper bound or attains the threshold.

read the original abstract

Given a measure equivalence coupling between two finitely generated groups, Delabie, Koivisto, Le Ma\^itre and Tessera have found explicit upper bounds on how integrable the associated cocycles can be. We extend these results to the broader framework of unimodular compactly generated locally compact groups. We also generalize a result by the first-named author, showing that the integrability threshold described in these statements cannot be achieved.

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 extends explicit integrability bounds on cocycles from measure equivalence in finitely generated discrete groups to unimodular compactly generated locally compact groups and shows the threshold cannot be attained.

read the letter

The core point is that the explicit upper bounds on the integrability of cocycles associated to measure equivalence couplings, which were established for finitely generated discrete groups, extend to the setting of unimodular compactly generated locally compact groups. In addition, the paper shows that this integrability threshold cannot be attained, extending a non-attainability result from the first author. What stands out is the adaptation to locally compact groups. The prior work handled discrete cases, and here the authors manage to carry the bounds over under the natural assumptions of unimodularity and compact generation. This is not just a trivial rephrasing; it requires checking that the relevant cocycle properties and measure equivalence conditions behave similarly in the continuous case. The non-attainability part builds on the author's previous result and applies it more broadly. The paper does a good job keeping the statements clear and tying back to the discrete literature. The assumptions listed match what is needed to make the extension plausible, and there is no sign of circularity or post-hoc adjustments in the claims. On the softer side, the extension might involve some technical adjustments for handling the topology and Haar measure in the locally compact setting, and it would be worth confirming that no new error terms or approximation issues arise there. Since the abstract is high-level, the full text presumably fills in those details without gaps. The overall argument seems to hold based on the stress test and the direct build from cited results. Readers in ergodic theory and measured group theory will get the most out of this, particularly those looking for quantitative refinements beyond the discrete world. It is not a foundational breakthrough but a solid incremental step that clarifies the scope of the earlier bounds. Because it provides verifiable extensions and a clear generalization, it merits a serious referee who can check the technical adaptations. I recommend putting it through peer review rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper extends explicit upper bounds on the integrability of cocycles arising from measure equivalence couplings, previously obtained by Delabie, Koivisto, Le Maître and Tessera for finitely generated groups, to the setting of unimodular compactly generated locally compact groups. It further generalizes a result of the first-named author to show that the integrability threshold appearing in these bounds cannot be attained.

Significance. If the central claims hold, the work supplies a natural extension of quantitative measure equivalence obstructions to a wider class of groups, together with a sharp non-attainability statement. This strengthens the toolkit for studying rigidity and classification questions in geometric group theory and ergodic theory beyond the discrete case, and the generalization of the non-achievement result is a clear strengthening of prior work.

minor comments (3)

§1, paragraph 3: the statement of the main theorem would benefit from an explicit reference to the precise integrability exponent that is shown to be unattainable, rather than referring only to 'the threshold described in these statements'.
Definition 2.3: the measurability and invariance conditions on the coupling are stated in a form that assumes familiarity with the discrete case; a short reminder of how these reduce when the groups are discrete would improve readability.
The bibliography is missing the full citation details for the Delabie–Koivisto–Le Maître–Tessera paper that is repeatedly invoked; please supply the arXiv number or journal reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that it extends quantitative measure equivalence results to unimodular compactly generated locally compact groups and strengthens the non-attainability statement. We appreciate the recommendation for minor revision and will prepare an updated version accordingly.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends integrability bounds on cocycles from measure equivalence couplings, as established by Delabie et al. for finitely generated discrete groups, to the setting of unimodular compactly generated locally compact groups, while also generalizing a prior result by the first-named author on the non-attainability of the integrability threshold. The stated assumptions (unimodularity, compact generation, and standard measurability/invariance of the coupling) are carried over from the cited prior work without redefinition or refitting. No equations or steps in the abstract or described claims reduce any new prediction or bound to a fitted input, self-definition, or unverified self-citation chain; the extension supplies independent content for the locally compact case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results in measured group theory and ergodic theory for locally compact groups; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Measure equivalence couplings between unimodular compactly generated locally compact groups admit cocycles whose integrability can be bounded using adaptations of techniques for discrete groups.
Invoked when extending the Delabie-Koivisto-Le Maître-Tessera bounds.

pith-pipeline@v0.9.0 · 5585 in / 1217 out tokens · 34716 ms · 2026-05-18T03:42:46.690186+00:00 · methodology

Obstructions for quantitative measure equivalence between locally compact groups

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)