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arxiv: 2605.18433 · v1 · pith:QR3XKLNUnew · submitted 2026-05-18 · 🧮 math.DS · math.GR· math.OA

Coamenability and strong ergodicity

Ben Hayes This is my paper

Pith reviewed 2026-05-19 23:36 UTC · model grok-4.3

classification 🧮 math.DS math.GRmath.OA
keywords coamenabilitystrong ergodicityergodic relationsprobability measure-preservinggroup actionsdynamical systemsergodic components
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The pith

For coamenable inclusions of ergodic probability measure-preserving relations, strong ergodicity holds for one exactly when it holds for the other.

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

The paper establishes that strong ergodicity transfers in both directions across a coamenable inclusion of two ergodic relations that each preserve a probability measure. If the larger relation is strongly ergodic then so is the smaller one, and the converse also holds under these conditions. This equivalence matters because strong ergodicity captures a rigid form of mixing that is hard to verify directly, and the transfer lets one check the property on whichever side of the inclusion is easier to analyze. The result further shows that when the larger relation is strongly ergodic, the smaller relation decomposes into countably many strongly ergodic pieces even if it is not itself ergodic. A direct consequence is that coamenable subgroup actions inherit this decomposition and strong ergodicity from the larger group action.

Core claim

For a coamenable inclusion S ≤ R of ergodic probability measure-preserving relations, R is strongly ergodic if and only if S is strongly ergodic. When the inclusion is coamenable and R is strongly ergodic, the relation S has at most countably many ergodic components, each of which is strongly ergodic.

What carries the argument

Coamenable inclusion of relations, which carries the equivalence of strong ergodicity between the sub-relation and the larger relation.

If this is right

  • A strongly ergodic action of a group Gamma on a probability space restricts to countably many strongly ergodic ergodic components under any coamenable subgroup Lambda.
  • Strong ergodicity of the larger relation forces the smaller relation to have only countably many ergodic pieces when the inclusion is coamenable.
  • Verification of strong ergodicity can be moved from one relation to the other whenever the inclusion satisfies the coamenability condition.

Where Pith is reading between the lines

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

  • The transfer property may reduce questions about strong ergodicity for complicated relations to simpler coamenable sub-relations that are easier to check directly.
  • Similar equivalences could be explored for other rigidity properties such as spectral gap or property (T) under the same coamenability assumption.
  • Concrete group examples with known coamenable subgroups, such as certain lattices or amenable extensions, provide natural test cases for the decomposition into strongly ergodic components.

Load-bearing premise

The inclusion of the smaller relation into the larger one must be coamenable.

What would settle it

An explicit pair of ergodic probability measure-preserving relations with a non-coamenable inclusion where one side is strongly ergodic and the other is not.

read the original abstract

Following methods of Bannon-Marrakchi-Ozawa, we show that for coamenable inclusion $\mathcal{S}\leq \mathcal{R}$ of ergodic, probability measure-preserving relations, we have that $\mathcal{R}$ is strongly ergodic if and only if $\mathcal{S}$ is strongly ergodic. More general results are given when $\mathcal{S}\leq \mathcal{R}$ is coamenable, $\mathcal{R}$ is strongly ergodic, but we do not assume ergodicity of $\mathcal{S}$. As a consequence, if $\Lambda\leq \Gamma$ is a coamenable inclusion of groups, then any strongly ergodic $\Gamma$ action has countably many ergodic components for the $\Lambda$ action, each of which is strongly ergodic.

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 / 3 minor

Summary. The paper claims that for a coamenable inclusion S ≤ R of ergodic probability measure-preserving equivalence relations, R is strongly ergodic if and only if S is. It also establishes more general results when R is strongly ergodic but S need not be ergodic, by decomposing into ergodic components of S and transferring the property via coamenability. As a corollary, for a coamenable inclusion Λ ≤ Γ of groups, any strongly ergodic Γ-action has countably many ergodic components under the Λ-action, each of which is strongly ergodic.

Significance. If the result holds, it provides a useful transfer principle for strong ergodicity across coamenable inclusions of equivalence relations by adapting Bannon-Marrakchi-Ozawa techniques. The group-action corollary strengthens connections between relation theory and group actions, offering a tool for analyzing ergodic decompositions in rigidity and classification problems in dynamical systems.

minor comments (3)
  1. The abstract states the main equivalence clearly but does not indicate the measure space or the precise definition of coamenability used; adding one sentence on these would improve accessibility.
  2. In the general (non-ergodic S) case, the decomposition into ergodic components is mentioned but the notation for the components could be introduced earlier to clarify the transfer argument.
  3. The group-action corollary is stated concisely; a brief remark on how the relation inclusion is induced by the group inclusion would help readers unfamiliar with the correspondence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper, recognition of its significance, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external methods

full rationale

The paper states it follows methods of Bannon-Marrakchi-Ozawa to prove that for coamenable inclusions S ≤ R of ergodic pmp relations, R is strongly ergodic iff S is. The more general case decomposes S into ergodic components and transfers via coamenability, with a group-action corollary. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear; the argument relies on cited external techniques and stated assumptions without reducing the central equivalence to its own inputs by construction. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background in ergodic theory and operator algebras plus the coamenability assumption and the methods of the cited paper; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Coamenability of the inclusion S ≤ R
    Invoked as the key hypothesis enabling the equivalence of strong ergodicity.
  • domain assumption Relations are ergodic and probability measure-preserving
    Stated in the main claim; standard setup in the field.

pith-pipeline@v0.9.0 · 5648 in / 1270 out tokens · 27163 ms · 2026-05-19T23:36:47.982702+00:00 · methodology

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Reference graph

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