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arxiv: 2410.16480 · v2 · submitted 2024-10-21 · 🧮 math.DS · math.FA· math.GR· math.OA· math.PR

Coamenability and cospectral radius for orbit equivalence relations

Ben Hayes This is my paper

Pith reviewed 2026-05-23 19:33 UTC · model grok-4.3

classification 🧮 math.DS math.FAmath.GRmath.OAmath.PR
keywords coamenabilitycospectral radiusorbit equivalence relationsinclusions of relationsrandom walksmeasure-preserving dynamics
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The pith

The cospectral radius of random walks on R-classes connects to coamenability of the inclusion S ≤ R.

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

The paper examines inclusions of discrete probability measure-preserving orbit equivalence relations S ≤ R. It links the cospectral radius of a random walk on the classes of R to whether the inclusion is coamenable. The work also gives several equivalent formulations for the notion of coamenability in this setting. These results build directly on the prior almost-sure existence of the cospectral radius.

Core claim

For inclusions S ≤ R of discrete pmp orbit equivalence relations, the cospectral radius of a random walk on the R-classes is connected to the coamenability of the inclusion, and coamenability itself admits several equivalent formulations.

What carries the argument

Coamenability of the inclusion S ≤ R, together with the cospectral radius of the random walk on the R-classes.

If this is right

  • Coamenability of inclusions can be checked through any of the equivalent formulations developed in the paper.
  • The cospectral radius provides one concrete way to detect coamenability for such inclusions.
  • The almost-sure existence result from prior work extends to the coamenability setting.
  • Equivalent definitions allow different techniques for studying when one relation is coamenable inside another.

Where Pith is reading between the lines

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

  • The connection may yield computable invariants for distinguishing non-coamenable inclusions in concrete dynamical systems.
  • The equivalent formulations could be adapted to study coamenability in related structures such as group actions or graphs.
  • If the cospectral radius equals a specific threshold, it might imply a form of relative amenability that interacts with other invariants like cost.

Load-bearing premise

The orbit equivalence relations are discrete and probability measure-preserving.

What would settle it

An explicit inclusion S ≤ R where the cospectral radius fails to satisfy any of the listed equivalent conditions for coamenability.

read the original abstract

We consider inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving orbit equivalence relations. In previous work with Ab\'{e}rt-Fra\c{c}zyk, we established the pointwise almost sure existence of the cospectral radius of a random walk on the $\mathcal{R}$-classes. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion $\mathcal{S}\leq \mathcal{R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.

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 / 2 minor

Summary. The paper considers inclusions S ≤ R of discrete probability measure-preserving orbit equivalence relations. Building on prior joint work establishing the pointwise almost sure existence of the cospectral radius for a random walk on the R-classes, it examines the relationship between this cospectral radius and the coamenability of the inclusion S ≤ R. The manuscript also conducts a systematic study of coamenability for inclusions of relations and establishes several equivalent formulations of the notion.

Significance. If the claimed equivalences and connections hold, the work would link spectral invariants of random walks on equivalence relations to coamenability, offering new tools for studying amenability phenomena in ergodic theory and von Neumann algebras. The provision of multiple equivalent characterizations strengthens the conceptual framework and could facilitate applications to orbit equivalence and group actions.

minor comments (2)
  1. [Abstract] The abstract references prior work with Abért-Frańczyk on the cospectral radius but does not specify the citation; the full manuscript should include a complete reference list and a brief recall of the relevant existence theorem to aid readers unfamiliar with the prior result.
  2. Notation for the inclusions S ≤ R and the random walk is introduced without an explicit reminder of the standing assumptions (discrete, pmp); a short preliminary section restating these hypotheses would improve accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments appear in the report, so we provide no point-by-point responses below. We remain available to address any questions or concerns the referee may have.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript cites prior joint work solely for the pointwise a.s. existence of the cospectral radius under the stated discrete pmp hypotheses; the new contributions are the investigation of its relation to coamenability and the independent derivation of several equivalent formulations of coamenability for inclusions of relations. No equation or central claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain. The cited existence result is an external theorem whose hypotheses match the current setting but whose proof is not reproduced or presupposed here.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or nonstandard axioms are visible.

axioms (1)
  • domain assumption Orbit equivalence relations arising from discrete pmp actions admit a well-defined cospectral radius for random walks.
    Invoked via reference to previous work with Abért-Frańczyk.

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Reference graph

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