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arxiv: 2606.04696 · v1 · pith:FKPBQCECnew · submitted 2026-06-03 · 🧮 math.DS · math.GR

The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups

Yair Hartman , Nadav Kalma This is my paper

Pith reviewed 2026-06-28 03:56 UTC · model grok-4.3

classification 🧮 math.DS math.GR
keywords stationary actionsNo-Core Principlestationary random subgroupsSchreier graphsendsBorel setsgroup actions
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The pith

For stationary actions, any positive-measure Borel set meeting almost every orbit in finitely many points is supported on the finite-orbit part.

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

The paper proves that stationary actions of countable groups obey a No-Core Principle. Specifically, a Borel set with positive measure that intersects almost every orbit finitely often must be supported, up to null sets, on the part of the space where orbits are finite. This extends a known regularity property from measure-preserving actions. The principle implies that the Schreier graph of a stationary random subgroup of a finitely generated group has 0, 1, 2, or infinitely many ends with probability one.

Core claim

If a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. For stationary random subgroups of finitely generated groups, this yields that their Schreier graphs have almost surely 0, 1, 2, or infinitely many ends. Boomerang subgroups, however, can be constructed in the free group on three generators with Schreier graphs having exactly any prescribed number of ends k ≥ 3.

What carries the argument

The No-Core Principle for stationary actions, which guarantees that finite intersections with orbits imply support on finite orbits for positive measure sets.

If this is right

  • The Schreier graphs of stationary random subgroups have 0, 1, 2 or infinitely many ends almost surely.
  • The probabilistic geometry of stationary random subgroups is more restricted than the topological geometry of Boomerang subgroups.
  • Boomerang subgroups of the free group on three generators exist with Schreier graphs having exactly k ends for every k ≥ 3 including infinity.

Where Pith is reading between the lines

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

  • The result may suggest similar regularity phenomena in other non-invariant measures on group actions.
  • Extensions could apply the principle to actions of uncountable groups or different types of measures.
  • The distinction between stationary and measure-preserving cases highlights how stationarity provides enough structure for this conclusion.

Load-bearing premise

The action is assumed to be stationary.

What would settle it

Finding a stationary action of a countable group and a positive measure Borel set that intersects almost every orbit in finitely many points but has positive measure on the infinite-orbit part would falsify the principle.

Figures

Figures reproduced from arXiv: 2606.04696 by Nadav Kalma, Yair Hartman.

Figure 1
Figure 1. Figure 1: If d(H, Hg0) > 2n, the balls Bn(H) and Bn(Hg0) are disjoint. Re￾moving both leaves at least (k − 1) + (k − 1) = 2k − 2 infinite components, contradicting the assumption that the graph has exactly k ends. Proof. Let E = {H ∈ SubΓ : 2 < |Ends(H)| < ∞}. Assume toward contradiction that ν(E) > 0. For k ≥ 3, set Ek = {H ∈ SubΓ : |Ends(H)| = k}. Since E = [∞ k=3 Ek, there exists k0 ≥ 3 such that ν(Ek0 ) > 0. For… view at source ↗
Figure 2
Figure 2. Figure 2: The rewiring construction of G ′ . The original x1-edges originating at the identity vertices (dashed gray) are removed. Instead, new x1-edges (solid red) are added to connect e (j) to x (j+1) 1 in the adjacent copy, linking the k components. Re-rooting the Schreier graph at the endpoint reached by the word g −pn corresponds to the subgroup g pn∆g −pn . Consider the path in G ′ starting at e 0 and labeled … view at source ↗
read the original abstract

We prove a No-Core Principle for stationary actions of countable groups. Namely, if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends to stationary actions a basic regularity phenomenon known for measure-preserving actions. We apply this principle to the geometry of Stationary Random Subgroups. For a finitely generated group, we prove that the Schreier graph of a stationary random subgroup has almost surely 0,1,2, or infinitely many ends. Finally, we contrast this probabilistic regularity with the topological notion of Boomerang subgroups: for every $k\geq 3$, including $k=\aleph_0$, we construct a Boomerang subgroup of $\mathbb{F}_3$ whose Schreier graph has exactly $k$ ends.

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 manuscript proves a No-Core Principle for stationary actions of countable groups: if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends a standard regularity fact from invariant measures. The principle is applied to stationary random subgroups (SRS) of finitely generated groups, showing that the associated Schreier graphs have almost surely 0, 1, 2, or infinitely many ends. The paper also constructs, for every k ≥ 3 (including k = ℵ₀), a Boomerang subgroup of F₃ whose Schreier graph has exactly k ends.

Significance. If the No-Core Principle holds, it supplies a useful regularity tool for stationary (as opposed to invariant) actions in ergodic theory and geometric group theory. The SRS application yields a clean probabilistic statement on the number of ends, while the Boomerang constructions demonstrate that arbitrary finite or countably infinite numbers of ends are realizable topologically. The manuscript ships a new principle together with two distinct applications, one probabilistic and one constructive, that contrast the two regimes.

minor comments (2)
  1. [Abstract] The abstract introduces the abbreviation SRS without spelling it out on first use, though the term is standard in the literature.
  2. [Abstract] The statement of the No-Core Principle in the abstract does not explicitly record the underlying probability space or the precise null-set convention; a parenthetical clarification would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the No-Core Principle and its applications, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a theorem extending a known regularity fact from invariant to stationary measures on countable-group actions, then applies it to ends of Schreier graphs and Boomerang constructions. No equations, definitions, or self-citations in the abstract or stated claims reduce the No-Core Principle or its consequences to fitted parameters, self-referential constructions, or load-bearing prior results by the same authors. The central claim is presented as a direct proof extending an external standard fact, with applications following independently; this matches the default expectation of a non-circular paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of descriptive set theory, Borel measurability, and the definition of stationarity for group actions; no free parameters, ad-hoc constants, or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of ZFC set theory together with the usual notions of Borel sigma-algebra and probability measures on Polish spaces.
    Invoked implicitly when speaking of Borel sets, positive measure, and almost-everywhere statements.
  • domain assumption The definition of a stationary action (existence of a stationary probability measure on the space).
    Central to the statement of the No-Core Principle.

pith-pipeline@v0.9.1-grok · 5675 in / 1425 out tokens · 45801 ms · 2026-06-28T03:56:49.328010+00:00 · methodology

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Reference graph

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