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arxiv: 2606.08279 · v1 · pith:5MDVI3KUnew · submitted 2026-06-06 · 🧮 math.CO · math.DS· math.GR· math.PR

Tracks on planar complexes and soficity

Hiroki Ishikura This is my paper

Pith reviewed 2026-06-27 19:16 UTC · model grok-4.3

classification 🧮 math.CO math.DSmath.GRmath.PR
keywords soficityBorel graphsplanar graphsequivalence relationsunimodular random graphssimplicial complexestracks
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The pith

Every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic.

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

The paper shows that probability-measure-preserving equivalence relations coming from locally finite Borel graphs with planar connected components are sofic. This includes the case of unimodular random planar graphs. The argument relies on approximating the associated Borel simplicial complexes using treeable covering spaces. These coverings are obtained from a canonical family of tracks defined on planar simplicial complexes. The result strengthens earlier findings by removing extra assumptions needed in previous work on planar maps.

Core claim

We show that every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic in the sense of Elek--Lippner. In particular, every unimodular random planar graph is sofic. To prove this, we investigate Borel simplicial complexes with planar components and approximate them by treeable covering spaces. To construct these coverings, we use a canonical family of tracks on planar simplicial complexes introduced by Dunwoody.

What carries the argument

Canonical family of tracks on planar simplicial complexes used to construct approximating treeable covering spaces.

If this is right

  • Unimodular random planar graphs are sofic.
  • Soficity holds for equivalence relations from planar-component Borel graphs without additional assumptions.
  • Borel simplicial complexes with planar components can be approximated by treeable coverings.

Where Pith is reading between the lines

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

  • The track-based approximation technique might generalize to graphs embeddable on surfaces of higher genus if similar track families exist.
  • Planarity here acts as a condition that permits a tree-like approximation in the measure theoretic setting.

Load-bearing premise

Borel simplicial complexes with planar components admit approximation by treeable covering spaces constructed from the canonical tracks.

What would settle it

A specific locally finite Borel graph with planar components whose generated equivalence relation fails to be sofic would serve as a counterexample.

read the original abstract

We show that every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic in the sense of Elek--Lippner. In particular, every unimodular random planar graph is sofic. This removes the additional assumptions in the works of Angel--Hutchcroft--Nachmias--Ray and Tim\'{a}r on the soficity of unimodular random planar maps and graphs. To prove this, we investigate Borel simplicial complexes with planar components and approximate them by treeable covering spaces. To construct these coverings, we use a canonical family of tracks on planar simplicial complexes introduced by Dunwoody.

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

1 major / 0 minor

Summary. The manuscript claims to prove that every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic in the sense of Elek--Lippner. In particular, every unimodular random planar graph is sofic. The proof investigates Borel simplicial complexes with planar components and approximates them by treeable covering spaces constructed from the canonical family of tracks on planar simplicial complexes introduced by Dunwoody. This removes additional assumptions from prior works on soficity of unimodular random planar maps and graphs.

Significance. If the central claim holds, the result is significant: it establishes soficity for all such planar pmp equivalence relations and unimodular random planar graphs without extra assumptions, using a combinatorial track construction to produce the required treeable Borel coverings. No machine-checked proofs or reproducible code are mentioned.

major comments (1)
  1. [construction of the treeable coverings (in the proof of the main result)] The reduction of Elek--Lippner soficity to the existence of treeable Borel coverings obtained from Dunwoody's tracks requires that the track selection (or covering maps) be Borel measurable on the standard probability space while preserving planarity and local finiteness of components. The combinatorial definition on countable planar complexes does not automatically yield a Borel object; the manuscript must supply an explicit measurable-selection or uniformization argument for this step, as it is load-bearing for the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying a key point that requires clarification in the proof. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [construction of the treeable coverings (in the proof of the main result)] The reduction of Elek--Lippner soficity to the existence of treeable Borel coverings obtained from Dunwoody's tracks requires that the track selection (or covering maps) be Borel measurable on the standard probability space while preserving planarity and local finiteness of components. The combinatorial definition on countable planar complexes does not automatically yield a Borel object; the manuscript must supply an explicit measurable-selection or uniformization argument for this step, as it is load-bearing for the main theorem.

    Authors: We agree that the manuscript must supply an explicit argument establishing Borel measurability of the track selection and the induced covering maps. While the input simplicial complexes are Borel by hypothesis and Dunwoody's tracks are defined combinatorially on each component, the global selection across the probability space requires a uniformization step to ensure the resulting objects remain Borel. In the revised manuscript we will add a dedicated subsection (immediately preceding the proof of the main theorem) that invokes the Jankov-von Neumann uniformization theorem on the analytic set of admissible track systems; the selection preserves planarity and local finiteness because these properties are preserved under the canonical Dunwoody construction on each finite planar complex. This makes the reduction to Elek--Lippner soficity fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; central approximation uses independent Dunwoody construction

full rationale

The derivation approximates Borel simplicial complexes with planar components by treeable coverings constructed from Dunwoody's canonical tracks. This step invokes an external combinatorial result from prior literature (Dunwoody) rather than any self-definition, fitted parameter renamed as prediction, or self-citation chain. No equation or claim reduces the soficity statement to its own inputs by construction. The cited tracks are treated as given external input, making the overall argument self-contained against that benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of Dunwoody's tracks to Borel planar complexes and on the Elek-Lippner definition of soficity; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The canonical family of tracks on planar simplicial complexes introduced by Dunwoody yields treeable covering spaces that approximate the given Borel complexes
    Invoked to construct the approximations used in the proof.

pith-pipeline@v0.9.1-grok · 5633 in / 1068 out tokens · 29697 ms · 2026-06-27T19:16:40.606120+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 8 canonical work pages · 3 internal anchors

  1. [1]

    Aldous and R

    D. Aldous and R. Lyons. Processes on Unimodular Random Networks. Electron. J. Probab. 12 (2007), no. 54, 1454--1508

    2007

  2. [2]

    Angel, T

    O. Angel, T. Hutchcroft, A. Nachmias and G. Ray. Hyperbolic and parabolic unimodular random maps. Geom. Funct. Anal. 28 (2018), no. 4, 879--942

    2018

  3. [3]

    Benjamini, R

    I. Benjamini, R. Lyons and O. Schramm. Unimodular random trees. Ergodic Theory Dynam. Systems 35 (2015), no. 2, 359--373

    2015

  4. [4]

    L. Bowen. Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102 (2003), 213--236

    2003

  5. [5]

    Bowen, M

    L. Bowen, M. Chapman, A. Lubotzky and T. Vidick. The Aldous--Lyons Conjecture I: Subgroup Tests. arXiv:2408.00110

    work page arXiv

  6. [6]

    Bowen, M

    L. Bowen, M. Chapman and T. Vidick. The Aldous--Lyons Conjecture II: Undecidability. arXiv:2501.00173

    work page arXiv

  7. [7]

    Bowen and R

    L. Bowen and R. Tucker-Drob. Superrigidity, measure equivalence, and weak Pinsker entropy, Groups Geom. Dyn. 16 (2022), no. 1, 247--286

    2022

  8. [8]

    Carderi, D

    A. Carderi, D. Gaboriau and M. de la Salle. Non-standard limits of graphs and some orbit equivalence invariants. Ann. H. Lebesgue 4 (2021), 1235--1293

    2021

  9. [9]

    Carmesin, M

    J. Carmesin, M. Hamann and B. Miraftab. Canonical trees of tree decompositions. J. Combin. Theory Ser. B 152 (2022), 1--26

    2022

  10. [10]

    R. Chen. On the canonical Dicks-Dunwoody structure tree. https://rynchn.github.io/math/dicks-dunwoody.pdf

  11. [11]

    R. Chen, A. Poulin, R. Tao and A. Tserunyan, Tree-like graphings, wallings, and median graphings of equivalence relations , Forum Math. Sigma 13 (2025), e64

    2025

  12. [12]

    Conley, D

    C.T. Conley, D. Gaboriau, A.S. Marks and R.D. Tucker-Drob. One-ended spanning subforests and treeability of groups. arXiv:2104.07431

    work page arXiv

  13. [13]

    Dicks and M.J

    W. Dicks and M.J. Dunwoody. Groups Acting on Graphs. Cambridge Stud. Adv. Math., 17, Cambridge University Press, Cambridge, 1989

    1989

  14. [14]

    Planar graphs and covers

    M.J. Dunwoody. Planar graphs and covers. arXiv:0708.0920

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Dunwoody

    M.J. Dunwoody. The accessibility of finitely presented groups. Invent. Math. 81 (1985), no. 3, 449--457

    1985

  16. [16]

    Orbit equivalence and sofic approximation

    K. Dykema, D. Kerr and M. Pichot. Orbit equivalence and sofic approximation. arXiv:1102.2556

    work page internal anchor Pith review Pith/arXiv arXiv

  17. [17]

    G. Elek. On the limit of large girth graph sequences. Combinatorica 30 (2010), no. 5, 553--563

    2010

  18. [18]

    Elek and G

    G. Elek and G. Lippner. Sofic equivalence relations. J. Funct. Anal. 258 (2010), no. 5, 1692--1708

    2010

  19. [19]

    Gaboriau

    D. Gaboriau. Examples of groups that are measure equivalent to the free group. Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1809--1827

    2005

  20. [20]

    Gromov, Endomorphisms of symbolic algebraic varieties

    M. Gromov, Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1 (1999), no. 2, 109--197

    1999

  21. [21]

    Jardón-Sánchez

    H. Jardón-Sánchez. Applications of tree decompositions and accessibility to treeability of Borel graphs. arXiv:2308.13087

    work page arXiv

  22. [22]

    Accessibility, planar graphs, and quasi-isometries

    J.P. MacManus. Accessibility, planar graphs, and quasi-isometries. arXiv:2310.15242

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    Tim\' a r

    \' A . Tim\' a r. Unimodular random one-ended planar graphs are sofic. Combin. Probab. Comput. 32 (2023), no. 6, 851--858

    2023

  24. [24]

    O. Solé-Pi. Minor-excluded graphs and soficity. arXiv:2508.06731

    work page arXiv

  25. [25]

    B. Weiss. Sofic groups and dynamical systems (Ergodic theory and harmonic analysis, Mumbai, 1999) Sankhyā Ser. A. 62 (2000), no. 3, 350--359

    1999