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arxiv: 2310.15242 · v6 · pith:ETMKPTA7new · submitted 2023-10-23 · 🧮 math.GR · math.CO· math.MG

Accessibility, planar graphs, and quasi-isometries

Joseph Paul MacManus This is my paper

Pith reviewed 2026-05-24 07:05 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.MG
keywords accessibilityquasi-isometryplanar graphsquasi-transitive graphsfinitely generated groupsfree productssurface groupsCayley graphs
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The pith

A connected locally finite quasi-transitive graph quasi-isometric to a planar graph must be accessible, so finitely generated groups with this property are virtually free products of free and surface groups.

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

The paper shows that any connected, locally finite, quasi-transitive graph quasi-isometric to a planar graph is accessible. Accessibility means the graph decomposes along finite cuts into one-ended or finite pieces in a controlled manner. Because quasi-isometries preserve this property under quasi-transitivity, the result classifies all finitely generated groups quasi-isometric to planar graphs: each such group is virtually a free product of free groups and surface groups. These groups therefore virtually admit planar Cayley graphs. A reader cares because the statement gives an exact large-scale geometric characterization of which groups resemble planar graphs.

Core claim

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such a group is virtually a free product of free and surface groups, and thus virtually admits a planar Cayley graph.

What carries the argument

Quasi-isometry from a quasi-transitive locally finite graph to a planar graph, which forces accessibility via preservation of finite cuts and ends.

If this is right

  • The graph decomposes along finite vertex cuts into accessible one-ended components.
  • Any finitely generated group quasi-isometric to a planar graph is virtually a free product of free groups and surface groups.
  • Such a group virtually possesses a planar Cayley graph.
  • The classification is complete for all finitely generated groups with this quasi-isometry property.

Where Pith is reading between the lines

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

  • Dropping quasi-transitivity may allow inaccessible graphs that are still quasi-isometric to planar graphs.
  • The result suggests checking whether quasi-isometries to graphs on higher-genus surfaces yield analogous classifications.
  • Accessibility here may interact with the number of ends of the group in ways not fully spelled out.

Load-bearing premise

The graph must be quasi-transitive so that its automorphisms make the accessibility property translate into a group-theoretic classification.

What would settle it

A connected locally finite quasi-transitive graph that is quasi-isometric to a planar graph yet fails to be accessible would disprove the claim.

Figures

Figures reproduced from arXiv: 2310.15242 by Joseph Paul MacManus.

Figure 1
Figure 1. Figure 1: One-ended planar graph which is not quasi-isometric to a complete Riemannian plane. It is not immediately clear whether or not this graph is quasi-isometric to a quasi-transitive graph. We now look towards the headline result, Theorem A. The following discussion will be mainly centered around finitely generated groups, but everything is equally applicable to quasi-transitive graphs. It will be instructive … view at source ↗
Figure 2
Figure 2. Figure 2: Constructing a 2-connected planar super-graph. The marked vertices are depicted in red. Let Π = S r Πr. Let Γ ′ denote the subdivision of Γ, where each edge is divided into three edges. Clearly Γ is homeomorphic to Γ ′ , and the natural map Γ → Γ ′ is a (3, 0)-quasi-isometry. We have a natural inclusion Γ ′ ,→ Π which is an isometry onto its image (indeed, the new paths added to Π) create no new shortcuts.… view at source ↗
Figure 3
Figure 3. Figure 3: The ray α2 is forced to diverge from f due to the Jordan curve theo￾rem. Lemma 2.5. There exists some uniform constant r ≥ 0 such that every finite face cycle of Γ has length at most n. Proof. If Γ has no infinite face then choose x ∈ Γ arbitrarily. Otherwise, apply Lemma 2.4 and choose x ∈ Γ such that x lies at least M = 1000λ 1000Br from this infinite face, where r is the constant given in Lemma 2.2 [PI… view at source ↗
Figure 4
Figure 4. Figure 4: Our choice of the paths βi traces out a Jordan curve separating x from infinity. Suppose now we translate this figure somewhere using our quasi-action. Fix g ∈ G. To ease notation, let us denote the quasi-isometry φg with the following shorthand: φg(x) = x ′ . We have that α ′ i ∩ α ′ j = ∅ for all i ̸= j, and also that (1) β ′ i ∩ (α ′ i+2 ∪ δ ′ i+2) = ∅ for each i = 1, 2, 3. Each α ′ i ∪δ ′ i is still a … view at source ↗
Figure 5
Figure 5. Figure 5: Constructing the coned-off complex KH. Here, H = {A, B}. The coned-off complex will, in general, be locally infinite. Its purpose is to provide a lens with which we may inspect the ring BHK, via the following. Proposition 4.10. Let K be a connected 2-dimensional simplicial complex and H a system of subgraphs. If every Y ∈ H has infinite 0-skeleton, then there is a natural isomorphism BHK ∼= B(KH). If K is … view at source ↗
Figure 6
Figure 6. Figure 6: The subgraph Λ1 is highlighted in red. . . . . . . Λ2 ⊂ Γ2 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The simple body and adjoined cacti of a facial subgraph of an almost 2-connected planar graph. Each adjoined cactus has bounded diameter. for some uniform ε > 0. Let V ∈ DΠ be the unique face such that U ⊃ V ⊃ W. Let x ∈ F(U). Then there exists a path of length at most ε in Γ connecting x to some point in F(W). But clearly this path must intersect F(V ). Since Π is quasi-isometrically embedded in Γ, there … view at source ↗
Figure 9
Figure 9. Figure 9: Decomposing the path p into the pi and qi . Fix i, and to ease notation let x = xi , y = yi , q = qi . We have that dΓ(x, φ(Y )) = dΓ(y, φ(Y )) = ε. Let q ′ = ψ(q), which is a path connecting x ′ := ψ(x) to y ′ := ψ(y). We have that q ′ is a contained in the complement of the ε ′ -neighbourhood of Y , where ε ′ = 1 λ ε − λ > 2λ 2 > 0. We also have that x ′ , y′ ∈ BX(Y ; r), where r = λε + λ. Choose a, b ∈ … view at source ↗
Figure 11
Figure 11. Figure 11: If x1, x2, x3 pairwise lie on subfaces of U0, but share no common face, then a tripod as above must exist in Γ. By Propositions 1.10, 1.11, we lose nothing by assuming that Γ is bounded valence, or that the quasi-isometries are continuous. The reason we want φ to be surjective is so we can apply Lemma 6.10 later in this section. Note that there is an induced quasi-action of G upon Γ (see Definition 1.14).… view at source ↗
Figure 14
Figure 14. Figure 14: Jordan curve J separating v0 from z. The purple region is disjoint from V . Proof. In what follows, we will suppress ϑ from our notation for the sake of readability. That is, we will identify Γ with its ϑ-image in S 2 . We first show that V is locally connected, i.e. that for every x ∈ V and every open subset W ⊂ S 2 containing x, there exists an open subset O ⊂ W with x ∈ O and O ∩ V connected. The only … view at source ↗
Figure 15
Figure 15. Figure 15: The figure Z (dashed) drawn in the plane. Let Bx = BΓ(x; r), By = BΓ(y; r) be the closed r-balls about x and y. Let U1, . . . , Un ∈ DΓ be those faces which intersect Bx. Let V be the connected component of S 2 − S i Ui which contains y, and thus also contains all of By. Then by Lemma 7.7 we have that ∂V contains a simple closed curve L 3 which separates Bx from By. Note that L ⊂ Γ. We will modify L and f… view at source ↗
Figure 17
Figure 17. Figure 17: Any path through Γ between αi and αj must pass through λ 2 distinct βk paths different from the αi and αj . (2): Let S ≥ 0 be sufficiently large so that BΛ(z, s) separates ends in Λ for every s ≥ S, z ∈ Λ. Since Λ admits a cobounded quasi-action, such an S certainly exists. Fix s ≥ S. Let x, y ∈ Λ such that dΛ(x, y) > 2s. We have that both B1 := BΛ(x, s) and B2 := BΛ(y, s) separate ends in Λ. We have that… view at source ↗
Figure 18
Figure 18. Figure 18: Construction of Θ. (1) ϑ(a), ϑ(b) lie in distinct components of S 2 − ϑ(ℓ), and (2) {a, b} ∩ BΛ(ℓ; R) = ∅. Let βi be a segment of ℓ connecting αi to αi+1, which is otherwise disjoint from all the αj . Note that βi sits outside of the λ 2 -neighbourhood of αi+2 in Γ, by an easy application of the Jordan curve theorem, since any path between these two curves must intersect either αi or αi+1, or a bounded ne… view at source ↗
read the original abstract

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such a group is virtually a free product of free and surface groups, and thus virtually admits a planar Cayley graph.

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 that a connected, locally finite, quasi-transitive graph quasi-isometric to a planar graph is necessarily accessible. This yields a classification of finitely generated groups quasi-isometric to planar graphs: such groups are virtually free products of free groups and surface groups, and thus virtually admit planar Cayley graphs.

Significance. If the result holds, the classification supplies a complete structural description for the quasi-isometry class of planar graphs within geometric group theory. It combines accessibility splittings with planarity constraints on the factors, extending known results on groups with planar Cayley graphs and providing a falsifiable group-theoretic consequence (virtual planarity of a Cayley graph) that aligns with standard techniques in the area.

minor comments (2)
  1. The abstract is concise but could explicitly reference the main theorem number for easier navigation from the introduction.
  2. Notation for quasi-transitivity and accessibility should be introduced with a brief reminder of the standard definitions in §1 to aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a theorem establishing accessibility for connected locally finite quasi-transitive graphs quasi-isometric to planar graphs, then derives a group classification as a corollary. The provided abstract and reader summary contain no equations, fitted parameters, self-citations invoked as load-bearing uniqueness theorems, or renamings of known results that reduce the central claim to its inputs by construction. The logical chain is presented as a direct implication under explicit hypotheses (quasi-transitivity, local finiteness), consistent with standard geometric group theory techniques without internal reduction to self-definition or prior author results that would force the outcome. This is the expected outcome for a self-contained proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions of quasi-isometry, accessibility, quasi-transitivity, and planarity in graphs together with the correspondence between groups and their Cayley graphs; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • standard math Quasi-isometry is an equivalence relation that preserves coarse geometric properties such as accessibility.
    Invoked implicitly when passing from the graph statement to the group classification.
  • standard math Finitely generated groups act properly and cocompactly on their Cayley graphs.
    Used to translate the graph result into a statement about groups.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Relative accessibility for graphs

    math.CO 2026-05 unverdicted novelty 7.0

    Relative accessibility in graphs is defined relative to peripheral systems, characterized via Boolean ring subrings for quasi-transitive graphs, and shown to match the group-theoretic version while being quasi-isometr...

  2. Almost planar finitely presented groups

    math.GR 2026-05 unverdicted novelty 7.0

    Finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs; k-planar coarsely simply connected quasi-transitive graphs are quasi-isometric to planar graphs.

Reference graph

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