arxiv: 2605.03040 · v1 · submitted 2026-05-04 · 🧮 math.GR · math.CO· math.MG

Recognition: unknown

Almost planar finitely presented groups

Davide Spriano, John M. Mackay, Joseph P. MacManus

Pith reviewed 2026-05-08 02:00 UTC · model grok-4.3

classification 🧮 math.GR math.COmath.MG

keywords finitely presented groupsk-planar graphsCayley graphsquasi-isometricplanar graphscoarsely simply connectedquasi-transitive graphsgroup theory

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The pith

Finitely presented groups with k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs.

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

The paper proves that if a finitely presented group has a Cayley graph that can be drawn in the plane with a bounded number of crossings per edge, then it admits a finite-index subgroup whose Cayley graph is planar. A broader result shows that any k-planar graph that is coarsely simply connected, connected, locally finite, and quasi-transitive is quasi-isometric to a planar graph. This answers a special case of a question posed by Georgakopoulos and Papasoglu. A sympathetic reader would care because it reduces the study of certain almost-planar groups to the better-understood class of groups with planar Cayley graphs.

Core claim

Finitely presented groups that admit k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, a k-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph.

What carries the argument

The k-planar embedding of the Cayley graph, combined with finite-index subgroup selection and quasi-isometry to planar graphs.

If this is right

Such groups are virtually equivalent to groups whose Cayley graphs embed in the plane without crossings.
The quasi-isometry type of these graphs can be represented by a planar graph.
Questions about the large-scale geometry of k-planar groups reduce to questions about planar graphs in the finitely presented case.

Where Pith is reading between the lines

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

This suggests that crossing number bounds on Cayley graphs can be eliminated by passing to a finite cover.
The result may extend to other notions of almost-planarity if the coarse simple connectedness condition can be relaxed.
It opens the possibility of classifying groups by the minimal k for which their Cayley graphs are k-planar.

Load-bearing premise

The graphs must be coarsely simply connected and the groups must be finitely presented with k-planar Cayley graphs.

What would settle it

A counterexample would be a finitely presented group whose k-planar Cayley graph has no finite-index subgroup with a planar Cayley graph, or a k-planar coarsely simply connected quasi-transitive graph that is not quasi-isometric to any planar graph.

Figures

Figures reproduced from arXiv: 2605.03040 by Davide Spriano, John M. Mackay, Joseph P. MacManus.

**Figure 1.** Figure 1: slips through the net of all existing obstructions described above; it has asymptotic dimension equal to 2, its Poincaré profiles all satisfy the bound of Theorem 1.2 [HM25, Thm. 1.17], and it admits a non-constant bounded harmonic function with finite Dirichlet energy (as does every non-elementary hyperbolic group [Anc07]). It is this particularly troublesome group which motivated the toolbox developed in… view at source ↗

**Figure 2.** Figure 2: Folds create non-stability: the map f, which folds the plane onto itself, is not stable because there is a modification f ′ that agrees with f outside the red ball and that avoids f(x). 1.3. Questions. We conclude this introduction by stating two existential questions. A negative answer to both of these questions would settle Conjecture 1.1 in its entirety. Firstly, as was remarked above, Theorem D impli… view at source ↗

**Figure 3.** Figure 3: Proving that spheres are uniformly visual in P. 3.3. The Varopoulos inequality. We will make use of an isoperimetric inequality of Varopoulos, as it appears in [SC95, Thm. 2.1]. Given a vertextransitive graph Γ, let γΓ(n) denote the growth function of Γ; that is, the number of vertices in any n-ball. Given a subset Ω ⊂ V (Γ), we denote by ∂Ω the set of vertices which lie at a distance of exactly 1 from Ω.… view at source ↗

**Figure 4.** Figure 4: Proof of Lemma 4.10. While we know by Lemma 4.9 that local modifications are still regular maps, we do not quite yet have the uniformity required to be able to compose local modifications on a sufficiently sparse set and still preserve regularity. For this, we will use the following quantitative elaboration. Lemma 4.10. Let f ′ be a ε-local modification of f at x0, such that f(x0) ̸∈ f ′ (X). Then there ex… view at source ↗

read the original abstract

We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely simply connected graphs: a $k$-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar 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.

Desk Editor's Note private letter to a colleague

This paper shows finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs, and proves a special case of the Georgakopoulos-Papasoglu question for coarsely simply connected graphs.

read the letter

This paper shows that finitely presented groups admitting k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. It also proves that any k-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph, settling the Georgakopoulos-Papasoglu question in that restricted setting. The finite presentation is used to pass from the group to a finite-index subgroup, while coarse simple connectedness controls the global topology enough for the quasi-isometry to hold. The hypotheses are stated explicitly, so the reduction from k-planar to planar follows directly from the graph theorem applied to the Cayley graph. The work is clean on its own terms and organizes a useful implication without overclaiming generality. The proof strategy appears to combine crossing-number control with quasi-isometry techniques that preserve the relevant coarse properties. No circularity or hidden fitting shows up in the argument as described. One limitation is the restriction to coarsely simply connected graphs and to finitely presented groups; those conditions are necessary for the current argument and are not hidden, but they leave the broader question open. The dependence of the quasi-isometry constants on k is probably tracked but not a central issue. Readers working in geometric group theory on Cayley graphs, planarity, or quasi-isometries will find the result directly useful. It is a targeted theorem rather than a broad survey, so it fits best for specialists who already follow questions about embeddability of group actions. The paper deserves a serious referee because it supplies a new theorem that answers a cited open question with a clear statement and explicit conditions. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper proves that finitely presented groups admitting k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, it shows that any k-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph, resolving a question of Georgakopoulos and Papasoglu in this special case.

Significance. If the central claims hold, the result is a solid contribution to geometric group theory. It reduces the study of groups with k-planar Cayley graphs to the planar case via finite-index subgroups and provides a quasi-isometry theorem for a natural class of graphs. The direct proof under explicit hypotheses (without ad-hoc parameters) is a strength, and the finite-presentation hypothesis aligns well with coarse simple connectedness of Cayley graphs.

minor comments (3)

§1 (Introduction): Expand the discussion of why coarse simple connectedness is the key additional hypothesis that allows the quasi-isometry to a planar graph; a brief counterexample sketch for the non-coarsely-simply-connected case would clarify the necessity of the assumption.
Definition of k-planarity (likely §2): Confirm that the embedding definition used here is equivalent to the standard one in the literature (e.g., at most k crossings per edge in a drawing in the plane); if it differs, state the difference explicitly.
Proof of the general graph theorem (likely §3 or §4): The quasi-isometry construction should include explicit (even if not optimal) constants depending on k and the coarse simple connectedness parameters; this would make the result more usable for applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending minor revision. The summary accurately captures the main results, and we appreciate the recognition of the direct proof and alignment with the finite-presentation hypothesis.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states and proves a theorem: finitely presented groups admitting k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs, plus the general result that any k-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph. This rests directly on the listed hypotheses (finite presentation, k-planarity, coarse simple connectedness, local finiteness, connectedness, quasi-transitivity) without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations whose content is merely renamed. No equations or steps equate a claimed output to an input by construction; the argument is a direct proof under the stated conditions and is therefore independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5351 in / 1124 out tokens · 41777 ms · 2026-05-08T02:00:46.259881+00:00 · methodology

discussion (0)

Reference graph

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