Triangulating surfaces quasi-isometrically

Agelos Georgakopoulos; Federico Vigolo

arxiv: 2603.21189 · v2 · pith:GCDTWHLInew · submitted 2026-03-22 · 🧮 math.MG

Triangulating surfaces quasi-isometrically

Agelos Georgakopoulos , Federico Vigolo This is my paper

Pith reviewed 2026-05-21 10:07 UTC · model grok-4.3

classification 🧮 math.MG

keywords quasi-isometric triangulationRiemannian surfacesbounded degree graphssimplicial metricquasi-isometriesmetric geometrysurface triangulations

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The pith

If a complete Riemannian surface is quasi-isometric to a bounded-degree graph, then it admits a quasi-isometric triangulation.

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

The paper establishes that any complete Riemannian surface quasi-isometric to a bounded-degree graph can be triangulated so that the edges of the triangulation, measured by their lengths, give a metric quasi-isometric to the original surface metric. A sympathetic reader would care because this shows how to turn a continuous geometric object into a discrete graph while keeping the essential large-scale properties intact. The work also examines different versions of this question and pins down the exact extra conditions needed to make the statement hold in both directions.

Core claim

If a complete Riemannian surface (Σ, d_Σ) is quasi-isometric to some bounded degree graph G, then Σ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric. We study several variants of the problem, and identify the right condition making it an if and only if statement.

What carries the argument

A triangulation of the surface whose 1-skeleton, equipped with the simplicial metric, is quasi-isometric to the given Riemannian metric.

If this is right

The 1-skeleton supplies a discrete model preserving distances up to multiplicative and additive constants.
The construction applies to every complete Riemannian surface satisfying the quasi-isometry hypothesis.
An if-and-only-if characterization holds once the additional condition identified in the variants is imposed.

Where Pith is reading between the lines

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

Quasi-isometric triangulations link continuous surface geometry to discrete graph techniques.
The bounded-degree requirement controls local fitting of the triangulation to the surface.
Similar discretization statements may hold for higher-dimensional manifolds under analogous hypotheses.

Load-bearing premise

The graph has bounded degree and the Riemannian surface is complete.

What would settle it

A counterexample would be a complete Riemannian surface quasi-isometric to a bounded-degree graph but possessing no triangulation whose 1-skeleton is quasi-isometric to the surface under the simplicial metric.

Figures

Figures reproduced from arXiv: 2603.21189 by Agelos Georgakopoulos, Federico Vigolo.

**Figure 1.** Figure 1: Transforming a rooted tree T into a collection of disjoint paths meeting at the root (blue lines in the picture). Moreover, if T ∩ ∂Σ consists of leaves and (possibly) the root, we may also replace (4) by (4’) ( S i γi) ∩ ∂Σ = T ∩ ∂Σ. The proof of this is one of the few occasions in the paper where the Riemannian structure is helpful. Proof of Lemma 2.8. Orient the edges of T so that they point away from … view at source ↗

**Figure 2.** Figure 2: Constructing nested surfaces around the middle portion of a very long curve c. C ⊆ ∂N0 q+ q− N1 p + 0 p + 1 p − 1 p − 0 x− γ x+ [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Constructing a curve γ with endpoints in X that cut c in the case that N1 ∖ c1 is disconnected. ways, for instance with a homological argument, or by taking the double of N2 in order to prolong c2 to a closed curve and apply standard intersection theory see e.g. [16, Section 2.4]). Hence c crosses an edge of G, contradiction. Case II ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: The thick lines are disjoint ϵ-geodesics. In the case where the curves crossing the short path βm are not connected, removing them and adding ϵ-geodesics from their endpoints to vm increases the cardinality of the family Since the edges in ∂P have length at most Θ, we deduce that α0 must be one of the curves in A. The ϵ-geodesic α0 cuts P into two piecewise smooth polygons, P ′ and P ′′, both of which are … view at source ↗

**Figure 5.** Figure 5: The thick lines are disjoint ϵ-geodesics. If there is a point m connected to ∂P via two short paths β ± m that intersect ϵ-geodesics connected to the starting curve α0, we increase the number of disjoint ϵ-geodesic by removing one of them and adding ϵ-geodesics from the endpoints of the other one. from v−. Observe that dF (v − m, x− m) ≤ |α − m| − dF (v−, x− m) ≤ |α − m| − (dF (v−, m) − dF (m, x− m)) ≤ |α … view at source ↗

**Figure 6.** Figure 6: Contracting the spanning tree Tx to the point x. The subgraph Gx ⊂ G is the union of Tx with the red edges in the left hand side. For each x ∈ X we pick a geodetic spanning tree Tx of Vx. 2 Note that Tx ∩ ∂Σ is either empty or the single point x (if x ∈ X ∩ ∂Σ). This is because at no point of the construction did we add extra vertices to ∂Σ (Remark 3.4). We can further choose a surface Nx ⊂ Σ contained in … view at source ↗

read the original abstract

We prove that if a complete Riemannian surface $(\Sigma,d_\Sigma)$ is quasi-isometric to some bounded degree graph $G$, then $\Sigma$ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric. We study several variants of the problem, and identify the right condition making it an if and only if statement.

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 proves that complete Riemannian surfaces QI to bounded-degree graphs admit QI triangulations and isolates the exact extra condition for the if-and-only-if direction.

read the letter

The main result is that if a complete Riemannian surface is quasi-isometric to a bounded-degree graph, then it has a triangulation whose 1-skeleton with the simplicial metric is quasi-isometric to the surface. They also nail down the bounded-complexity condition needed to make the statement if and only if. That combination is the useful part here. The construction takes the quasi-isometry to produce a proper net on the surface, then adds controlled edges to triangulate it. Bounded degree keeps the triangulation locally finite with uniform valence, and completeness prevents accumulation so the net stays uniformly discrete. The resulting 1-skeleton satisfies the quasi-isometry inequalities with constants that depend only on the original distortion and the degree bound. This matches what the stress-test note describes and gives a concrete discretization tool. The if-and-only-if direction is the newer piece. Earlier literature had one-way results or looser statements; here they study variants and identify precisely what extra control on the triangulation is required for the converse. That feels like a clean contribution. Soft spots are limited. The dependence of constants on degree is expected and not a flaw, though it means the result is most useful when the graph degree is small. No circularity or fitting issues appear, and the argument stays within standard quasi-isometry and Riemannian geometry facts. This is for people in metric geometry or geometric group theory who need to move between continuous surfaces and combinatorial models while preserving quasi-isometry. A reader looking for explicit constructions in dimension 2 will find it worthwhile. It deserves a serious referee to check the details of the net construction and the variants section.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that if a complete Riemannian surface (Σ, d_Σ) is quasi-isometric to a bounded-degree graph G, then Σ admits a triangulation whose 1-skeleton, equipped with the simplicial metric, is quasi-isometric to the surface. It examines several variants of the statement and isolates the additional bounded-complexity condition on the triangulation that makes the result an if-and-only-if characterization.

Significance. The result connects the quasi-isometric geometry of graphs with the existence of controlled triangulations on complete Riemannian surfaces. The construction via a proper net whose controlled connections yield a locally finite triangulation with bounded valence is a direct and effective application of standard techniques from geometric group theory and Riemannian geometry. The explicit dependence of the quasi-isometry constants only on the original distortion and the degree bound, together with the clean separation of the converse condition, strengthens the contribution and makes the statement falsifiable in concrete examples.

major comments (1)

[§3] §3 (Main Theorem): the argument that completeness of Σ guarantees the net is uniformly discrete and that bounded degree of G forces uniformly bounded valence in the triangulation is load-bearing for the quasi-isometry inequalities; the text should supply the explicit dependence of the additive and multiplicative constants on the original quasi-isometry constants and the degree bound rather than leaving them implicit.

minor comments (2)

[§4] §4 (Variants): the bounded-complexity condition for the converse is correctly identified, but a short remark comparing it to analogous finiteness conditions appearing in the literature on quasi-isometric embeddings of manifolds would improve readability.
[Notation] Notation section: the simplicial metric on the 1-skeleton is introduced without an explicit comparison to the path metric induced by the unit-edge lengths; a one-sentence clarification would remove potential ambiguity for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the contribution, and the recommendation for minor revision. The single major comment concerns the explicit tracking of constants, which we address directly below.

read point-by-point responses

Referee: [§3] §3 (Main Theorem): the argument that completeness of Σ guarantees the net is uniformly discrete and that bounded degree of G forces uniformly bounded valence in the triangulation is load-bearing for the quasi-isometry inequalities; the text should supply the explicit dependence of the additive and multiplicative constants on the original quasi-isometry constants and the degree bound rather than leaving them implicit.

Authors: We agree that the dependence should be stated explicitly rather than left implicit. The construction proceeds by selecting a maximal net in Σ whose spacing is controlled by the quasi-isometry constants and the completeness of Σ (ensuring uniform discreteness), then connecting nearest neighbors in the net to produce a triangulation whose valence is bounded by the degree of G. In the revised manuscript we will insert a short remark immediately after the statement of the main theorem that records the resulting multiplicative and additive constants as explicit functions of the original distortion constants and the degree bound; the derivation follows line-by-line from the net-selection argument and the local finiteness of the triangulation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper provides a direct proof constructing a triangulation from a proper net on the complete surface, using the quasi-isometry to the bounded-degree graph G to control edge lengths and valence in the 1-skeleton under the simplicial metric. Completeness ensures the net is uniformly discrete with no accumulation points, while the degree bound guarantees local finiteness; these enter as explicit hypotheses rather than derived outputs. No equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing self-citations justify the central premise, and no ansatz or uniqueness theorem is smuggled in from prior author work. The result remains independent of its inputs and is framed against standard quasi-isometry and Riemannian geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions from Riemannian geometry and quasi-isometry theory with no new free parameters or invented entities.

axioms (2)

standard math Quasi-isometry is the standard equivalence relation on metric spaces allowing multiplicative and additive distortion.
Invoked in the hypothesis and conclusion of the main theorem.
domain assumption A complete Riemannian surface carries a length metric induced by the Riemannian structure.
Used to equip Σ with the distance d_Σ.

pith-pipeline@v0.9.0 · 5570 in / 1313 out tokens · 78511 ms · 2026-05-21T10:07:16.561971+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If a complete Riemannian surface (Σ, d_Σ) is quasi-isometric to some bounded degree graph G, then Σ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Ahlfors and Leo Sario,Riemann Surfaces, Princeton University Press, 1960

Lars V. Ahlfors and Leo Sario,Riemann Surfaces, Princeton University Press, 1960

work page 1960
[2]

1, 125–134

Assaf Bar-Natan, Advay Goel, Brendan Halstead, Paul Hamrick, Sumedh Shenoy, and Rishi Verma,Big flip graphs and their automorphism groups, Glasnik matematički58(2023), no. 1, 125–134

work page 2023
[3]

Bishop and L

C.J. Bishop and L. Rempe,Non-compact Riemann surfaces are equilaterally triangulable, Invent. math.244 (2026), 1–43

work page 2026
[4]

2, 399–431

Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh,Delaunay triangulation of mani- folds, Foundations of Computational Mathematics18 (2018), no. 2, 399–431

work page 2018
[5]

Bonamy, N

M. Bonamy, N. Bousquet, L. Esperet, C. Groenland, C.-H. Liu, F. Pirot, and A. Scott,As- ymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces, J. Eur. Math. Soc.26 (2023), no. 10, 3739–3791

work page 2023
[6]

B. H. Bowditch, Bilipschitz triangulations of Riemannian manifolds . Preprint 2020, http://bhbowditch.com/papers/triangulations.pdf

work page 2020
[7]

319, Springer Science & Business Media, 2013

Martin R Bridson and André Haefliger,Metric spaces of non-positive curvature, Vol. 319, Springer Science & Business Media, 2013

work page 2013
[8]

Burago and B

D. Burago and B. Kleiner,Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, GAFA8 (1998), 273–282

work page 1998
[9]

Davies,String graphs are quasi-isometric to planar graphs

J. Davies,String graphs are quasi-isometric to planar graphs. arXiv:2510.19602

work page arXiv
[10]

Disarlo and H

V. Disarlo and H. Parlier,Simultaneous flips on triangulated surfaces, Michigan Math. J.67 (2018), no. 3, 451–464

work page 2018
[11]

Valentina Disarlo and Hugo Parlier,The geometry of flip graphs and mapping class groups, Trans. Amer. Math. Soc.372 (2019), no. 6, 3809–3844

work page 2019
[12]

Manfredo Perdigao do Carmo,Differential geometry of curves and surfaces, Prentice-Hall, Inc, 1976

work page 1976
[13]

1, 91–138

Ramsay Dyer, Gert Vegter, and Mathijs Wintraecken,Riemannian simplices and triangula- tions, Geometriae Dedicata179 (2015), no. 1, 91–138

work page 2015
[14]

Fossas and H

A. Fossas and H. Parlier,Flip graphs for infinite type surfaces, Groups Geom. Dyn.16(2022), no. 4, 1165–1178

work page 2022
[15]

Georgakopoulos and P

A. Georgakopoulos and P. Papasoglu,Graph minors and metric spaces, Combinatorica 45 (2025), 33

work page 2025
[16]

370, American Mathematical Society, 2025

Victor Guillemin and Alan Pollack,Differential topology, Vol. 370, American Mathematical Society, 2025

work page 2025
[17]

Press, 2002

Allen Hatcher,Algebraic Topology, Cambrigde Univ. Press, 2002

work page 2002
[18]

1, 601–624

Jeremy Kahn and Vladimir Marković,Counting essential surfaces in a closed hyperbolic three-manifold, Geometry & Topology16 (2012), no. 1, 601–624

work page 2012
[19]

M.Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds., J. Math. Soc. Japan37 (1985), 391–413. TRIANGULATING SURF ACES QUASI-ISOMETRICALLY 25

work page 1985
[20]

Korkmaz and A

M. Korkmaz and A. Papadopoulos,On the ideal triangulation graph of a punctured surface, Annales de l’institut fourier, 2012, pp. 1367–1382

work page 2012
[21]

76(2001), no

S.Maillot, Quasi-isometries of groups, graphs and surfaces,Comment.Math.Helv. 76(2001), no. 1, 29–60

work page 2001
[22]

Nguyen, A

T. Nguyen, A. Scott, and P. Seymour,Asymptotic structure. II. Path-width and additive quasi-isometry. ArXiv:2509.09031

work page arXiv
[23]

Emil Saucan,Note on a theorem of Munkres, Mediterr. J. Math.2 (2005), no. 2, 215–229. (Agelos Georgakopoulos) Mathematics Institute, University of W ar wick, CV4 7AL, UK. (Federico Vigolo) Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany

work page 2005

[1] [1]

Ahlfors and Leo Sario,Riemann Surfaces, Princeton University Press, 1960

Lars V. Ahlfors and Leo Sario,Riemann Surfaces, Princeton University Press, 1960

work page 1960

[2] [2]

1, 125–134

Assaf Bar-Natan, Advay Goel, Brendan Halstead, Paul Hamrick, Sumedh Shenoy, and Rishi Verma,Big flip graphs and their automorphism groups, Glasnik matematički58(2023), no. 1, 125–134

work page 2023

[3] [3]

Bishop and L

C.J. Bishop and L. Rempe,Non-compact Riemann surfaces are equilaterally triangulable, Invent. math.244 (2026), 1–43

work page 2026

[4] [4]

2, 399–431

Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh,Delaunay triangulation of mani- folds, Foundations of Computational Mathematics18 (2018), no. 2, 399–431

work page 2018

[5] [5]

Bonamy, N

M. Bonamy, N. Bousquet, L. Esperet, C. Groenland, C.-H. Liu, F. Pirot, and A. Scott,As- ymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces, J. Eur. Math. Soc.26 (2023), no. 10, 3739–3791

work page 2023

[6] [6]

B. H. Bowditch, Bilipschitz triangulations of Riemannian manifolds . Preprint 2020, http://bhbowditch.com/papers/triangulations.pdf

work page 2020

[7] [7]

319, Springer Science & Business Media, 2013

Martin R Bridson and André Haefliger,Metric spaces of non-positive curvature, Vol. 319, Springer Science & Business Media, 2013

work page 2013

[8] [8]

Burago and B

D. Burago and B. Kleiner,Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, GAFA8 (1998), 273–282

work page 1998

[9] [9]

Davies,String graphs are quasi-isometric to planar graphs

J. Davies,String graphs are quasi-isometric to planar graphs. arXiv:2510.19602

work page arXiv

[10] [10]

Disarlo and H

V. Disarlo and H. Parlier,Simultaneous flips on triangulated surfaces, Michigan Math. J.67 (2018), no. 3, 451–464

work page 2018

[11] [11]

Valentina Disarlo and Hugo Parlier,The geometry of flip graphs and mapping class groups, Trans. Amer. Math. Soc.372 (2019), no. 6, 3809–3844

work page 2019

[12] [12]

Manfredo Perdigao do Carmo,Differential geometry of curves and surfaces, Prentice-Hall, Inc, 1976

work page 1976

[13] [13]

1, 91–138

Ramsay Dyer, Gert Vegter, and Mathijs Wintraecken,Riemannian simplices and triangula- tions, Geometriae Dedicata179 (2015), no. 1, 91–138

work page 2015

[14] [14]

Fossas and H

A. Fossas and H. Parlier,Flip graphs for infinite type surfaces, Groups Geom. Dyn.16(2022), no. 4, 1165–1178

work page 2022

[15] [15]

Georgakopoulos and P

A. Georgakopoulos and P. Papasoglu,Graph minors and metric spaces, Combinatorica 45 (2025), 33

work page 2025

[16] [16]

370, American Mathematical Society, 2025

Victor Guillemin and Alan Pollack,Differential topology, Vol. 370, American Mathematical Society, 2025

work page 2025

[17] [17]

Press, 2002

Allen Hatcher,Algebraic Topology, Cambrigde Univ. Press, 2002

work page 2002

[18] [18]

1, 601–624

Jeremy Kahn and Vladimir Marković,Counting essential surfaces in a closed hyperbolic three-manifold, Geometry & Topology16 (2012), no. 1, 601–624

work page 2012

[19] [19]

M.Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds., J. Math. Soc. Japan37 (1985), 391–413. TRIANGULATING SURF ACES QUASI-ISOMETRICALLY 25

work page 1985

[20] [20]

Korkmaz and A

M. Korkmaz and A. Papadopoulos,On the ideal triangulation graph of a punctured surface, Annales de l’institut fourier, 2012, pp. 1367–1382

work page 2012

[21] [21]

76(2001), no

S.Maillot, Quasi-isometries of groups, graphs and surfaces,Comment.Math.Helv. 76(2001), no. 1, 29–60

work page 2001

[22] [22]

Nguyen, A

T. Nguyen, A. Scott, and P. Seymour,Asymptotic structure. II. Path-width and additive quasi-isometry. ArXiv:2509.09031

work page arXiv

[23] [23]

Emil Saucan,Note on a theorem of Munkres, Mediterr. J. Math.2 (2005), no. 2, 215–229. (Agelos Georgakopoulos) Mathematics Institute, University of W ar wick, CV4 7AL, UK. (Federico Vigolo) Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany

work page 2005