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RAAGedy right-angled Coxeter groups

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Pith's one-line read The paper gives graph-level criteria that decide, for every triangle-free CFS graph with at most 10 vertices, whether its right-angled Coxeter group is quasiisometric to a right-angled Artin group.

desk verdict Nearly settles the small-graph RAAGedy classification with genuinely reusable tools; the ≤10-vertex completeness claim depends on an unpinned computation. read the letter →

arxiv 2506.16789 v1 pith:D6C34OGC submitted 2025-06-20 math.GR

Christopher H. Cashen , Pallavi Dani , Alexandra Edletzberger , Annette Karrer This is my paper · ORCID
classification math.GR MSC 20F6520F55
keywords right-angledCoxetergroupArtinquasiisometryCFSgraphmaximalproductregionJSJdecompositionmodificationoperationscoarseneardouble
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks which right-angled Coxeter groups (RACGs) are quasiisometric to some right-angled Artin group (RAAG), a property the authors call RAAGedy. It gives criteria on the presentation graph that decide this question, and when the graph is RAAGedy the criteria also produce a presentation graph for a quasiisometric RAAG. The positive direction uses three graph operations that change the presentation graph without changing the quasiisometry type of the group: link doubling, plus two new operations called cloning and unfolding. The negative direction gives several geometric obstructions, translated into graphical conditions, coming from Morse boundaries, JSJ decompositions, ladders in the maximal product region graph, and compliant cycles. A computer enumeration of all triangle-free CFS graphs with at most 10 vertices completely settles the motivating question in that range, leaving exactly 8 eleven-vertex graphs unresolved.

What carries the argument

The load-bearing objects are three graph modification operations: link doubling, which replaces the graph by one whose Coxeter group is a finite-index subgroup of the original; cloning, which adds a twin of a vertex that is a satellite of at least two other vertices and leaves the quasiisometry type unchanged; and unfolding, which rewires along separating joins and also preserves the quasiisometry type. The twin graph, the quotient of the presentation graph by equal-link classes, carries the recognition algorithm for coarse near doubles. On the obstruction side, the maximal product region graph, whose vertices are maximal standard product regions and whose edges record intersections that are standard product regions, is the key object: a quasiisometry between universal covers induces an isomorphism of these decorated graphs, so the 1-bottleneck property of RAAG MPRGs and the contrary existence of wide ladders in RACG MPRGs become quasiisometry obstructions. Compliant subcomplexes, built inductively from maximal products by projections and line times bushy-tree factors, provide a second obstruction mechanism tailored to the projection-diameter dichotomy of RAAGs.

What would settle it

Re-run the paper's enumeration with independently versioned code on all triangle-free CFS graphs with at most 10 vertices; any graph for which neither the positive nor the negative criteria fire would refute the completeness claim. For the geometric engine, exhibit a quasiisometry between a Davis complex and a Salvetti complex whose induced map on maximal product regions is not an isomorphism of the decorated MPRG.

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Extended reading notes

Core claim

The central claim is that for triangle-free CFS graphs without separating cliques, being RAAGedy is detectable at the level of the presentation graph. A graph is RAAGedy if it can be transformed, by link doubling, cloning, and unfolding, into a coarse near double or into a graph satisfying the Dani-Levcovitz conditions; the coarse near double condition is recognized from the twin graph and simplifies to: no unclonable singletons, or all unclonable singletons contained in the set of one vertex and its satellites, or in the set of two adjacent vertices and their satellites. In the converse direction, the paper shows several graph conditions force non-RAAGedy: presence of stable cycles or connected Morse boundary after iterated link doubling; rigid or hanging vertices in the JSJ graph of cylinders incompatible with RAAG JSJ decompositions; a ladder in the maximal product region graph that violates the 1-bottleneck property of RAAG MPRGs; and compliant cycles whose accumulated closest-point projections are too large. Together with the enumeration, these criteria completely answer the motivating question for all triangle-free CFS graphs with at most 10 vertices.

Load-bearing premise

The negative criteria rest on Oh's theorem that a quasiisometry between universal covers of compact weakly special square complexes induces an isomorphism of their decorated maximal product region graphs; if that invariance fails for Davis and Salvetti complexes, the ladder and compliant-cycle obstructions do not follow.

Editorial extensions

If this is right

  • For every triangle-free CFS graph with at most 10 vertices, RAAGediness is decided by the paper's criteria, and in the positive case the criteria produce an explicit presentation graph for a quasiisometric RAAG.
  • Cloning and unfolding are new quasiisometry-preserving graph operations, so any graph invariant invariant under all three operations is a quasiisometry invariant of the corresponding RACG.
  • The ladder obstruction shows that the maximal product region graph of a RACG can be a quasitree without the precise 1-bottleneck structure of a RAAG MPRG, so the MPRG carries information finer than its quasiisometry type.
  • The compliant-cycle obstruction subsumes the JSJ-based obstructions of no cycles of cuts and no virtually Z2 edge incident to a rigid non-Z2 vertex, giving a single mechanism for many non-RAAGedy examples.
  • The complete answer for up to 10 vertices and the list of 8 unresolved 11-vertex graphs give a concrete finite testbed for further questions, such as whether RAAGediness is constructible by coning from a square.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the quoted invariance theorem for maximal product region graphs holds at the stated generality, the ladder and compliant-cycle obstructions should apply beyond 2-dimensional Davis and Salvetti complexes; testing them on higher-dimensional weakly special square complexes would be a natural extension.
  • The 8 unresolved 11-vertex graphs are the obvious next targets: either a new positive operation or a new obstruction will be needed, and those graphs are the minimal places to look for it.
  • The computer examples with deeply buried stable cycles suggest there may be no uniform bound on how many link doublings are needed to expose a stable cycle; if so, no finite search over link doubles can certify absence of the Morse-boundary obstruction, making the decomposition-sequence criterion potentially necessary.
  • Cloning and unfolding are only known to preserve quasiisometry type, not commensurability, so groups shown RAAGedy through them may be quasiisometric to a RAAG without admitting a finite-index RAAG subgroup; determining when these operations can be upgraded to commensurability is a testable next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies which right-angled Coxeter groups (RACGs) are quasiisometric to some right-angled Artin group (RAAG), focusing on triangle-free CFS presentation graphs. It introduces two new graph operations, cloning and unfolding, which change the presentation graph without changing the quasiisometry type of the corresponding RACG, and combines them with the existing link doubling operation. On the positive side, it gives criteria, such as coarse near doubles (Theorem 4.16) and transformations into known RAAGedy types (criterion (ΞR)), for deciding that a graph is RAAGedy and for producing a RAAG presentation graph. On the negative side, it develops obstructions from Morse boundaries, JSJ decompositions, maximal product region graphs (ladders), and compliant cycles. The paper reports a computer enumeration of triangle-free CFS graphs and claims that the criteria completely answer the motivating question for graphs with at most 10 vertices, with 8 unresolved 11-vertex graphs.

Significance. If the results hold, this is a substantial contribution to the quasiisometric classification of RACGs. The paper provides practical, graphically verifiable criteria for a problem that previously had only partial answers, and the new invariants (MPRG ladders, compliant cycles) go beyond existing RAAG rigidity tools. The constructive nature of the positive criteria and the large collection of worked examples are valuable. The paper is also unusually explicit: the main proofs are written out in detail, the operations are precisely defined, and the code is made available. The completeness claim for small graphs is a natural and useful deliverable, provided the computational component is made fully reproducible.

major comments (2)
  1. [§3.3, Table 1, Figure 1, and Abstract] The headline claim that the motivating question is completely answered for triangle-free CFS graphs with at most 10 vertices is an output of the authors' code, but the manuscript gives only aggregate counts in Table 1 and region counts in Figure 1. No commit hash, archived input/output data, per-graph classification file, or independent verification procedure is provided. Because the implemented criteria include iterated link doubling (up to depth three), cloning, unfolding, near-double recognition, ladder detection via Theorem 6.16, and compliant-cycle detection via Theorem 7.5, a single coding error could flip the classification of one graph and invalidate the completeness claim. This is load-bearing; the paper should be accompanied by a fixed, versioned computational artifact with explicit inputs and outputs, or by an independent machine-checkable certificate.
  2. [§2.6, §6, and §7] The ladder and compliant-cycle obstructions both pass through Oh's maximal-product-region invariance (Theorem 2.22 and Corollary 2.23), which is quoted for compact weakly special* square complexes. The paper applies this to Davis and Salvetti complexes, noting that walls are 2-sided, but does not spell out why the finite-index cover passage in Definition 2.18 leaves the decorated maximal-product-region graph unchanged for exactly the complexes used later. This is not an observed error, but it is a correctness-risk point in a load-bearing dependency; please add an explicit verification or a precise citation covering the Davis and Salvetti cases.
minor comments (5)
  1. [Abstract] The word 'motiving' in the abstract should be 'motivating'.
  2. [§1.2] In the list of non-RAAGedy criteria, 'crietria' should be 'criteria'; also, the acronyms (CC') and (CC) are visually similar and should be cross-referenced explicitly where each is introduced.
  3. [Figure 1] The caption says the region labels are list items from Section 1.2, but the figure has no legend; please add a legend or define the labels directly in the caption.
  4. [Example 4.18 and Figure 5] The text refers to red and blue vertices in Figure 5; if the figure appears in grayscale, add a shading or symbol key.
  5. [§3.3] The statement about the horizontal line in Table 1 and the forced bipartiteness of graphs with more than (n-1)^2/4 + 1 edges would benefit from a brief explanation in the text, since the table alone does not make this clear.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reasoning: the positive and negative criteria reduce to independently proved graph operations and external quasiisometry invariants; only minor self-citations to the authors' thesis work appear, and the unpinned computer enumeration is a reproducibility/correctness issue, not circularity.

full rationale

The paper's derivation chain is not circular. The positive direction (CND, XiR) rests on graph operations—link doubling, cloning, unfolding—whose quasiisometry preservation is proved in the paper via independent tree lemmas (Lemmas 4.11 and 4.22) and Proposition 2.10's tree-of-spaces criterion. The final step to RAAGedy uses classical commensurability results of Davis–Januszkiewicz (Theorem 2.4) and Dani–Levocovitz, plus the purely graph-theoretic near-double recognition (Proposition 4.7). No parameter is fitted to data, and no 'prediction' is a renamed input. The negative direction uses external invariants: Oh's MPRG invariance (Theorem 2.22, Corollary 2.23), stability theorems (Theorem 2.13, Theorem 5.8), and JSJ cylinder theory. The JSJ description of RACGs is cited to Edletzberger's thesis [40], one of the authors, and several results are also cited to [41]; these are self-citations, but they are parameter-free theorems about graph-theoretic decompositions whose assumptions do not include the target RAAGedy conclusion, so they count as independent support rather than circular premises. The paper even supplies proofs for the main uses, such as Lemma 5.13 and Theorem 6.1. The text explicitly marks Theorem 5.16 as an omitted proof (special case of Theorem 7.5) and references forthcoming work of Cordes–Karrer–Ruane; neither is load-bearing for the central claims. The at-most-10-vertices completeness assertion depends on the authors' code with no commit hash or per-graph data, but that is a reproducibility/correctness risk, not a circular reduction: the enumeration and checks implement the proved criteria rather than encoding the answer. Overall, there is no step where Eq. X = Eq. Y by construction or where a fitted parameter is renamed a prediction, so circularity is minimal; score 2 reflects only the minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; this is a structural graph and group theory paper. The new graph operations (cloning, unfolding) and objects (coarse near doubles, ladders, compliant subcomplexes) are definitions with proofs of quasiisometry invariance, not postulated entities with independent falsifiable handles. The main external dependencies are listed in the axioms.

assumptions (5)
  • standard math CAT(0) cube complex machinery: gate projections, Bridge Lemma (Proposition 2.31), and coarse intersection results for standard subcomplexes.
    Used throughout Sections 2.7, 2.8, and Section 7 to compute projections and coarse intersections in Davis and Salvetti complexes; cited as known results, not proved here.
  • domain assumption Oh's Theorem 2.22 and Corollary 2.23: a quasiisometry between universal covers of compact weakly special square complexes induces a bijection or isomorphism of maximal standard product region graphs that respects decorations.
    This is the bridge converting the MPRG, ladder, and compliant-cycle obstructions into quasiisometry invariants; the paper relies on it without reproving it.
  • domain assumption Restriction to triangle-free, incomplete graphs without separating cliques after finite-index reduction by Grushko-Stallings-Dunwoody decomposition and removal of clique factors.
    The paper states this reduction in Section 1.1 and uses it throughout; the complete small-graph decision is for this class only.
  • standard math JSJ theory over 2-ended subgroups: existence of JSJ tree and graph of cylinders and quasiisometry invariance of the JSJ tree of cylinders.
    Used in Section 5.3 and Corollary 5.14 to convert JSJ graph features into quasiisometry obstructions.
  • standard math Theorem 2.13 on Morse subsets of 1-ended RAAGs and the equivalence between stable subgroups of RACGs and square-complete square-free subgraphs (Theorem 2.14).
    Used for Proposition 2.16 and the stable-cycle obstruction in Section 5.2.

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Pith. "Pith review of RAAGedy right-angled Coxeter groups." pith.science (2026). https://pith.science/paper/D6C34OGC

@misc{pith2026250616789,
  author       = {Pith},
  title        = {Pith review of: RAAGedy right-angled Coxeter groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6C34OGC}},
  note         = {Machine review of arXiv:2506.16789}
}
read the original abstract

We give criteria for deciding whether or not a triangle-free simple graph is the presentation graph of a right-angled Coxeter group that is quasiisometric to some right-angled Artin group, and, if so, producing a presentation graph for such a right-angled Artin group. We introduce two new graph modification operations, cloning and unfolding, to go along with an existing operation called link doubling. These operations change the presentation graph but not the quasiisometry type of the resulting group. We give criteria on the graph that imply it can be transformed by these operations into a graph that is recognizable as presenting a right-angled Coxeter group commensurable to a right-angled Artin group. In the converse direction we derive coarse geometric obstructions to being quasiisometric to a right-angled Artin group, first by specializing existing results from the literature to this setting, then by developing new approaches using configurations of maximal product regions. In all cases we give sufficient graphical conditions that imply these geometric obstructions. We implemented our criteria on a computer and applied them to an enumeration of small graphs. Our methods completely answer the motiving question when the graph has at most 10 vertices.

Figures

Figures reproduced from arXiv: 2506.16789 by the authors.

Figure 1
Figure 1. The 3938 isomorphism types of triangle-free CFS graphs with at most 11 vertices. There are exactly 8 graphs, all with 11 vertices, for which we do not know if they are RAAGedy/non￾RAAGedy. They are listed in Section 8. Preexisting work covers only the ‘planar’ part of the diagram, the regions labelled ‘DL’ and ‘DJ’, and a handful of isolated examples, so the figure shows that our new constructions vastly improve the… view at source ↗
Figure 2
Figure 2. Russell, Spriano, and Tran do something similar [80, Proposition 7.6] [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. is the smallest example of a triangle-free strongly CFS graph such that for every vertex v the graph Γ ´ tvu is not strongly CFS. Thus, Γ cannot be built by coning from a square while remaining strongly CFS at each step. ˛ [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: First link doubling example. In Proposition 4.7 below, we describe twin modules as even/odd according to their orders. We refer to the link lkpMq of a twin module M, which we take to mean lkpMq :“ lkRpΓqpMq, that is, the link of the vertex M in the twin graph RpΓq (rec…
Figure 5
Figure 5. Figure 5: Some coarse near doubles Γ (top row) and graphs ∆ (bottom row) such that WΓ is quasiisometric to A∆, respectively. Example 4.18. The top row of [PITH_FULL_IMAGE:figures/full_fig_p036_5.png]
Figure 6
Figure 6. Figure 6: Unfolding visualized as a continuous deformation. Γ 1 is obtained from Γ by unfolding as in Proposition 4.19 with A :“ tau, B :“ tbu, C :“ tcu, E :“ A \ B \ C, F :“ tf0, f1u, G :“ tg0, g1u, and H :“ th0, h1u, so WΓ and WΓ1 are quasiisometric. ˛ In the previous example …
Figure 7
Figure 7. Figure 7: Unfolding example. As in the previous example, the sets for Proposition 4.19 are indicated by their lowercase vertex labels, and we conclude that WΓ is quasiisometric to WΓ1 [PITH_FULL_IMAGE:figures/full_fig_p037_7.png]
Figure 8
Figure 8. Figure 8: Phase 1 coarse map ϕ1, with a0 red, b0 blue, c green, d olive Vertices with an incoming c–edge have outgoing ta0, . . . , amu–edges, but no incident tb0, . . . , bn, du–edges. Vertices with an incoming d–edge have outgoing tb0, . . . , bnu–edges, but no incident ta0, .…
Figure 9
Figure 9. Figure 9: Choices of donors for phase 2 map. Now we confirm the claim that these four types of donations do not interfere with one another. Given a vertex u, let δpuq denote its donor, if any. Suppose v is a predecessor of w and let w 1 be the immediate predecessor of w. For the…
Figure 10
Figure 10. Figure 10: Graphs showing that unfolding is not a quasiisometry when |C| ą 1. As in Example 4.21, the labelling of vertices in [PITH_FULL_IMAGE:figures/full_fig_p042_10.png]
Figure 11
Figure 11. Figure 11: Two graphs in the same Ξ component. Γ is not a coarse near double: 5, 6, 7, and 8 are all unclonable singletons. It does not admit a FIDL–Λ since it is not bipartite. Γ 1 is a near double: it has a pair of adjacent singletons, 4 and 5, and one additional singleton, 8,…
Figure 12
Figure 12. Figure 12: A spider attacking a small graph. We claim that mpΓq consists of one component isomorphic to mpΓ0q, and one component consisting of mpΓ1q with two additional leaf vertices attached, coming from the two possible squares that use the segment p0, 1, 2q of Γ0 and one of t…
Figure 13
Figure 13. Figure 13: A graph with a deeply buried stable cycle for Exam￾ple 5.6 We also know an example where a stable cycle appears only after performing three link doubles, so it seems unlikely that there should be any universal bound on how deeply stable cycles can be buried. These exa…
Figure 14
Figure 14. Figure 14: The blue/violet subgraph corresponds to a cylinder in the JSJ graph of cylinders of the RACG. The red/violet subgraph corresponds to a non-virtually–Z 2 rigid vertex. Their intersection (violet) corresponds to a virtually–Z 2 edge. We state another kind of obstruction…
Figure 15
Figure 15. Figure 15: Prototypical ladder. A variant is shown in [PITH_FULL_IMAGE:figures/full_fig_p056_15.png]
Figure 16
Figure 16. Figure 16: A non-prototypical example of a ladder Definition 6.13 (wide ladder). A wide ladder L in a graph Π is a graph satisfying: (a) There is a connected graph C, each vertex of which is associated to a connected subgraph Q of Π, and the subgraphs associated with adjacent ve…
Figure 17
Figure 17. Figure 17: In each picture Q is blue, Q1 is red, any edges in their intersection are violet. These satisfy Definition 6.13 (b) and (c) but not (d), and there is a vertex v whose star separates their union into multiple non-singleton components. Lemma 6.14. A graph containing a w…
Figure 18
Figure 18. Figure 18: The graph Γ of Example 6.17 with its maximal thick joins colored and RicΓ “ WΓzΠΓ with vertices of matching colors. There is one edge of Γ that does not belong to any thick join, and each of its endpoints belongs to a unique maximal join, corresponding to the two ends…
Figure 19
Figure 19. Figure 19: Link doubling of Γ in example Example 6.17. Note also that WΓ is strongly CFS, has no 2–ended splittings, contains no compliant cycle as in Theorem 7.5, and has totally disconnected Morse boundary, by Proposition 5.9, so contains no 1–ended stable subgroups. The ladde…
Figure 20
Figure 20. Figure 20: Fundamental domain for the action of the subgroup on the MPRG of Example 6.17. stable subgroups. It has an edge not contained in a thick join. Link double over the two vertices of this edge, as in the previous example. Γ :“ 0 2 3 1 4 5 6 7 8 D˝ p80q ˝ D˝ 7 ÝÑ 100 300 …
Figure 21
Figure 21. Figure 21: The Γ of Example 6.18 and an iterated link double. The fundamental domain for the action of the finite-index subgroup is shown in [PITH_FULL_IMAGE:figures/full_fig_p060_21.png]
Figure 22
Figure 22. Figure 22: Take Q to be the entire fundamental domain. There are choices r “ 201 and s “ 200 that satisfy Theorem 6.16 for this Q, so Γ is not RAAGedy. 501 401 500 400 301 300 100 000 100 000 200 500 400 300 201 100 000 301 501 401 600 100 000 200 400 300 201 601 100 000 301 401…
Figure 23
Figure 23. Figure 23: Quasiisometry carrying compliant cycle into a RAAG. Each compliant set Xi on the left, visualized as complementary regions of a tree in the plane, is sent by ϕ close to a compliant subcomplex Yi . Consecutive Yi , Yi`1 may no longer come B–close to each other, but the…
Figure 24
Figure 24. Figure 24: and [PITH_FULL_IMAGE:figures/full_fig_p067_24.png]
Figure 25
Figure 25. Figure 25: Σˆ P for r “ 2 and P as in [PITH_FULL_IMAGE:figures/full_fig_p068_25.png]
Figure 26
Figure 26. Figure 26: Some CFS graphs with compliant cycles where the Si are pairs and P is an anticlique. The graph in Figure 26a is planar, so the fact that it is not RAAGedy could have also been deduced by applying a theorem of Nguyen and Tran [73]. The graph in Figure 26b is nonplanar,…
Figure 27
Figure 27. Figure 27: An example Γ such that Theorem 7.5 can be satisfied by a single compliant subset S0 and a single path P “ P0. Example 7.10. Consider the graph Γ of [PITH_FULL_IMAGE:figures/full_fig_p071_27.png]
Figure 28
Figure 28. Figure 28: A graph with no compliant cycle with connected P, but having a compliant cycle consisting of two compliant sets and two connecting paths. t2, 6u ˚ t0, 4, 5u and T 1 :“ t0, 4u ˚ t1, 2, 3, 6u, which are the two maximal thick joins that intersect S0 in a non-clique. Both…
Figure 29
Figure 29. Figure 29: A graph with a compliant cycle where we need to recognize different quasiisometry orbits to verify the hypotheses of Theorem 7.5 are satisfied. Example 7.12. Consider the graph Γ of [PITH_FULL_IMAGE:figures/full_fig_p072_29.png]
Figure 30
Figure 30. Figure 30: A graph Γ and RicΓ such that RicΓ contains a cut vertex of ΠΓ. The corresponding splitting of Γ is illustrated by the red/violet and blue/violet subgraphs. The cut vertex property is used to distinguish quasiisometry orbits of maximal product regions in the constructi…
Figure 31
Figure 31. Figure 31: An example where taking a link double is necessary to satisfy the conditions of Theorem 7.5. Example 7.14 (Compliant cycle vs Morse boundary.). The graph of Figure 26a has a compliant cycle, but its Morse boundary is totally disconnected, by Corollary 5.11. The graph …
Figure 32
Figure 32. Figure 32: Graph with a stable cycle but no compliant cycle. 8. Further questions Question 8.1. Quasiisometry versus commensurability: ‚ We have only shown that cloning and unfolding produce groups quasiisometric to the one we started with. Are they actually commensurable? ‚ Doe…

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