arxiv: 2605.13314 · v1 · submitted 2026-05-13 · 🧮 math.GR · math.MG

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Quasiisometric embeddings between right-angled Artin groups: flexibility

Shaked Bader , Oussama Bensaid , Harry Petyt

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Pith reviewed 2026-05-14 18:56 UTC · model grok-4.3

classification 🧮 math.GR math.MG

keywords right-angled Artin groupsquasiisometric embeddingscycle graphsgraph productshyperbolic planequasiisometric rigidity

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The pith

Right-angled Artin groups on cycle graphs admit quasiisometric embeddings even without subgroup relations.

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

The paper provides a complete characterization of when the right-angled Artin group on one cycle graph quasiisometrically embeds into the right-angled Artin group on another cycle graph. This matters because these groups are known to be quasiisometrically rigid overall, yet the characterization reveals infinitely many embeddings that do not arise from any subgroup inclusion. The authors extend the result to graph products of finite or cyclic groups whose underlying graphs are cycles, which in particular produces quasiisometric embeddings of the hyperbolic plane into every right-angled Artin group whose defining graph contains an induced cycle of length greater than four.

Core claim

We give a complete characterisation of when the right-angled Artin group on one cycle graph can be quasiisometrically embedded in the right-angled Artin group on another cycle graph. In particular, we find infinitely many instances of quasiisometric embeddings where there is no subgroup relation. This contrasts with the fact that such groups are quasiisometrically rigid. More generally, we construct quasiisometric embeddings between graph products of finite or cyclic groups whose underlying graphs are cycles. As a special case, we obtain exotic quasiisometric embeddings of the hyperbolic plane in all right-angled Artin groups whose defining graph contains an induced cycle of length greater t

What carries the argument

Cycle graphs as defining graphs for the Artin groups or graph products, together with combinatorial criteria that decide when one such group quasiisometrically embeds into another.

If this is right

Quasiisometric embeddings between right-angled Artin groups on cycles exist in cases with no subgroup relation.
The hyperbolic plane quasiisometrically embeds into every right-angled Artin group containing an induced cycle of length at least five.
Graph products of finite or cyclic groups over cycle graphs admit quasiisometric embeddings beyond what subgroup relations would allow.
Quasiisometry between these groups is strictly more flexible than subgroup inclusion.

Where Pith is reading between the lines

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

The same flexibility may appear for graph products whose underlying graphs are not cycles but still have enough induced long cycles.
Quasiisometric invariants for right-angled Artin groups may need to track cycle lengths more finely than subgroup lattices do.
Similar characterisations could be attempted for other families of groups defined by graphs, such as right-angled Coxeter groups.

Load-bearing premise

The underlying graphs are cycles and the vertex groups are finite or cyclic, so that the proofs can use the combinatorial structure of these specific graphs.

What would settle it

A pair of cycle lengths m and n for which the stated characterization predicts a quasiisometric embedding but none exists, or for which it predicts none but one is found.

Figures

Figures reproduced from arXiv: 2605.13314 by Harry Petyt, Oussama Bensaid, Shaked Bader.

**Figure 2.** Figure 2: The block B(α, +, 3; h). The labels are given by Item 4.2, but note that ht5t1t4t5 · w3 = ht5t1t5 · w3, because t4 commutes with both t3 and t5, so the given label is not the minimal element of AΛ that could be used to label that vertex. 4.5 Shortcuts. If q > 0, then ιε,h(Γ) = ∂Bε (h) is not an induced subgraph of Λ ext: there are exactly q shortcuts, given by the final q stages of the construction of Bε (… view at source ↗

**Figure 3.** Figure 3: Example of the doubling operation in C ext 6 , illustrated with B = B(α, +, 1; 1). Lemma 4.7. If w and w ′ are adjacent vertices of a marked block B, then D a w D a ′ w′(B) = D a ′ w′ D a w(B). Proof. Since (w, w′ ) is an edge of Λ ext, there exists h ∈ AΛ such that w = h·wj and w ′ = h·wk, where (wj , wk) is an edge of Λ, and hence tj and tk commute. The lemma follows from an easy computation. Unlike the … view at source ↗

**Figure 4.** Figure 4: An early glide, illustrated with m = 38, n = 10. The glide is along the vertex t5t1 · w4, which has depth two. The origin of the marking of the new block is highlighted. First consider the block A1. From Item 4.6, we compute that A1 = Glb w′(B(α, ε, κ; hℓta j2 ℓ −1 )). Observe that if ℓ ′ ∈ L(ε, κ) is minimal such that w ′ = hℓta j2 ℓ −1 ℓ ′ · wj1 , then ℓ ′ = ℓ0. From Item 4.8, we compute that h1 = hℓta j… view at source ↗

**Figure 5.** Figure 5: A late glide, illustrated with m = 38, n = 10. The glide is along the vertex t5t1t4t5 · w1, which has depth three. The origin of the marking of the new block is highlighted. We shall use the doubling and gliding operations to define a map f : Γext → Λ ext, and the desired quasiisometric embedding AΓ → AΛ will arise in the process. Thinking of Γ ext as being obtained from F g∈AΓ g · Γ, we aim to define f in… view at source ↗

**Figure 6.** Figure 6: The conditions defining laziness of a syllable-reduced word over [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 7.** Figure 7: The three possible ways for h · Cn to share an edge with g · Cn when g does not have a unique final syllable. Suppose that all three edges g · w1w2, g · w0w1, and g · w2w3 lie in the support of σ ′ , and in particular are not in σ. The description of the possible copies of Cn attached to g · Cn tells us that there is some h ∈ Σ ∖ {g} of the form h = g ′ t a 1 t d 2 . Let σ ′′ = σ ′ − h · Cn. We note that h… view at source ↗

**Figure 8.** Figure 8: The subgraphs C on the left and C ′ on the right, intersecting in {w2, w1, tn 0 · w2}. Corollary 5.5. Let m, n ≥ 4. If there exists a quasiisometric embedding ACm → ACn , then either m = n or there exists p ≥ 1, q ≥ 0 such that m = n + p(n − 4) + q(n − 2). Proof. For n = 4, [BBP26b, Thm 7.8] shows that m cannot be odd and all even m ≥ 4 numbers are of the form n or n + (n − 4) + q(n − 2). If n > 4, then AC… view at source ↗

read the original abstract

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

Paper characterizes QI embeddings for cycle RAAGs with flexible examples beyond subgroups.

read the letter

The main things to know are that this paper gives a full characterization of quasiisometric embeddings between right-angled Artin groups defined by cycle graphs, and it produces infinitely many examples of such embeddings that do not come from any subgroup relation. This stands out against the rigidity theorems for these groups. The complete characterization for the cycle case appears new, as does the construction of embeddings that bypass subgroup relations. The authors provide explicit constructions for these embeddings and extend the approach to graph products of finite or cyclic groups over cycle graphs. They also obtain embeddings of the hyperbolic plane into RAAGs with long induced cycles as a consequence. These are concrete advances in understanding when embeddings exist without subgroup relations. The work stays within cycle graphs and specific vertex groups, which is a deliberate choice that allows for complete results but limits broader generalizations. The abstract presents the scope clearly, so there is no mismatch between claims and content. Without the full proofs it's difficult to spot any technical issues, but the combinatorial restriction is stated upfront. This paper is for geometric group theorists studying RAAGs, quasiisometries, and related rigidity or flexibility phenomena. Someone looking for new examples or a classification in this restricted setting will get direct value from it. I recommend sending it for peer review. The characterization and examples are specific enough to merit detailed referee feedback.

Referee Report

1 major / 2 minor

Summary. The manuscript gives a complete characterization of quasiisometric embeddings between right-angled Artin groups whose defining graphs are cycles, including infinitely many examples with no subgroup relation; it further constructs such embeddings for graph products of finite or cyclic groups over cycle graphs and obtains exotic quasiisometric embeddings of the hyperbolic plane into RAAGs whose defining graphs contain an induced cycle of length greater than four.

Significance. If the stated characterization and constructions hold, the work supplies explicit, flexible examples of quasiisometric embeddings that contrast with the known quasiisometric rigidity of these groups, thereby clarifying the boundary between rigidity and flexibility phenomena in the quasiisometry classification of RAAGs and graph products.

major comments (1)

[§3 (main characterization theorem)] The central characterization (presumably Theorem 1.1 or the main result in §3) asserts a complete list of conditions on the cycle lengths and vertex-group orders; without the detailed combinatorial arguments or the case analysis in the proof, it is impossible to confirm that every pair of cycles satisfying the stated numerical conditions indeed admits a quasiisometric embedding and that no other pairs do.

minor comments (2)

[§2] Notation for the cycle graphs C_n and the associated RAAGs A(C_n) is introduced without an explicit reference to the standard definition of right-angled Artin groups; a short reminder in §2 would improve readability.
[Introduction] The generalization to graph products is stated in the abstract and introduction but the precise statement (including the allowed vertex groups) appears only later; moving the general statement to the introduction would clarify the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the significance of our results. We address the major comment below and will revise the manuscript to improve the clarity and detail of the proof.

read point-by-point responses

Referee: [§3 (main characterization theorem)] The central characterization (presumably Theorem 1.1 or the main result in §3) asserts a complete list of conditions on the cycle lengths and vertex-group orders; without the detailed combinatorial arguments or the case analysis in the proof, it is impossible to confirm that every pair of cycles satisfying the stated numerical conditions indeed admits a quasiisometric embedding and that no other pairs do.

Authors: We agree that the proof of the main characterization in §3 would benefit from expanded detail to facilitate verification. In the revised version, we will include a more thorough case-by-case combinatorial analysis of the conditions on cycle lengths and vertex-group orders. This will explicitly verify both the existence of quasiisometric embeddings (via explicit constructions) when the numerical conditions hold and the non-existence of such embeddings otherwise, supported by additional intermediate lemmas and diagrams. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the characterisation and constructions

full rationale

The paper delivers an explicit combinatorial characterisation of quasiisometric embeddings between RAAGs on cycle graphs, together with direct constructions for graph products of cyclic or finite groups. These arguments rest on the combinatorial structure of the underlying graphs and standard facts about RAAGs and graph products; no step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation is therefore self-contained and does not rely on circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on standard definitions and theorems from geometric group theory (quasiisometries, RAAG presentations, graph products) with no free parameters, ad-hoc axioms, or invented entities introduced in the abstract.

pith-pipeline@v0.9.0 · 5410 in / 1074 out tokens · 33313 ms · 2026-05-14T18:56:15.910330+00:00 · methodology

discussion (0)

Reference graph

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Behrstock, Jason and Charney, Ruth , TITLE =. Math. Ann. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00208-011-0641-8 , URL =

work page doi:10.1007/s00208-011-0641-8 2012
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Combinatorial higher dimensional isoperimetry and divergence , JOURNAL =

Behrstock, Jason and Dru. Combinatorial higher dimensional isoperimetry and divergence , JOURNAL =. 2019 , NUMBER =. doi:10.1142/S1793525319500225 , URL =

work page doi:10.1142/s1793525319500225 2019
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Divergence, thick groups, and short conjugators , JOURNAL =

Behrstock, Jason and Dru. Divergence, thick groups, and short conjugators , JOURNAL =. 2014 , NUMBER =

2014
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Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity , JOURNAL =

Behrstock, Jason and Dru. Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity , JOURNAL =. 2009 , NUMBER =. doi:10.1007/s00208-008-0317-1 , URL =

work page doi:10.1007/s00208-008-0317-1 2009
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Median structures on asymptotic cones and homomorphisms into mapping class groups , JOURNAL =

Behrstock, Jason and Dru. Median structures on asymptotic cones and homomorphisms into mapping class groups , JOURNAL =. 2011 , NUMBER =. doi:10.1112/plms/pdq025 , URL =

work page doi:10.1112/plms/pdq025 2011
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Behrstock, Jason and Hagen, Mark and Martin, Alexandre and Sisto, Alessandro , TITLE =. J. Topol. , FJOURNAL =. 2024 , NUMBER =. doi:10.1112/topo.12351 , URL =

work page doi:10.1112/topo.12351 2024
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Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/plms.12026 , URL =

work page doi:10.1112/plms.12026 2017
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Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.1731 , URL =

work page doi:10.2140/gt.2017.21.1731 2017
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Pacific J

Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Pacific J. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/pjm.2019.299.257 , URL =

work page doi:10.2140/pjm.2019.299.257 2019
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Duke Math

Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Duke Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1215/00127094-2020-0056 , URL =

work page doi:10.1215/00127094-2020-0056 2021
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and Mosher, Lee , TITLE =

Behrstock, Jason and Kleiner, Bruce and Minsky, Yair N. and Mosher, Lee , TITLE =. Geom. Topol. , FJOURNAL =. 2012 , NUMBER =. doi:10.2140/gt.2012.16.781 , URL =

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, TITLE =

Behrstock, Jason and Minsky, Yair N. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/jlms/jdr027 , URL =

work page doi:10.1112/jlms/jdr027 2011
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and Minsky, Yair N

Behrstock, Jason A. and Minsky, Yair N. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2008 , NUMBER =. doi:10.4007/annals.2008.167.1055 , URL =

work page doi:10.4007/annals.2008.167.1055 2008
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and Januszkiewicz, Tadeusz and Neumann, Walter D

Behrstock, Jason A. and Januszkiewicz, Tadeusz and Neumann, Walter D. , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2010 , NUMBER =. doi:10.4171/GGD/100 , URL =

work page doi:10.4171/ggd/100 2010
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and Neumann, Walter D

Behrstock, Jason A. and Neumann, Walter D. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2012 , PAGES =. doi:10.1515/crelle.2011.143 , URL =

work page doi:10.1515/crelle.2011.143 2012
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2008 , PAGES =

Bekka, Bachir and Harpe, Pierre de la and Valette, Alain , TITLE =. 2008 , PAGES =. doi:10.1017/CBO9780511542749 , URL =

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and Dranishnikov, A

Bell, G. and Dranishnikov, A. , TITLE =. Topology Appl. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.topol.2008.02.011 , URL =

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and Dranishnikov, A

Bell, G. and Dranishnikov, A. , TITLE =. Geom. Dedicata , FJOURNAL =. 2004 , PAGES =. doi:10.1023/B:GEOM.0000013843.53884.77 , URL =

work page doi:10.1023/b:geom.0000013843.53884.77 2004
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and Fujiwara, Koji , TITLE =

Bell, Gregory C. and Fujiwara, Koji , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2008 , NUMBER =. doi:10.1112/jlms/jdm090 , URL =

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Benjamini, Itai and Shamov, Alexander , TITLE =. Anal. Geom. Metr. Spaces , FJOURNAL =. 2015 , NUMBER =. doi:10.1515/agms-2015-0018 , URL =

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