{"id":"73723393-b955-48e5-ba70-635db2947c0c","arxiv_id":"2608.09535","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every RAAG embedding into certain hierarchically hyperbolic groups forces a clique-graph embedding of its defining graph into a newly constructed expanded core graph.","lead":"This paper introduces a graph called the expanded core graph for a class of groups called hierarchically hyperbolic groups, and shows that every embedding of a right-angled Artin group factors through an intermediate group that the graph can detect. The result gives a common combinatorial obstruction to such embeddings across mapping class groups, right-angled Artin groups, and virtually compact special groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The factorization rests entirely on Proposition 2.10, quoted from the authors' own [OP26, Theorem 3.9]; since the present paper corrects statements from that predecessor and does not reprove it, the main theorems inherit any gap in that theorem.","rationale":"I read the full manuscript. The internal structure is coherent: definitions are precise, the class Ξ is verified on the claimed examples, Lemma 4.5 and Theorem 4.8 follow formally from the quoted Proposition 2.10, and the mapping class group and RAAG applications are consistent with the Kim–Koberda framework. I found no direct contradiction inside the paper. The weak point is not the present text but the undocumented load-bearing piggyback on [OP26, Theorem 3.9]. The reader's weakest_assumption names exactly this. Since the paper explicitly corrects related assertions from the same predecessor, a conservative stance—conditional acceptance pending a complete proof or an independent expert verification of Proposition 2.10—is appropriate. I do not see reason to upgrade or downgrade the reader's conditional verdict.","tokens_in":38451,"tokens_out":20030,"duration_ms":175486,"concrete_test":"Request the full proof of Proposition 2.10 from [OP26, Theorem 3.9] and re-run the argument of Proposition 3.13 with that proof in hand, checking that geometric irredundance suffices for the algebraic part when supports are nested or transverse. As a minimal test, verify that two strongly fully supported axial elements with one support properly nested in the other have sufficiently large powers generating a rank-two free group in every model HHG (e.g., in the relatively hyperbolic HHG of Proposition 3.9(3) and in a rich-family cubical structure); failure anywhere would invalidate Theorem 4.8.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.10 is used as a black box in exactly the places where the paper's novelty lives: Proposition 3.13 converts irredundant strongly supported collections into undistorted RAAGs; Lemma 4.5 and Theorem 4.8 factor arbitrary RAAG embeddings through such a subgroup; Theorem 5.3's converse constructs Ap(Λ) from vertices of the expanded core graph; Lemma 3.14 and Proposition 5.8 use its commutation consequences. The statement is nontrivial: it asserts that for any geometrically irredundant family of fully supported axial elements, the graph prescribed by orthogonality of supports is realized as a RAAG by sufficiently large powers, with no accidental relations, and with undistortedness under a bounded-orbit condition. No proof is given here. The paper itself (Remark 3.10, Appendix B) explicitly retracts claims from the same predecessor line—[OP26, Remark 6.12] and [AB23, Lemma 3.4]—so the reader cannot treat [OP26, Theorem 3.9] as having been vetted by this text. If that theorem has a hidden hypothesis or a gap, Proposition 3.13, Lemma 4.5, Theorem 4.8, Theorem 5.1, and Theorem 5.3 all fail. This is the single most load-bearing assumption; all other axioms (Ξ, Ω, expanded core graph) are clearly defined and internally used consistently.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the expanded core graph for hierarchically hyperbolic group structures in two axiom classes, Ξ and Ω, and uses it to study embeddings of right-angled Artin groups. The main structural result, Theorem 4.8, states that after replacing the standard generators of an embedded RAAG by suitable positive powers, the embedding factors through an intermediate RAAG generated by strongly fully supported axial elements; for the class Ξ this intermediate RAAG is quasi-isometrically embedded. The authors then prove that the extension graph of this intermediate RAAG embeds into the expanded core graph, yielding a Kim–Koberda-type obstruction in the clique graph of the expanded core graph (Theorem 5.1) and a complete equivalence in rank at most two (Theorem 5.3). They identify the expanded core graph with the curve disjointness graph for mapping class groups and with the extension graph for natural rich-family structures on RAAGs, and derive chromatic obstructions. The paper also proves permanence properties for finite direct products and relatively hyperbolic constructions, and includes an appendix comparing the metric and hierarchical orthogonal stabilizers, with a counterexample to a lemma of Abbott–Behrstock.","tokens_in":38765,"tokens_out":7148,"duration_ms":70712,"significance":"If the main theorems are correct, the paper gives a genuinely unifying framework: it recovers the Kim–Koberda obstruction for RAAGs and mapping class groups as special cases of a single HHG mechanism, and it introduces a useful new combinatorial invariant, the expanded core graph. The paper is carefully written in many local respects: the structural lemmas in Section 3 are proved in detail, the examples in Section 5 are worked out, and Appendix B is candid about corrections to previous work and about the difference between metric and hierarchical stabilizers. The central caveat is that the engine of the whole paper, Proposition 2.10, is imported without proof from the authors' own preprint [OP26, Theorem 3.9], and the main theorems inherit any gap in that result. The paper also gives only a very compressed proof of the Ω-analogue of the extension-graph embedding, which is load-bearing for the mapping class group application.","major_comments":[{"comment":"The paper's central factorization and all of its main consequences rest on [OP26, Theorem 3.9], quoted as Proposition 2.10, which is neither proved nor even sketched here. This theorem asserts that powers of a geometrically irredundant collection of fully supported axial elements generate the RAAG prescribed by orthogonality of supports, and that under a bounded-orbit hypothesis the embedding is quasi-isometric. It is used essentially in Proposition 3.13, Lemma 4.5, Theorem 4.8, Theorem 5.1, and Theorem 5.3. Because the present paper explicitly corrects other statements from the same predecessor line in Remark 3.10 and Appendix B, the reader cannot treat [OP26, Theorem 3.9] as already vetted by this text. Please provide a proof of Proposition 2.10 in this paper, or state the main theorems as conditional on that external result with a precise page-and-theorem reference and a clear indication of where the proof can be found.","section":"Section 2, Proposition 2.10"},{"comment":"The proof of Proposition 5.8 is a blanket reference to the proofs of Lemmas 4.5 and 4.7, Theorems 4.8, 5.1, and 5.3, with 'strongly fully supported' replaced by 'fully supported'. This is load-bearing for the mapping class group application and for the Ω-version of the rank-two criterion. Under the weaker Ω axioms one no longer has the pointwise triviality on orthogonal coordinates supplied by Lemma 3.3, so the proof of Lemma 4.5 does not formally transfer; one must instead use the bounded-orbit clause in Definition 2.9 and weak commutativity in Definition 5.4(3b) to carry out the construction of the intermediate RAAG and the extension-graph embedding. Please give the analogue of Lemma 4.5 for Ω in detail, or at least provide a precise statement of the modified claim and the places where the weaker axioms replace the strong-support axioms.","section":"Section 5, Proposition 5.8"},{"comment":"The paper does not prove that the class Ξ is nonempty beyond the examples in Proposition 3.9, and for the rich-family compact-special example the orthogonal decomposition and commutative properties are imported from [AB23, Proposition 3.10(2)]. Since Appendix B explicitly gives a counterexample to [AB23, Lemma 3.4] and discusses why the main results of [AB23] survive, the paper should clarify exactly which statements from [AB23] are being used and verify directly that those statements apply to the present strengthened F_U-stabilizers setting. This is not a fatal issue, but it is a point where the reader needs to be able to check that the axioms of Ξ are actually satisfied by the claimed examples.","section":"Sections 3.2 and 5.2.1"}],"minor_comments":[{"comment":"The title and abstract contain typographical errors: 'OR THOGONALITY' should be 'ORTHOGONALITY', and the phrase 'er-graph' in the introduction appears to be a typo. Please proofread the text carefully.","section":"Title and Abstract"},{"comment":"The symbol G is used both for the ambient group and for the set of minimal unbounded domains in the definition of the expanded core graph. This is confusing in statements such as 'for every U∈G'; please use a different notation, for example script G or G_min.","section":"Definition 4.1"},{"comment":"The sentence 'suitable positive powers of finitely many elements lie in any prescribed finite-index subgroup' is standard but should be justified, since the powers must be chosen for finitely many elements simultaneously; a short argument using the finite index would make the lemma self-contained.","section":"Lemma 2.16"},{"comment":"The verification that the standard HHG structure on MCG(S) belongs to Ω relies on facts about pure mapping classes summarized in [OP26, §7.1]. Please state the precise facts used, especially the description of active domains of pure mapping classes, so that the reader can verify Definition 5.4(2) without consulting another preprint.","section":"Proposition 5.9"},{"comment":"The verification that the adjoined bounded domains B_{n,m} satisfy all axioms of an HHS, in particular the bounded geodesic image axiom and the realization axiom, is compressed. Please expand this verification, since the example is used to refute [AB23, Lemma 3.4] and to motivate the strengthened F_U-stabilizers property.","section":"Appendix B, Example B.2"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unproved reliance on [OP26, Theorem 3.9]. If that theorem is correct, the paper's central claims are plausible and the contribution is significant; however, the referee cannot certify the main theorems without access to a proof of that result. The editors may wish to ensure that [OP26] is available in a verifiable form, or that the authors include a proof in this paper. The Ω-results are used for the mapping class group, so the terse proof of Proposition 5.8 should be expanded before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper deserves a serious referee. It does what it claims: it transplants the Kim–Koberda RAAG embedding obstruction into hierarchically hyperbolic groups, introduces the expanded core graph and the classes Ξ and Ω, proves an intermediate-RAAG factorization, and recovers the mapping class group and RAAG cases as special instances. The proofs are detailed and the paper is honest in the right way: Remark 3.10 corrects a claim from the authors' own [OP26], and Appendix B gives a concrete counterexample to [AB23, Lemma 3.4] while explaining why the main results of [AB23] survive. That is how you handle predecessor issues.\n\nThe soft spot is real but not fatal. The central factorization Theorem 4.8 rests on Proposition 2.10, quoted as [OP26, Theorem 3.9], which is not reproved here. Since the paper itself corrects statements from that same preprint, a referee cannot take the theorem on faith; she needs to check [OP26] or ask for a proof or an independent verification. If that theorem has a gap, the obstruction and rank-two criterion inherit the failure. The classes Ξ and Ω are also strong: they are verified for compact special groups, mapping class groups, and the stated permanence constructions, but the title's general \"hierarchically hyperbolic groups\" framing overstates the actual scope. And the rank-two criterion is, of course, only for rank at most two; beyond that, the clique-graph obstruction is the best they get.\n\nOn the evidence in front of me, the local mathematics is careful and consistent. I found no direct contradiction in the main line of argument, and the recovery of the curve graph and the extension graph as expanded core graphs in the examples is a good sanity check.\n\nVerdict: send it to peer review. The referee needs to be someone willing to read [OP26] alongside this paper, and the authors should expect to be asked for a proof or a precise citation of Proposition 2.10. But the contribution is substantial enough to warrant that referee time.\n\nBest,\n[Your name]","headline":"Genuine HHG extension of Kim–Koberda with detailed proofs and honest self-corrections, but the main theorems lean on an unproved black-box theorem from the authors' own earlier preprint.","tokens_in":39258,"tokens_out":2088,"would_cite":true,"duration_ms":22478,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every RAAG embedding into a hierarchically hyperbolic group in the paper's class $\\Xi$ factors, after passing to powers, through a quasi-isometrically embedded RAAG generated by axial elements; RAAG embeddability then reduces to the…","keywords":["right-angled Artin groups","hierarchically hyperbolic groups","expanded core graph","extension graph","RAAG embedding obstructions","mapping class groups","compact special groups","quasi-isometric embeddings"],"falsifier":"Test the engine directly: find a hierarchically hyperbolic structure in class $\\Xi$ and a finite graph $\\Lambda$ such that $A(\\Lambda)$ embeds into $G$ but $\\Lambda$ does not embed as an induced subgraph of the clique graph of the expanded core graph; that refutes the main obstruction. More decisively, look for a geometrically irredundant family of strongly fully supported axial elements on pairwise orthogonal domains whose sufficiently large powers fail to generate the prescribed RAAG, which would contradict the quoted undistorted-RAAG theorem the entire factorization relies on.","tokens_in":38198,"feed_emoji":"🧩","tokens_out":13761,"duration_ms":109871,"temperature":0.7,"pith_summary":"This paper claims that, for a broad class of hierarchically hyperbolic groups—geometric groups that include mapping class groups and compact special groups—the question of which right-angled Artin groups embed is controlled by a single combinatorial object: the expanded core graph of the chosen hierarchical structure. The authors prove that every injective homomorphism from a right-angled Artin group $A(\\Lambda)$ into such a group, after replacing each standard generator by a positive power, factors through an intermediate RAAG generated by axial elements fully supported on minimal unbounded domains; in the class modelled on compact special groups the intermediate RAAG is quasi-isometrically embedded. The extension graph of that intermediate RAAG embeds into the expanded core graph, which yields a concrete obstruction: if $A(\\Lambda)$ embeds, then $\\Lambda$ must be an induced subgraph of the clique graph of the expanded core graph. When the structure has rank at most two, this becomes an equivalence. The paper also shows that, for mapping class groups, the expanded core graph is the disjointness graph of essential curves, and for RAAGs with natural rich-family structures it recovers the extension graph, so the result generalises the classical embedding obstructions while reproducing them in their original settings.","feed_headline":"One graph decides which right-angled Artin groups embed","feed_subtitle":"In rank-two hierarchically hyperbolic groups, RAAG embeddability is equivalent to an induced-subgraph condition in the expanded core graph","key_machinery":"The expanded core graph $\\widehat G_{\\mathcal S}$ has as its vertices the axial directions carried by minimal unbounded domains: one vertex for each such domain when its hyperbolic space is a quasi-line, and one vertex per inequivalent axial direction otherwise; two vertices are adjacent when their domains are orthogonal. The mechanism that makes the graph relevant is the orthogonal decomposition property: high powers of any infinite-order element split into pairwise commuting axial factors, each fully supported on one active domain. Passing to a finite geometrically irredundant subcollection and applying the quoted undistorted-RAAG theorem produces the intermediate RAAG; the identification of axial directions with vertices of $\\widehat G_{\\mathcal S}$ is what embeds its extension graph into $\\widehat G_{\\mathcal S}$.","core_discovery":"The central claim is structural. For an HHG structure $(G,\\mathcal S)$ in the class $\\Xi$—or, more generally, for a group virtually admitting such a structure—every injective homomorphism $\\phi: A(\\Lambda)\\to G$ can be modified by replacing each standard generator $v$ with $\\phi(v)^N$ so that the resulting embedding lands in a subgroup $M\\cong A(\\Gamma)$ generated by strongly fully supported axial elements; in the $\\Xi$ case $M$ is undistorted in $G$. The defining graph $\\Gamma$ records orthogonality among the supporting domains of a geometrically irredundant collection of axial directions. The paper then proves that the extension graph $\\Gamma^e$ embeds as an induced subgraph of the expanded core graph $\\widehat G_{\\mathcal S}$, so the classical clique-graph obstruction for RAAGs transfers verbatim: $\\Lambda \\le (\\widehat G_{\\mathcal S})^k$. When the structure has rank at most two, the defining graph of every intermediate RAAG is triangle-free, the stronger extension-graph criterion applies, and $A(\\Lambda)\\le G$ is equivalent to $\\Lambda\\le \\widehat G_{\\mathcal S}$. In the standard mapping class group structure, the expanded core graph is the disjointness graph of essential curves; in rich-family structures on a RAAG, it embeds into the extension graph and coincides with it when the rich family contains all singletons.","pith_inferences":["If the rank-two completeness persists in higher rank, or for wider families of defining graphs, the expanded core graph would become a full RAAG-subgroup invariant for the classes $\\Xi$ and $\\Omega$; the paper records this as an open direction.","The existence of the purely algebraic class $\\Omega$ suggests the quasi-isometric embedding conclusion is not needed for the obstruction itself; the same combinatorics may hold for HHG structures where undistortedness fails but decomposition and commutation still hold.","A concrete testable prediction is that for graph products or right-angled Coxeter groups carrying natural HHG structures, the expanded core graph should embed into (or coincide with) the extension graph, giving explicit RAAG embedding obstructions that can be computed from the defining graph."],"forward_implications":["Virtually compact special groups admit finite graphs $\\Lambda_M$ of arbitrarily large girth such that $A(\\Lambda_M)$ does not embed into $G$.","For the standard mapping class group structure, the obstruction takes the form $A(\\Lambda)\\le G \\Rightarrow \\Lambda\\le C(S)^k$, recovering the curve-graph clique obstruction for essential curves.","For a RAAG $A(\\Gamma)$ with a rich-family structure, the expanded core graph embeds into the extension graph $\\Gamma^e$, and coincides with it when the rich family contains all singleton subgraphs.","The classes $\\Xi_{\\mathrm{cc}}$ and $\\Omega$ are preserved under finite-index restriction, finite direct products, and the standard relatively hyperbolic construction, so the obstructions persist for groups built from these operations.","In rank at most two, $A(\\Lambda)\\le G$ if and only if $\\Lambda$ is an induced subgraph of $\\widehat G_{\\mathcal S}$, making the criterion complete rather than merely obstructional."],"supporting_citations":[{"why":"Supplies the central undistorted-RAAG theorem (quoted as Proposition 2.10) that turns irredundant strongly fully supported axial elements into the intermediate RAAG; without it the factorization does not go through.","marker":"[OP26]"},{"why":"Gives the extension-graph theorem and clique-graph obstruction for RAAG-into-RAAG embeddings, which the paper transfers to the ambient HHG.","marker":"[KK13]"},{"why":"Provides the mapping-class-group clique obstruction and the chromatic argument that the paper recovers as special cases.","marker":"[KK14]"},{"why":"Introduces the metric orthogonal stabilizers $G_U$ and the structural hypotheses that the class $\\Xi$ adapts and strengthens.","marker":"[AB23]"},{"why":"Provides the hierarchically hyperbolic group framework, distance formula, realization, and the relatively hyperbolic and direct-product constructions used throughout.","marker":"[BHS19]"},{"why":"Supplies the axial-elliptic dichotomy, boundary automorphism facts, and coarse surjectivity used to analyze elements and axial directions.","marker":"[DHS17]"},{"why":"Used for acylindrically hyperbolic actions to produce infinitely many pairwise independent loxodromic elements and to detect commutation via centralizers.","marker":"[Osi16]"},{"why":"Establishes rich-family HHG structures on compact special groups, one of the two main classes that belong to $\\Xi$.","marker":"[BHS17]"}],"fun_headline_variants":["Expanded core graph solves RAAG embedding in rank two","New graph criterion for right-angled Artin group embeddings","Rank-two HHG RAAG embeddings decided by a single graph","One graph captures RAAG embeddings in low-rank HHGs","Expanded core graph: RAAG embedding test for HHGs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on a previously proven theorem, quoted without reproof, that sufficiently large powers of a geometrically irredundant collection of strongly fully supported axial elements generate exactly the right-angled Artin group prescribed by orthogonality among their supporting domains; if that theorem fails, the factorization and every embedding obstruction built on it fail as well.","fun_headline_variants_meta":{"raw":{"variants":["Expanded core graph solves RAAG embedding in rank two","New graph criterion for right-angled Artin group embeddings","Rank-two HHG RAAG embeddings decided by a single graph","One graph captures RAAG embeddings in low-rank HHGs","Expanded core graph: RAAG embedding test for HHGs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000196,"raw_usage":{"total_tokens":1398,"prompt_tokens":1021,"completion_tokens":377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":637,"completion_tokens_details":{"reasoning_tokens":294}},"tokens_in":637,"tokens_out":377,"duration_ms":3523,"temperature":1.0,"reasoning_tokens":294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:28:11.548797+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the engine directly: find a hierarchically hyperbolic structure in class $\\Xi$ and a finite graph $\\Lambda$ such that $A(\\Lambda)$ embeds into $G$ but $\\Lambda$ does not embed as an induced subgraph of the clique graph of the expanded core graph; that refutes the main obstruction. More decisively, look for a geometrically irredundant family of strongly fully supported axial elements on pairwise orthogonal domains whose sufficiently large powers fail to generate the prescribed RAAG, which would contradict the quoted undistorted-RAAG theorem the entire factorization relies on.","supporting_citations":[],"review_version":1}