{"id":"e142d182-1a1c-4c8a-b3de-d9d135e4930c","arxiv_id":"2506.03377","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A contractible complex generalizing the McCullough-Miller space is constructed for pure symmetric automorphism groups of right-angled Artin groups, with applications to cohomological dimension and l2-cohomology.","lead":"The paper builds a new geometric object, a contractible complex called MM_Gamma, on which the pure symmetric outer automorphism group of any right-angled Artin group acts with free abelian stabilizers. If the construction and proof hold, it gives a unified way to compute cohomological dimensions and l2-Betti numbers for these groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.16's passage from W' back to W is unproved: the added b^k terms contribute nonzero length changes under a partial conjugation, and the k-choice step is circular as written.","rationale":"I agree with the reader that Lemma 6.16 is the load-bearing step. The surrounding contractibility argument would work if a strictly reductive partial conjugation is guaranteed; Propositions 7.8-7.11 are structurally coherent. The obstacle is the passage from W' back to W. The displayed equality is false: a partial conjugation changes the length of each added b^k from k to k+2, so the W'-reductivity differs from the W-reductivity by minus twice the support size. The proof does not provide the needed comparison. A second defect is the k-choice statement, which is circular as written because W' depends on k. Both defects are repairable: one can compute the 2|A| correction explicitly and choose k larger than ∥u0∥_W before forming W'. Because the repair is local and does not touch the rest of the argument, a conditional verdict is appropriate rather than rejection.","tokens_in":34255,"tokens_out":24977,"duration_ms":277633,"concrete_test":"Perform the exact length bookkeeping in Lemma 6.16. For α_1 = C_a^A and W' = W ∪ {b^k}, verify the identity red_W(α_1^{-1}) = red_{W'}(α_1^{-1}) + 2|A|, and then redo the k-choice argument by fixing k > ∥u0∥_W before defining W'. If the corrected computation fails to force red_W(α_1^{-1}) > 0, the Existence Lemma and hence the contractibility proof collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The contractibility proof depends on Lemma 6.16, which must produce a partial conjugation with red_W > 0. The proof adds W' = W ∪ {b^k}, peak-reduces β^{-1} with respect to W', and then concludes from a strict decrease in W'-length that red_W(α_1^{-1}) > 0. This transition is not justified by the displayed equality in Lemma 6.16. For a partial conjugation α_1 = C_a^A, the b^k terms do not have constant contribution: |b^k|_{α_1^{-1}(VΓ)} = |α_1(b^k)|_{VΓ} = k+2 for every b ∈ A, so red_{W'}(α_1^{-1}) = red_W(α_1^{-1}) - 2|A|. The displayed equality is therefore false as written, and it also shifts from α_1^{-1} to α_1. Strict decrease in W' does imply strict decrease in W because the correction is negative, but the proof never states or proves this comparison. In addition, the argument ruling out a non-partial first factor asks to choose k ≥ ∥v∥_{W'}, which is circular because W' depends on k. Since Lemma 7.9 and Proposition 7.10 invoke Lemma 6.16, Theorem B is incomplete as written; the gap appears repairable, but the current text does not contain the repair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the McCullough-Miller complex from free groups to arbitrary right-angled Artin groups. It introduces Γ-valid based partitions, the Γ-Whitehead poset Wh_Γ, and the complex MM_Γ as the simplicial realization of marked vertex types modulo the relation generated by carried automorphisms. Theorem A states that ΣPOut(A_Γ) acts on MM_Γ with free abelian stabilizers and fundamental domain |Wh_Γ|. The bulk of the paper (Sections 5–7) develops compatibility, refinement, disjunction, and reductivity, and uses Day's peak-reduction theorem to prove the main contractibility result (Theorem B). Section 8 applies the action to compute the cohomological dimension and the ℓ²-Betti numbers of ΣPOut(A_Γ).","tokens_in":34405,"tokens_out":11814,"duration_ms":126340,"significance":"If Theorem B is established, the paper provides a Deligne-type contractible complex for ΣPOut(A_Γ) with free abelian cell stabilizers, a genuine extension of the free-group McCullough-Miller construction. The stabilizer computation in Lemma 5.3 and the translation of the combinatorial framework to RAAGs are carried out in detail, and Example 6.13 correctly identifies a defect in the original McCullough-Miller Lemma 3.8. The paper also credits concurrent work by Corrigan and Abgrall for Theorem C, so the main novelty is the complex itself and its contractibility proof.","major_comments":[{"comment":"The displayed chain red_W(α_1^{-1}) = ∥u0∥_W − ∥α_1^{-1}(u0)∥_W = ∥u0∥_{W'} − ∥α_1(u0)∥_{W'} > 0 is not justified. For a partial conjugation α_1 = C_a^A, the W'-reductivity satisfies red_{W'}(α_1^{-1}) = red_W(α_1^{-1}) − 2|A|, because each generator b^k with b∈A is sent to a^{-1} b^k a and gains two letters. The equality with ∥u0∥_{W'} − ∥α_1(u0)∥_{W'} is therefore false as written, and it also shifts from α_1^{-1} to α_1. The strict inequality red_{W'}(α_1^{-1}) > 0 that the peak-reduction argument actually yields does imply red_W(α_1^{-1}) > 0, but only through the missing comparison red_{W'}(α_1^{-1}) = red_W(α_1^{-1}) − 2|A|; this step must be supplied.","section":"Section 6.1, Lemma 6.16"},{"comment":"The argument ruling out a non-partial first factor is circular as written. The proof defines W' = W ∪ {b^k | b∈VΓ} before peak reduction, and then, after obtaining α_1, invokes a choice k ≥ ∥v∥_{W'}, with v never defined. Because W' itself contains b^k, the quantity ∥v∥_{W'} depends on the same k that is being chosen; choosing k after the factorization is produced is not legitimate, since the peak-reduction factorization depends on k. A repair must fix k in advance, or use a bound independent of k, and then rule out a non-partial α_1.","section":"Section 6.1, Lemma 6.16"},{"comment":"The contractibility proof of Theorem B depends directly on Lemma 6.16. Lemma 7.9 invokes Lemma 6.16 to produce a strictly reductive partial conjugation P, and Proposition 7.10 uses Lemma 7.9 to prove contractibility of T. Consequently the gap in Lemma 6.16 is load-bearing: Theorem B is not established by the current text. If the repairs indicated in the two preceding comments are made, the remaining structure of the proof appears coherent.","section":"Section 7, Lemma 7.9 and Proposition 7.10"}],"minor_comments":[{"comment":"The first sentence says 'In this subsection we prove Theorem D', but the subsection proves Theorem C; this should be corrected.","section":"Section 8.1"},{"comment":"In the statement of Theorem D, the first displayed equality has an unbalanced parenthesis: dimNG(Hi(MMΓ;N (ΣPOut(AΓ)) needs a closing parenthesis before the equals sign.","section":"Theorem D"},{"comment":"In the proof of Theorem A, the phrase 'For the statement about centralizers' should read 'For the statement about stabilizers'.","section":"Section 5, proof of Theorem A"},{"comment":"The sentence 'The complex MMΓ is the union of the starts of the nuclear vertices' should read 'the union of the stars of the nuclear vertices'.","section":"Section 7.2"},{"comment":"In the displayed calculation for αβ(c), the expression appears to interchange the roles of a and b: the definition gives the value b a c a^{-1} b^{-1} rather than a b c b^{-1} a^{-1} (depending on the convention for composing automorphisms). Please check and correct this display.","section":"Lemma 5.4"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper overlaps with Corrigan's concurrent work for Theorem C, but the authors are transparent about this. The main issue is the incomplete proof of Lemma 6.16; the gap appears local and repairable, so I recommend major revision rather than rejection. If the k-choice circularity cannot be repaired, the central contractibility claim would be in doubt."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a natural and worthwhile generalization of McCullough-Miller to RAAGs. The Gamma-Whitehead poset, the marked equivalence relation, and the complex MM_Gamma are genuinely new, and the paper is honest about Corrigan's independent proof of Theorem C. But the proof of Theorem B leans on Lemma 6.16 (the Existence Lemma), and the proof of that lemma has a gap you need to know before citing it.\n\nWhat is good. Theorem A is proven cleanly: Lemma 5.3 gives free abelian stabilizers, the compatibility analysis in Section 5 is careful, and the Fundamental Domain description is convincing. The factorization lemma and the adjacency-counter formalism are sensible, and Example 6.13 is a real service: it identifies a genuine problem in McCullough-Miller's original Lemma 3.8. The citation practice is honest: Theorem C is explicitly credited to Corrigan, and the relation to Abgrall's work is stated.\n\nWhere it slips. Lemma 6.16. The proof adds W' = W ∪ {b^k | b in VΓ} and peak-reduces beta^{-1} with respect to W'. It wants to show the first factor alpha_1 is a strictly reductive partial conjugation. Two things are not justified as written. First, the displayed equality red_W(alpha_1^{-1}) = ||u0||_W - ||alpha_1^{-1}(u0)||_W = ||u0||_{W'} - ||alpha_1(u0)||_{W'} does not follow: it shifts from alpha_1^{-1} to alpha_1, and it treats the added b^k words as if they canceled uniformly. For a partial conjugation C_a^A, each b in A contributes -2 to red_{W'}, so the correct relation is red_W(alpha_1^{-1}) = red_{W'}(alpha_1^{-1}) + 2|A|. That inequality would actually save the implication from red_{W'} > 0 to red_W > 0, but the proof never states or proves it. Second, the argument that alpha_1 must be a partial conjugation asks to choose k >= ||v||_{W'}, which is circular because W' depends on k; the peak reduction is done after W' is fixed, and you cannot then choose k to rule out the first factor.\n\nWhere that leaves us. Lemma 7.9 and Proposition 7.10 invoke Lemma 6.16, so the contractibility proof is incomplete as written. The gap looks repairable, but the repair is not in the paper. Theorem C and Theorem D are mostly independent of this gap, and Theorem C is already covered by Corrigan.\n\nWho this is for. Geometric group theorists working on RAAG automorphism groups, cohomological dimension, and ell^2 invariants. It deserves a serious referee: the construction and the program are solid, and the flaw is local and plausibly fixable. I would send it to review, but I would not cite Theorem B until the Existence Lemma is repaired.","headline":"A valuable RAAG analogue of McCullough–Miller with a real gap in the Existence Lemma; Theorem B is unproved as written but the construction deserves a careful referee.","tokens_in":35082,"tokens_out":5568,"would_cite":false,"duration_ms":59422,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20J06","20F36","57M07","55P20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper builds a contractible complex for pure symmetric automorphism groups of every right-angled Artin group, and uses it to compute cohomological dimension and ℓ²-Betti numbers.","keywords":["right-angled Artin groups","pure symmetric automorphisms","MM-type complex","Whitehead poset","partial conjugations","cohomological dimension","ℓ2-Betti numbers"],"falsifier":"One concrete check is to compute, for the edgeless graph with four vertices and the word $w = x^2 b x^{-2} c$ used in the paper's Example 6.13, whether a peak-reduction factorization of a height-lowering $\\beta^{-1}$ with respect to $W' = W \\cup \\{b^k\\}$ can have a first factor $\\alpha_1$ with $\\mathrm{red}_{W'}(\\alpha_1^{-1}) > 0$ but $\\mathrm{red}_W(\\alpha_1^{-1}) \\leq 0$; if such a factor exists, the displayed equality in Lemma 6.16 fails and the contractibility proof needs a missing argument.","tokens_in":33925,"feed_emoji":"📐","tokens_out":17846,"duration_ms":179423,"temperature":0.7,"pith_summary":"Right-angled Artin groups interpolate between free groups and free abelian groups, and their pure symmetric outer automorphism groups $\\Sigma\\mathrm{POut}(A_\\Gamma)$ generalize the objects studied for free groups. The paper's central claim is that for every finite simplicial graph $\\Gamma$ there is a simplicial complex $\\mathrm{MM}_\\Gamma$, built from the $\\Gamma$-Whitehead poset, on which $\\Sigma\\mathrm{POut}(A_\\Gamma)$ acts with free abelian cell stabilizers and with fundamental domain the simplicial realization of the Whitehead poset, and that $\\mathrm{MM}_\\Gamma$ is contractible. The value of that claim is that these groups gain a Deligne-type complex, the kind of object that has powered cohomological-dimension, cohomology-ring, and $\\ell^2$-Betti-number computations for free groups. The paper also proves a formula for the cohomological dimension of $\\Sigma\\mathrm{POut}(A_\\Gamma)$ in terms of the rank of Whitehead elements, and matches von Neumann dimensions of the complex's cohomology with those of the non-nuclear part of the Whitehead poset.","feed_headline":"Contractible complex found for all RAAG automorphism groups","feed_subtitle":"Free abelian stabilizers turn the Whitehead poset into the group's cohomology data.","key_machinery":"The central object is the $\\Gamma$-Whitehead poset $\\mathrm{Wh}_\\Gamma$: its elements are $\\Gamma$-vertex types, i.e. families of pairwise compatible based partitions of the generator set whose petals are unions of connected components of $\\Gamma$ minus the star of the operative generator. The complex $\\mathrm{MM}_\\Gamma$ is the simplicial realization of the poset of marked vertex types, equivalently the coset poset of the stabilizers of marked types. The proof that it is contractible proceeds by adding stars of nuclear vertices in increasing height, measured by cyclic word lengths of a fixed finite word set; the notion of reductivity records how much an automorphism decreases that height. The technical engine is an adjacency-counter formula for the length change of a partial conjugation, a peak-reduction statement for automorphisms of right-angled Artin groups, and an Existence Lemma producing a strictly reductive partial conjugation when height is not minimal. The final homotopy equivalences come from refinement and disjunction operations on based partitions and from standard poset lemmas for cone decompositions.","core_discovery":"The paper's discovery is that the classical free-group complex construction survives when the graph $\\Gamma$ is arbitrary, provided the petals of the partitions are required to be unions of connected components of $\\Gamma - \\mathrm{st}(a)$, and compatibility is defined through a crossing condition. With this added structure, the marked poset is still a coset poset for stabilizers generated by partial conjugations, and the stabilizers are free abelian. Theorem A states that $\\Sigma\\mathrm{POut}(A_\\Gamma)$ acts on $\\mathrm{MM}_\\Gamma$ with free abelian cell stabilizers and a strong fundamental domain. Theorem B, the main result, asserts that $\\mathrm{MM}_\\Gamma$ is contractible, making it a Deligne-type complex for $\\Sigma\\mathrm{POut}(A_\\Gamma)$. The paper then proves that the cohomological dimension of $\\Sigma\\mathrm{POut}(A_\\Gamma)$ equals the maximum rank $r(A)$ over the $\\Gamma$-Whitehead poset, realized by a free abelian subgroup, and that the von Neumann dimensions of the cohomology of $\\mathrm{MM}_\\Gamma$ match those determined by the non-nuclear part of the Whitehead poset, with a stronger $\\ell^2$-cohomology statement under an extra centralizer condition.","pith_inferences":["Because all cell stabilizers are free abelian, the action makes $\\mathrm{MM}_\\Gamma$ a natural model for the classifying space for abelian subgroups of $\\Sigma\\mathrm{POut}(A_\\Gamma)$, so it can feed equivariant cohomology computations beyond ordinary cohomological dimension.","The height filtration used in the contractibility proof is a natural place to attach the homological finiteness invariants of the group, in analogy with how the free-group complex was used for those invariants; the paper does not pursue this.","The extra centralizer condition that separates the two parts of Theorem D is already noted in the paper to fail in general, so the weaker von Neumann-dimension statement is likely the right general formulation; identifying exactly which graphs satisfy the stronger condition would be a concrete follow-up.","A comparison with cube-complex models for the larger symmetric outer automorphism group would clarify whether the poset-based model and the cube-based model carry the same equivariant cohomology."],"forward_implications":["The group $\\Sigma\\mathrm{POut}(A_\\Gamma)$ gets a contractible complex with free abelian stabilizers, so $\\mathrm{MM}_\\Gamma$ can serve as a Deligne-type model for computing its cohomology.","The cohomological dimension of $\\Sigma\\mathrm{POut}(A_\\Gamma)$ is exactly the maximum rank $r(A)$ over the $\\Gamma$-Whitehead poset, and this dimension is realized by a free abelian subgroup.","The von Neumann dimension of $H^i(\\mathrm{MM}_\\Gamma; \\mathcal{N}(\\Sigma\\mathrm{POut}(A_\\Gamma)))$ equals that of $\\mathcal{N}(\\Sigma\\mathrm{POut}(A_\\Gamma)) \\otimes H^{i-1}(|\\mathrm{Wh}_\\Gamma^0|)$, and under the centralizer condition the $\\ell^2$-cohomology decomposes as $\\ell^2(G) \\otimes H^{i-1}(|\\mathrm{Wh}_\\Gamma^0|)$.","For an edgeless graph, $\\mathrm{MM}_\\Gamma$ specializes to the classical free-group complex, so the results reproduce the known statements for $\\Sigma\\mathrm{POut}(F_n)$.","Since every right-angled Artin group is isomorphic to $\\Sigma\\mathrm{POut}(A_\\Delta)$ for some graph $\\Delta$, the construction gives a uniform Deligne-type model covering all right-angled Artin groups."],"supporting_citations":[{"why":"Supplies the original free-group complex construction and the overall contractibility strategy being generalized.","marker":"[19]"},{"why":"Supplies the peak-reduction and adjacency-counter lemmas for automorphisms of right-angled Artin groups used in the reductivity arguments.","marker":"[10]"},{"why":"Establishes that partial conjugations generate the group, which underlies carriers, stabilizers, and rank.","marker":"[17]"},{"why":"Gives the finite presentation whose commutativity relations are used to identify free abelian stabilizers.","marker":"[24]"},{"why":"Refines the presentation of the group, providing the relations used in the compatibility arguments.","marker":"[16]"},{"why":"Provides the poset-lemma criteria that turn refinements and disjunctions into homotopy equivalences.","marker":"[20]"},{"why":"Provides the framework for computing $\\ell^2$-Betti numbers from an action on a contractible complex with cone fundamental domain, used in Theorem D.","marker":"[18]"},{"why":"Supplies the cohomological-dimension bound for group actions on acyclic complexes used in Theorem C.","marker":"[4]"},{"why":"Gives the free-group cohomological-dimension theorem that Theorem C generalizes.","marker":"[5]"}],"fun_headline_variants":["McCullough-Miller complex generalized to all RAAGs","Contractible complex for all RAAG automorphism groups","RAAG automorphism groups get Deligne-type complex","Cohomology of RAAG automorphism groups via new complex","Free abelian stabilizers in RAAG automorphism complex"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The contractibility proof rests on the Existence Lemma, which assumes that after adjoining powers $b^k$ of every generator to the word set and peak-reducing a height-lowering automorphism, the first factor is a partial conjugation whose length decrease is already positive on the original word set; the displayed equality comparing the two length functions is not justified, because the extra $b^k$ powers can change by unequal amounts under partial conjugations.","fun_headline_variants_meta":{"raw":{"variants":["McCullough-Miller complex generalized to all RAAGs","Contractible complex for all RAAG automorphism groups","RAAG automorphism groups get Deligne-type complex","Cohomology of RAAG automorphism groups via new complex","Free abelian stabilizers in RAAG automorphism complex"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000633,"raw_usage":{"total_tokens":2881,"prompt_tokens":867,"completion_tokens":2014,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1932}},"tokens_in":483,"tokens_out":2014,"duration_ms":13980,"temperature":1.0,"reasoning_tokens":1932,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:05:45.889322+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to compute, for the edgeless graph with four vertices and the word $w = x^2 b x^{-2} c$ used in the paper's Example 6.13, whether a peak-reduction factorization of a height-lowering $\\beta^{-1}$ with respect to $W' = W \\cup \\{b^k\\}$ can have a first factor $\\alpha_1$ with $\\mathrm{red}_{W'}(\\alpha_1^{-1}) > 0$ but $\\mathrm{red}_W(\\alpha_1^{-1}) \\leq 0$; if such a factor exists, the displayed equality in Lemma 6.16 fails and the contractibility proof needs a missing argument.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the peak-reduction and adjacency-counter lemmas for automorphisms of right-angled Artin groups used in the reductivity arguments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that partial conjugations generate the group, which underlies carriers, stabilizers, and rank."},{"cited_title":"Quillen,Homotopy properties of the poset of nontrivialp-subgroups of a group, Adv","cited_arxiv_id":null,"evidence_quote":"Provides the poset-lemma criteria that turn refinements and disjunctions into homotopy equivalences."},{"cited_title":"McCammond and J","cited_arxiv_id":null,"evidence_quote":"Provides the framework for computing $\\ell^2$-Betti numbers from an action on a contractible complex with cone fundamental domain, used in Theorem D."},{"cited_title":"BrownCohomology of Groups, Graduate Texts in Mathematics, Springer, 1982","cited_arxiv_id":null,"evidence_quote":"Supplies the cohomological-dimension bound for group actions on acyclic complexes used in Theorem C."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the free-group cohomological-dimension theorem that Theorem C generalizes."}],"review_version":1}