{"id":"90d7247c-06b3-45d6-b8e6-284057f26709","arxiv_id":"2411.15434","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"2-dimensional Shephard groups act cocompactly on a CAT(0) complex, are acylindrically hyperbolic and relatively hyperbolic, and this yields residual finiteness for many 2-dimensional Artin groups.","lead":"This paper studies 2-dimensional Shephard groups, quotients of Artin groups where standard generators have finite order. It constructs a CAT(0) complex for these groups and derives that a broad class of 2-dimensional Artin groups are residually finite, a property that was open for most such groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.3's key geometric claim—that a simple loop with non-multiple runs must enclose an eac-cell with consistent orientation—is stated without proof and is load-bearing for Theorem B; if it fails, the CAT(0) construction collapses.","rationale":"The reader's weakest assumption identifies exactly the step I would stress-test: Proposition 4.3 is the root of the main CAT(0) construction, and its proof relies on a geometric assertion about simple loops enclosing e_ac-cells that is not fully justified. I agree with the conditional verdict: the paper is substantial, the argument is mostly coherent, and I found no explicit counterexample, but the gap sits close enough to the central claim that the proof should be completed before the results are taken as fully established. I did not find grounds for REJECT, since the assertion is plausible: in a graph of cyclic factors, a simple cycle avoiding p- and r-multiple runs must use the (ac)^q relation to close, and the Euclidean enumeration proposed above is a practical way to test the claim before investing in a full proof. Therefore the appropriate verdict remains conditional, unchanged from the reader's assessment.","tokens_in":33530,"tokens_out":25516,"duration_ms":264507,"concrete_test":"Enumerate simple cycles of length < 2q in the Euclidean case Sh(3,6,3): take the Cayley graph of Delta(3,3,3) (the 1-skeleton of the tiling T(3,6,3)), list all simple cycles of length at most 11, and record for each cycle the maximal a-run and c-run lengths and the number of enclosed hexagons (the cells mapping to e_ac). If any cycle has every run length congruent to 1 or 2 modulo 3 but encloses zero hexagons, the geometric step in Prop 4.3 is false; if every such cycle encloses at least one hexagon, the step is verified in the base Euclidean case, and the gap can likely be patched by a planar Euler-characteristic argument for general (p,q,r).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.3 is the linchpin of Theorem B: Theorem 5.8 reduces the CAT(1) condition at an edge-vertex to the assertion that every embedded closed loop in bTheta_Lambda has edge length at least 2m_ij, and that assertion is exactly Prop 4.3. The proof of Prop 4.3 contains one genuinely unproved geometric step: after choosing a 2-chain R in C_2 with d_2(R) = rho, it is asserted that simplicity of the lifted loop, together with 0 < i_j < p_{s_j}, implies that the disk bounded by the loop contains a cell mapped to e_ac, and that all such cells are traversed with the same orientation, so n_ac is nonzero. This is not a purely algebraic consequence: R is not unique, since one may add the nonzero 2-cycle described in Lemma 3.1 and change n_ac, so the proof must rely on the specific planar disk enclosed by the loop. The Jordan-curve/enclosed-cell argument is only sketched for a Cayley 2-complex that is built from the planar tiling T(p,2q,r) by attaching extra cells. If this geometric claim fails, the syllable-length bound fails, and with it the CAT(0) development in Theorem B and the downstream Theorems C, D, F, and G.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies 2-dimensional Shephard groups, quotients of 2-dimensional Artin groups in which each standard generator is killed by a finite power. The author first analyses dihedral Shephard groups: Theorem A exhibits each infinite Sh(p,q,r) as a Z-central extension of an infinite triangle group (or a finite-index subgroup thereof) with infinite Euler class, yielding non-CAT(0), virtual nilpotency in the Euclidean case, and commensurability to a uniform lattice in the universal cover of PSL2(R) in the hyperbolic case. The central construction is Theorem B: for every 2-dimensional extended presentation graph Γ, the edge parabolics embed and ShΓ acts cocompactly on a piecewise Euclidean CAT(0) complex ΘΓ whose cell stabilizers are conjugates of spherical-type Shephard subgroups. The remaining results derive global consequences: acylindrical hyperbolicity (Theorem C), relative hyperbolicity (Theorem D), and residual finiteness of a broad class of 2-dimensional Artin groups (Theorem G), the latter via a geodesic-preserving quotient argument from the Deligne complex to ΘΓ.","tokens_in":33771,"tokens_out":11478,"duration_ms":116328,"significance":"If correct, Theorem B is a substantial structural result: it gives a Deligne-complex analogue for 2-dimensional Shephard groups and transfers the machinery of CAT(0) geometry to groups with torsion in the generating set. The consequences (relative hyperbolicity of Shephard groups where the associated Artin groups are typically not relatively hyperbolic, and residual finiteness for many 2-dimensional Artin groups) are novel and interesting. The paper is largely self-contained: Sections 2 and 3 give a coherent cohomological and geometric description of dihedral Shephard groups, and the arguments are modular. The main risk is concentrated in a small number of geometric lemmas: the syllable-length bound (Prop. 4.3), the convexity and fine-graph estimate (Lemma 7.5), and the hyperbolicity criterion (Lemma 7.3). These lemmas are load-bearing for Theorem B and for the later theorems, so they must be fully repaired before the paper can be accepted.","major_comments":[{"comment":"In the proof of Proposition 4.3, after reducing to a simple loop γ0, the paper asserts that the lifted simple loop must enclose at least one cell of the Cayley 2-complex projecting to e_ac and that every such cell is traversed with a consistent positive orientation, so n_ac ≠ 0. This is the only step that rules out w0 = e, and it is not justified: the 2-chain R with d2(R) = ρ is not canonical, since Lemma 3.1 supplies a nonzero 2-cycle that can be added to R and changes n_ac. The argument therefore must use the specific planar disk enclosed by the simple loop and compare the planar tiling T(p,2q,r) with the Cayley 2-complex obtained by attaching extra cells. The manuscript does not provide that comparison. Since Proposition 4.3 is exactly the input used in Theorem 5.8 to bound the girth of bΘΛ by 2m_ij, this gap is load-bearing for Theorem B and hence for Theorems C, D, F, and G.","section":"§4, Proposition 4.3 (proof)"},{"comment":"The proof of Lemma 7.5 asserts without proof that the closed star St(x) is convex because it is built from Euclidean right triangles with an acute angle at x. A union of triangles around a vertex is not convex in general, even if the local angle at x is acute; convexity requires control of the total angle around x and of the way the triangles are glued. The subsequent construction of the closest-point projection ρ onto St(x) and the inequality ℓ_x(y,z) ≤ C d_x(y,z) + D depend on this convexity. If St(x) is not convex, the comparison between the induced metric on ∂St(x) and the graph distance fails, and with it the proof that Y is fine in Theorem D.","section":"§7, Lemma 7.5 (proof)"},{"comment":"Lemma 7.3 applies the Flat Plane Theorem to the CAT(0) complex ΘΓ to conclude hyperbolicity from the absence of an isometrically embedded plane. The standard formulations of that theorem require additional hypotheses, typically geodesic completeness or properness of the space and/or properness of the action, and none of these is verified for ΘΓ. In fact, when an edge group She is infinite, the stabilizer of the corresponding vertex is infinite, so the action is not proper. The subsequent assertion that a flat plane \"must be a subcomplex\" is also not automatic for an isometric copy of E^2 in a CAT(0) cell complex. Since Lemma 7.3 is the only bridge from non-positive curvature to hyperbolicity of ΘΓ used in Theorem D, this needs a complete justification.","section":"§7, Lemma 7.3 (proof)"}],"minor_comments":[{"comment":"The sentence \"the syllable length of this word is n = ℓ(γ\" is missing a closing parenthesis; it should read \"n = ℓ(γ)\".","section":"§5, Theorem 5.8 proof"},{"comment":"The sentence \"This implies in particular that α ∉ C; otherwise...\" is logically garbled. What is needed is that the image of α remains outside the image of C in the quotient, i.e. ρ(α) ∉ ρ(C), which follows from φ(α) ∉ φ(C) because φ factors through the quotient map. The argument should be rewritten accordingly.","section":"§8, Lemma 8.6 proof"},{"comment":"In the proof of Corollary E, the phrase \"the peripheral subgroups of ShΛ\" is undefined; it should refer either to the peripheral subgroups of ShΓ or to the edge groups She corresponding to edges e of Γ.","section":"§7, Corollary E proof"},{"comment":"In the proof of Theorem D, the symbol VK(x) should be V_x(Y), and earlier in the same proof \"VK(x) is locally finite\" repeats the same typo.","section":"§7, Theorem D proof"},{"comment":"The notation \"When h = 1, M ∼= Z2\" is ambiguous: the torsion-free finite-index subgroup M is isomorphic to Z^2, not to the cyclic group Z/2Z. This should be clarified for the reader.","section":"§3.2, after Proposition 3.7"}],"recommendation":"major_revision","confidential_remarks":"The paper depends on the author's earlier preprint [Gol23] for the CAT(1) property of finite edge complexes and for the finite dihedral cases. The editor may wish to confirm the publication status of [Gol23]; if it remains unpublished, the dependency should be made explicit and the needed finite cases should be reproved or cited with a full statement. The three major comments above are the main obstacles to acceptance; if they can be repaired, the paper would make a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a serious piece of work. For 2-dimensional Shephard groups it constructs a CAT(0) complex analogous to the Deligne complex (Theorem B), proves a clean dihedral classification (Theorem A), and derives genuinely new consequences: relative hyperbolicity (Theorem D) and residual finiteness for a broad class of 2-dimensional Artin groups (Theorem G). The central-extension view of dihedral Shephard groups is elegant and well executed, and the applications are substantial. If the main theorem holds, this is an important advance, not an incremental one.\n\nThe soft spot is exactly where the stress-test puts it. Proposition 4.3 is the load-bearing syllable-length lemma, and its proof contains one asserted geometric step: after lifting the closed loop to the Cayley 2-complex, it is claimed that simplicity plus the exponent bounds forces the enclosed disk to contain a cell mapping to e_ac, and that all such cells are traversed with the same orientation, giving n_ac nonzero. That is not a purely algebraic consequence—the 2-chain R is not unique, since one can add the cycle from Lemma 3.1. The argument needs the actual planar disk bounded by the loop, not an arbitrary chain. This is a genuine gap as written, and since Theorem B and everything downstream depends on it, the paper is not yet in final form.\n\nThe other issues are minor by comparison. Lemma 7.5 asserts convexity of closed stars without proof; in this piecewise-Euclidean setting it may be true, but it needs justification. Lemma 7.3 invokes the Flat Plane Theorem; the space is CAT(0) and the action is cocompact, but the completeness/geodesic-completeness point should be stated explicitly.\n\nNone of this looks fatal. The surrounding machinery is detailed, the central-extension computations are concrete, and the intended geometric argument for Prop 4.3 is plausible given the tiling picture. The paper is not sloppy; it is a strong draft with one load-bearing step that needs to be filled in.\n\nI would send this to a serious referee. The verdict should be conditional/major revision: ask for a complete proof of the geometric claim in Prop 4.3 and a word on the convexity of closed stars. If those are supplied, the results are likely correct and significant. This is a paper for geometric group theorists working on Artin/Shephard groups and CAT(0) geometry.","headline":"A substantial, mostly careful paper whose main CAT(0) theorem rests on one currently unproved geometric claim in Prop 4.3; worth serious refereeing, but not ready as is.","tokens_in":34321,"tokens_out":1901,"would_cite":true,"duration_ms":21418,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","20F36","20F55","20E26"],"pacs":[],"model":"deepseek-v4-flash","headline":"A 2-dimensional Shephard group can fail to be CAT(0) while still acting cocompactly on a CAT(0) complex built from its spherical-type subgroups.","keywords":["Shephard groups","Artin groups","CAT(0) complexes","Deligne complex","relative hyperbolicity","acylindrical hyperbolicity","residual finiteness","central extensions"],"falsifier":"A direct computer search for a cyclically reduced word of syllable length $< 2q$ in $Sh(3,6,4)$ that equals the identity, with exponents not multiples of the generator orders, would refute Proposition 4.3 and with it the CAT(0) construction; no such word should exist if the paper is right.","tokens_in":33288,"feed_emoji":"📐","tokens_out":12136,"duration_ms":105402,"temperature":0.7,"pith_summary":"The paper investigates Shephard groups — quotients of 2-dimensional Artin groups obtained by forcing the standard generators to have finite order — and shows that the CAT(0) geometry of the Artin world survives only in a qualified form. A dihedral Shephard group $Sh(p,q,r)$ with $1/p+2/q+1/r\\le 1$ is never CAT(0), and any 2-dimensional Shephard group containing such an edge subgroup inherits that obstruction. Yet Theorem B gives every 2-dimensional Shephard group a cocompact action on a piecewise Euclidean CAT(0) complex $\\Theta_\\Gamma$ built from its spherical-type subgroups, namely the subgraphs whose associated Coxeter group is finite. This complex supports acylindrical hyperbolicity, relative hyperbolicity for hyperbolic-type graphs, and — as an application back to Artin groups — residual finiteness for a broad class of 2-dimensional Artin groups.","feed_headline":"Every 2D Shephard group acts cocompactly on a CAT(0) complex","feed_subtitle":"The new complex yields relative and acylindrical hyperbolicity, and residual finiteness for many Artin groups.","key_machinery":"The load-bearing object is the dihedral Shephard group $Sh(p,q,r)$ together with its coset geometry $\\widehat{\\Theta}(p,q,r)$. Each infinite such group is realized as a $\\mathbb{Z}$-central extension of a triangle group, or of a finite-index subgroup of one, whose Euler class has infinite order; this realizes the group as a uniform lattice in $\\mathrm{Isom}(\\widetilde{SL_2\\mathbb{R}})$ when $1/p+2/q+1/r<1$. The proof of Theorem B then rests on Proposition 4.3: a cyclically reduced word equal to the identity, with no exponent a multiple of the corresponding generator order, must have syllable length at least $2q$. That bound makes every local development of the complex of groups CAT(1), hence the global development $\\Theta_\\Gamma$ is CAT(0) by the standard complex-of-groups criterion.","core_discovery":"The central claim is Theorem B: for every 2-dimensional extended presentation graph $\\Gamma$, each edge subgroup embeds in $Sh_\\Gamma$, and $Sh_\\Gamma$ acts cocompactly on a CAT(0) piecewise Euclidean cell complex $\\Theta_\\Gamma$ whose cell stabilizers are the conjugates of $Sh_\\Lambda$ for spherical-type subgraphs $\\Lambda$. The construction mirrors the Deligne complex for Artin groups, using cosets of spherical-type subgroups as vertices. The negative counterpart, Theorem A, says a dihedral Shephard group $Sh(p,q,r)$ with $1/p+2/q+1/r\\le 1$ cannot act properly by semi-simple isometries on any CAT(0) space: at equality it is commensurable to the 3-dimensional integer Heisenberg group, and in the hyperbolic case to a universal central extension of a hyperbolic surface group, realized as a uniform lattice in $\\mathrm{Isom}(\\widetilde{SL_2\\mathbb{R}})$. The paper's interpretation is that infinite dihedral edge subgroups act as 'poison subgroups' for CAT(0) geometry, yet they are exactly the stabilizers appearing in the new complex, so the construction locates the obstruction rather than removing it.","pith_inferences":["Going beyond the paper, Lemma 8.7's persistence of geodesics under the quotient $\\Phi_\\Gamma\\to\\Theta_{\\Gamma(k)}$ suggests a wider route to residual finiteness: any family of geodesics that survives such quotients transfers residual finiteness from Shephard groups back to Artin groups, so the class in Theorem G is probably not maximal.","The paper proves only that word-hyperbolicity of $W_\\Gamma$ forces $\\Theta_\\Gamma$ to be hyperbolic; if an isometric embedding of the Davis complex into $\\Theta_\\Gamma$ exists, the converse would follow and the relative-hyperbolicity theorem would become a sharp analogue of the corresponding Artin-group result rather than one direction.","The 'poison subgroup' reading suggests a recognition principle the paper does not state: a 2-dimensional Shephard group is CAT(0) only if every embedded infinite dihedral edge subgroup has at least one generator of infinite order, since the finite-label examples show the obstruction is local and hereditary."],"forward_implications":["Every edge subgroup of a 2-dimensional Shephard group embeds, so infinite dihedral edge subgroups are provably poison subgroups and the CAT(0) obstruction of Theorem A applies to any 2-dimensional Shephard group containing one.","For irreducible 2-dimensional Shephard groups with at least three generators and an infinite edge subgroup in every connected component, the cocompact CAT(0) action yields acylindrical hyperbolicity (Theorem C).","For hyperbolic-type 2-dimensional graphs, the action on $\\Theta_\\Gamma$ gives relative hyperbolicity with the infinite spherical-type edge groups as peripheral subgroups; if all edge groups are finite, $Sh_\\Gamma$ is hyperbolic (Theorem D).","Relative hyperbolicity transfers solvable word problem, the Tits alternative, finite asymptotic dimension, and the rapid decay property to these Shephard groups, with biautomaticity unless some edge satisfies $1/p_i+2/m_{ij}+1/p_j=1$ (Corollary E).","For triangle-free graphs with no 4-cycle all of whose edges are labeled 2, both the Shephard group and the corresponding Artin group are residually finite (Corollary F and Theorem G)."],"supporting_citations":[{"why":"Supplies the syllable-length method for dihedral Artin groups that Proposition 4.3 adapts to the torsion setting.","marker":"[AS83]"},{"why":"Provides the Deligne-complex construction and the link-metric criteria used to show the development is CAT(0).","marker":"[CD95]"},{"why":"Proved CAT(0) for the Coxeter-like Shephard groups and handled the finite dihedral case; this paper extends beyond that class.","marker":"[Gol23]"},{"why":"Gives the CAT(0) acylindrical-hyperbolicity criterion whose proof Theorem C modifies for Shephard groups.","marker":"[Vas22]"},{"why":"Contains the embedded-loop argument that Proposition 4.3 follows and the link characterization used for hyperbolic-type graphs.","marker":"[Cri05]"},{"why":"Supplies the complex-of-groups development theorem, CAT(1)-link criteria, and fixed-point results used throughout.","marker":"[BH13]"},{"why":"Gives the torsion-free finite-index subgroup of the triangle group and the central-extension presentation underlying Theorem A.","marker":"[Mil75]"},{"why":"Provides the lens-space cohomology computation identifying the Euler class as having infinite order.","marker":"[Hat02]"},{"why":"Yields the relatively geometric action criterion that turns relative hyperbolicity into residual finiteness.","marker":"[EG22]"},{"why":"Shows dihedral Artin groups are virtually $\\mathbb{Z}\\times F_n$, the product separability fact behind Lemma 8.6.","marker":"[HJP16]"}],"fun_headline_variants":["2D Shephard groups act cocompactly on a new CAT(0) complex","Shephard groups: new CAT(0) complex yields relative hyperbolicity","Residual finiteness for many 2D Artin groups via Shephard quotients","CAT(0) complex for Shephard groups despite dihedral obstructions","2D Shephard groups: no CAT(0) space but cocompact CAT(0) action"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The geometric step in Proposition 4.3 — that a simple loop in the coset geometry must enclose a cell projecting to the relator $e_{ac}$ and traverse it with consistent orientation, forcing a nonzero coefficient — is asserted without a full proof; the entire CAT(0) complex construction and every downstream theorem depend on that claim.","fun_headline_variants_meta":{"raw":{"variants":["2D Shephard groups act cocompactly on a new CAT(0) complex","Shephard groups: new CAT(0) complex yields relative hyperbolicity","Residual finiteness for many 2D Artin groups via Shephard quotients","CAT(0) complex for Shephard groups despite dihedral obstructions","2D Shephard groups: no CAT(0) space but cocompact CAT(0) action"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001209,"raw_usage":{"total_tokens":4978,"prompt_tokens":942,"completion_tokens":4036,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":3921}},"tokens_in":558,"tokens_out":4036,"duration_ms":28151,"temperature":1.0,"reasoning_tokens":3921,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:21:22.984409+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computer search for a cyclically reduced word of syllable length $< 2q$ in $Sh(3,6,4)$ that equals the identity, with exponents not multiples of the generator orders, would refute Proposition 4.3 and with it the CAT(0) construction; no such word should exist if the paper is right.","supporting_citations":[],"review_version":1}