{"id":"df93bb86-8746-4f55-9d13-07b825731ed4","arxiv_id":"2504.14192","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For one-vertex tubular groups that are both CAT(0) and free-by-cyclic, virtual specialness always holds, yet cocompact cubulation often fails; three related open questions are answered negatively.","lead":"This paper settles four open questions about 'tubular' groups, families built from torus and cylinder pieces. It shows some of these groups are geometrically nice in one sense but not another, and gives explicit counterexamples that close off previously open conjectures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6's absolutized law-of-cosines criterion is false as stated; because Lemma 4.4 and Theorem 4.5 depend on it, the proof of the main CAT(0)-vs-special equivalence has a repairable gap.","rationale":"I read the paper in good faith. The central claim is that for this one-vertex tubular family, CAT(0) coincides with virtual specialness, and the proof strategy is: detect CAT(0) by Lemma 2.3/Theorem 2.6, detect virtual specialness by Woodhouse/Wise criteria, and equate the two via Lemma 4.4/Theorem 4.5. The reader's weakest assumption is correct and I can sharpen it: Theorem 2.6 is not merely missing a sign hypothesis, it is false. The concrete counterexample above shows an internal inconsistency with Lemma 2.3. The main theorem likely survives: in the G({p_i,q_i}) coordinates the exact identity gives 2α(1−cosφ)=0, so CAT(0) still forces α=0; similarly Lemma 4.4's conclusion follows from the exact law. Thus the right verdict is the reader's CONDITIONAL: the paper's conclusions are probably correct, but the proof as written relies on a false theorem and must be repaired. I do not see a more severe objection: the use of Woodhouse's non-dilation criterion and L²-Betti arguments are standard external tools, and the counterexamples to the named questions rest on separate, coherent arguments. No ad hominem; no manufactured concern.","tokens_in":24226,"tokens_out":20579,"duration_ms":181955,"concrete_test":"Verify the counterexample numerically: with v1=(1,0), w1=(0,1), v2=(2,5), w2=(3,-3), and B1=(1,0), B2=(-11/38, sqrt(1-(11/38)^2)), compute ∥B1∥²=∥B2∥²=1 and ∥2B1+5B2∥²=∥3B1-3B2∥²=441/19, while Theorem 2.6's equation 29-20cosφ=18-18cosφ has no solution in (0,π). Then redo Lemma 4.4 using ∥xB1+yB2∥²=L²(x²+y²+2xy cosφ) in place of the absolutized identity, and check whether Eq1 plus equal-norm equations still forces a_i b_i = c_i d_i.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is Theorem 2.6. For vectors B1,B2 of equal length L with angle φ, the exact norm is ∥xB1+yB2∥² = L²(x²+y²+2xy cosφ), not L²(|x|²+|y|²−2|xy|cosφ). The absolutized formula is correct only when xy≤0, and one φ cannot serve both signs. This is not a harmless typo: the theorem's 'if and only if' is false. Take v1=(1,0), w1=(0,1), v2=(2,5), w2=(3,−3). Let B1=(1,0), B2=(−11/38, √(1−(11/38)²)). Then ∥B1∥=∥B2∥=1 and ∥2B1+5B2∥² = ∥3B1−3B2∥² = 441/19, so by the paper's own Lemma 2.3 the group G=⟨Z²,s1,s2 | s1v1s1^{-1}=w1, s2v2s2^{-1}=w2⟩ is CAT(0). But Theorem 2.6's condition for i=2 reads 29−20cosφ = 18−18cosφ, forcing cosφ=11/2, impossible. Since Lemma 4.4 and Theorem 4.5 invoke Theorem 2.6 directly, the proof chain for Theorem 1.7 and hence Theorem 1.5 is broken as written. The damage is contained: in the G({p_i,q_i}) coordinates the correct law reduces to α(1−cosφ)=0, yielding the same integer conditions, and Lemma 4.4 can be rerun with the exact sign. The gap is therefore repairable, but the stated general theorem needs correction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies tubular groups that are free-by-cyclic, with three main threads. First, it gives algebraic criteria for when a one-vertex tubular group is CAT(0), free-by-cyclic, and virtually special, and uses them to prove that a CAT(0) free-by-cyclic tubular group with one vertex is virtually special (Theorem 1.7). For the family G({p_i,q_i}) = F_n ⋊_φ Z it obtains a precise trichotomy: CAT(0), virtual specialness, and two explicit integer conditions are equivalent, while cocompact cubulation is strictly rarer (Theorems 1.4 and 1.5). Second, it constructs amalgams of free-by-cyclic groups along cyclic subgroups that are not virtually free-by-cyclic, answering a question of Hagen--Wise negatively. Third, it shows that the Gersten group is cyclic-subgroup-separable but fails property (VRC), answering a question of Minasyan. The main technical engine is a criterion, Theorem 2.6, for CAT(0)-ness of multiple HNN extensions of Z^2.","tokens_in":24384,"tokens_out":26203,"duration_ms":239375,"significance":"If the main theorems hold, the paper is valuable: it supplies explicit, checkable algebraic conditions for a natural family of one-vertex tubular groups, proves a clean dichotomy between virtual specialness and cocompact cubulation, and gives negative answers to three published questions. The proofs are largely concrete and use established external criteria (Button, Wise, Woodhouse, Bridson--Haefliger, Minasyan) rather than opaque or circular arguments. The family G({p_i,q_i}) is a useful testing ground for the relationship among CAT(0), RFRS, virtual specialness, and cocompact cubulation. The paper also contains several instructive counterexamples, including the non-(VRC) cyclic-subgroup-separable example.","major_comments":[{"comment":"The stated iff in Theorem 2.6 is false as written because the law of cosines is absolutized incorrectly. For vectors B1, B2 of equal length with angle φ, the exact identity is ∥xB1 + yB2∥^2 = L^2(x^2 + y^2 + 2xy cos φ), not L^2(|x|^2 + |y|^2 - 2|xy| cos φ); the latter holds only when xy ≤ 0, and one common φ cannot absorb mixed signs across different i. A concrete counterexample is v1=(1,0), w1=(0,1), v2=(2,5), w2=(3,-3). With B1=(1,0) and B2=(-11/38, √(1-(11/38)^2)), one has ∥B1∥=∥B2∥=1 and ∥2B1+5B2∥^2 = ∥3B1-3B2∥^2 = 441/19, so Lemma 2.3 gives a CAT(0) group, but the paper's equation for i=2 becomes 29-20cos φ = 18-18cos φ, forcing cos φ = 11/2, impossible. The correct equation is 29+20cos φ = 18-18cos φ, solved by cos φ = -11/38. Since Lemma 4.4 and Theorem 4.9 invoke Theorem 2.6 directly, the proof chain for Theorems 1.5 and 1.7 is not valid as written. The applications appear repairable: in the G({p_i,q_i}) coordinates the exact equation reduces to a nonzero factor times (1-cos φ)=0, so the stated integer conditions survive. The authors should replace Theorem 2.6 by a correct sign-aware statement, e.g. an existential condition over B1,B2 using the exact quadratic form, and rerun the affected arguments.","section":"Theorem 2.6 / Theorem 2.1(1)"},{"comment":"The proof of Lemma 4.4 as printed mixes the exact left-hand side with the absolutized right-hand side supplied by Theorem 2.6. With the correct law of cosines, after using Eq1, the equality becomes approximately Δ(1-cos φ)=0 (up to the sign of k1k2), which still forces a_i b_i = c_i d_i because φ∈(0,π). Thus the conclusion is correct, but the current proof is not a proof as written. Theorem 4.9 has the same dependency: its displayed equivalence uses the absolutized formula, and although the final integer condition is correct in this family, the derivation must be redone with the exact quadratic form. I consider this a repairable gap rather than a fatal one, but it is load-bearing for the main equivalence and must be fixed before publication.","section":"Lemma 4.4 and Theorem 4.9"},{"comment":"Example 5.6 is not a valid demonstration that condition (2) of Theorem 5.1 is insufficient. The group G1 = ⟨a,s | s a s^{-1} = a^{-1}⟩ is not free-by-cyclic: in any homomorphism to Z, the relation forces 2φ(a)=0, hence φ(a)=0, so no homomorphism is nonzero on the edge group ⟨a⟩. Thus Theorem 5.1, which assumes free-by-cyclic factors, does not apply. Moreover, the text says that a commutes with every element of ⟨s^2,t⟩ and then claims that a and [s^2,t] generate a free subgroup; if a commutes with both s^2 and t, it commutes with their commutator, so a and [s^2,t] generate an abelian group, not a free group. The subsequent sentence about φ(a)=1 and φ([s^2,t])=1 also cannot establish that the kernel is non-free in the way stated. This example should be removed or replaced with a correct one; it is not used in the proof of the main negative answer in Example 5.19/5.20.","section":"Example 5.6"}],"minor_comments":[{"comment":"In the first case of the proof, 'after passing to a finite index subgroup of Z' is imprecise; the argument should explicitly pass to the preimage of mZ in G1 or replace the stable letter by a suitable power.","section":"Theorem 5.1 proof"},{"comment":"The phrase 'virtually non-cocompact special group' is awkward and potentially confusing; it should be rephrased as 'virtually special but not virtually cocompactly cubulated' or similar.","section":"Corollary 4.8"},{"comment":"The first sentence contains a typo: 'RFRF' should be 'RFRS'.","section":"Section 7"},{"comment":"If G1 is intended to be the infinite dihedral group, the presentation is missing the relation s^2=1; as written s has infinite order.","section":"Example 5.6"}],"recommendation":"major_revision","confidential_remarks":"The Theorem 2.6 sign error is real and affects the proof chain of the main theorems, but the stress-test note and my own reading indicate that the advertised applications survive after a sign-aware rewrite. The paper should not be rejected on this basis, but it needs a careful revision of Section 2 and of every proof that invokes the absolutized law of cosines. Example 5.6 is also incorrect and should be fixed independently of the main results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious geometric group theory paper. It answers three named open questions (Lyman, Hagen–Wise, Minasyan), partially answers Brady–Soroko, and gives a clean dichotomy for a natural family of free-by-cyclic tubular groups. The main theorem says that for G({p_i,q_i}), virtually special is equivalent to CAT(0), and both are equivalent to either all p_i = -q_i or a single quadratic condition. That is a real result, and the counterexamples in Sections 5 and 6 are concrete and credible. The paper is well organized, and the authors credit ideas properly, including Douba for Lemma 5.18.\n\nThe flaw is real. Theorem 2.6 is not true as stated. The absolutized law-of-cosines formula with a common angle φ is only valid when x_i y_i ≤ 0; one angle cannot absorb mixed signs. The stress-test counterexample works: for v1=(1,0), w1=(0,1), v2=(2,5), w2=(3,-3), there is an invertible matrix making the relevant lengths equal, so Lemma 2.3 gives CAT(0), yet Theorem 2.6's condition forces cosφ=11/2. So the theorem's iff is false.\n\nBecause Lemma 4.4 and Theorem 4.5 use Theorem 2.6, the proof chain for Theorem 1.7 and hence Theorem 1.5 is broken as written. But the damage is contained. In the G({p_i,q_i}) coordinates, the correct law of cosines reduces to (p_i+q_i)(q_i+p_1-q_1-p_i)=0, which is exactly the paper's condition. I reran Lemma 4.4 with the honest sign and got (Δa·T)(1−cosφ)=0, which still forces a_i b_i = c_i d_i. So the main results survive; the paper needs a corrected Theorem 2.6 and a rewritten proof of Lemma 4.4. There are also minor presentation errors: two different presentations are called the Gersten group (tct^{-1}=ca² versus ba²), and Example 5.6 has a garbled line.\n\nThis paper is for people working on free-by-cyclic groups, tubular groups, and cubulation. It deserves a serious referee. The results are likely correct and the questions are important. The referee should require fixing Theorem 2.6, updating Lemma 4.4's proof, and clearing up the presentation errors. I would accept after those revisions.","headline":"Solid paper with a real but repairable flaw in the CAT(0) detection lemma; the main theorem survives a sign-error check.","tokens_in":25198,"tokens_out":9770,"would_cite":true,"duration_ms":70376,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E26","20E06","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For free-by-cyclic groups with a one-vertex tubular presentation, the paper proves that being CAT(0) and being virtually special are the same property, and that cocompact cubulation is strictly stronger.","keywords":["free-by-cyclic groups","tubular groups","virtually special groups","CAT(0) groups","cocompact cubulation","property (VRC)","cyclic subgroup separability","RFRS"],"falsifier":"Construct $G = \\langle \\mathbb{Z}^2, s_1, s_2 \\mid s_1(1,0)s_1^{-1} = (0,1),\\ s_2(2,1)s_2^{-1} = (1,-1)\\rangle$. Taking $B_1 = (1,0)$ and $B_2 = (-1/2,\\sqrt{3}/2)$ gives an invertible $A = [B_1\\ B_2]$ with $\\lVert A v_1\\rVert = \\lVert A w_1\\rVert = 1$ and $\\lVert A v_2\\rVert = \\lVert A w_2\\rVert = \\sqrt{3}$, so Lemma 2.3 makes $G$ CAT(0); yet Theorem 2.6's equation for $i=2$ reads $5 - 4\\cos\\varphi = 2 - 2\\cos\\varphi$, which has no solution in $(0,\\pi)$. This contradicts Theorem 2.6 as stated, and checking the free-by-cyclic instances of Theorems 4.5 and 4.9 would show whether the sign issue can actually arise there.","tokens_in":23776,"feed_emoji":"🧊","tokens_out":27907,"duration_ms":199715,"temperature":0.7,"pith_summary":"Free-by-cyclic groups that are tubular — built by gluing one flat $\\mathbb{Z}^2$ vertex to itself along cyclic subgroups — sit at the intersection of several open questions about CAT(0) geometry and cube complexes. The paper's central theorem states that for the one-vertex family $G(\\{p_i,q_i\\}) = F_n \\rtimes_\\varphi \\mathbb{Z}$ with $a_i \\mapsto a^{p_i} a_i a^{q_i}$, being CAT(0) is exactly the same as being virtually special, and both are decided by an integer condition: either every $p_i = -q_i$, or one index $s$ forces $q_i(q_i+p_s-q_s) = p_i(p_i-p_s+q_s)$ for all $i$. From there the paper shows that many of these CAT(0) groups still cannot virtually act freely and cocompactly on a CAT(0) cube complex, answering one of [33]'s questions negatively; that amalgams of free-by-cyclic groups along cyclic subgroups need not be virtually free-by-cyclic, answering a question of [23, Remark 3.6]; and that a classical rank-three free-by-cyclic example is cyclic-subgroup-separable but fails property (VRC), answering [34, Question 11.6]. If correct, these results draw a sharp line: CAT(0) geometry and virtual specialness coincide here, while cocompact cubulation asks for more.","feed_headline":"CAT(0) equals virtually special in one-vertex tubular groups","feed_subtitle":"Integer parameters decide CAT(0) and virtual specialness; several examples resist cocompact cubulation.","key_machinery":"The load-bearing object is the one-vertex tubular group $G = \\langle \\mathbb{Z}^2, s_i \\mid s_i v_i s_i^{-1} = w_i\\rangle$, a multiple HNN extension with $\\mathbb{Z}^2$ vertex stabilizer and cyclic edge groups. Two detectors drive the argument. For CAT(0): by the flat torus theorem and the gluing theorem of [10, Theorem II.11.18] (Lemma 2.3), $G$ is CAT(0) exactly when some $A \\in \\mathrm{GL}_2(\\mathbb{R})$ equalizes the translation lengths $\\lVert A v_i\\rVert = \\lVert A w_i\\rVert$; writing coordinates in the basis $\\{v_1,w_1\\}$ reduces this to a common-angle cosine equation. For virtual specialness: choose the equitable set $S = \\{w_1-v_1,\\ w_1+v_1\\}$; the determinant identities $\\det[w_1\\mp v_1, v_i] = \\pm\\det[w_1\\mp v_1, w_i]$ make every immersed wall non-dilated, so the criterion of [41, Theorem 1.1] upgrades a free finite-dimensional cubulation to virtual specialness. A third component, the criterion of [12, Theorem 2.1], characterizes free-by-cyclic tubular groups by the condition that all differences $v_i-w_i$ lie on one line avoiding the $v_i$; this links the CAT(0) angle equations to the integer quadratic relation in Theorem 1.5.","core_discovery":"In the paper's own terms, the central discovery is an equivalence (Theorem 1.5). For $G = G(\\{p_i,q_i\\}) = F_n \\rtimes_\\varphi \\mathbb{Z}$, the following are equivalent: $G$ is virtually special, $G$ is CAT(0), and the parameters satisfy either $p_i = -q_i$ for every $i$ or, for some $s$ with $p_s \\neq -q_s$, the relation $q_i(q_i+p_s-q_s) = p_i(p_i-p_s+q_s)$ holds for every $i$. This is obtained from a broader result (Theorem 4.5): every free-by-cyclic tubular group with a single $\\mathbb{Z}^2$ vertex that is CAT(0) is virtually special. The paper also establishes that the rank-3 examples $F_3 \\rtimes_\\Psi \\mathbb{Z}$ with $b \\mapsto a^m b a^m$ and $c \\mapsto a^n c a^n$ are virtually special for all integers $m,n$, but virtually act freely and cocompactly on a CAT(0) cube complex only when $|m| = |n|$; that the classical rank-three example amalgamated with $\\mathbb{Z}^2$ along a cyclic subgroup is a tubular group that is not virtually free-by-cyclic; and that this same classical example is cyclic-subgroup-separable but has no virtual retraction onto the cyclic subgroup $\\langle a\\rangle$.","pith_inferences":["The sign gap in Theorem 2.6 suggests that the CAT(0) criterion should be re-derived as the actual condition $(x_i^2 + y_i^2 - x_i'^2 - y_i'^2) + 2\\cos\\theta\\,(x_i y_i - x_i' y_i') = 0$ with one angle $\\theta$; redoing Lemma 4.4 and Theorem 4.9 under that correct equation would test whether the quadratic parameter relation survives unchanged.","Lemma 2.3's norm-equalization matrix is not special to $\\mathbb{Z}^2$: the same flat-torus argument gives a CAT(0) criterion for one-vertex tubular groups over $\\mathbb{Z}^n$, and one could ask whether Theorem 4.5's 'CAT(0) ⇒ virtually special' extends to that higher-rank setting.","The non-virtually-free-by-cyclic amalgam in Example 5.20 has vanishing $\\ell^2$-Betti numbers and is locally indicable, so it is a natural test case for whether the 'virtually free-by-cyclic' fibring criteria can be relaxed at all.","The equivalence of CAT(0) and virtual specialness for one-vertex tubular groups, if it survives the sign fix, is a test bed for Question 7.3: one could check whether all finitely generated subgroups of these groups are again CAT(0), which would close the loop on virtual RFRS."],"forward_implications":["In the one-vertex tubular family, the answer to [8, Section 9: Question 1] is positive: every CAT(0) group in the family is virtually special, hence virtually embeds in a right-angled Artin group.","[33, Question 1] is answered negatively: virtual specialness of these CAT(0) free-by-cyclic groups does not imply a cocompact free action on a CAT(0) cube complex, with |m| ≠ |n| giving explicit counterexamples.","The class of virtually free-by-cyclic groups is not closed under amalgamation along cyclic subgroups: the classical rank-three example amalgamated with Z² (or with F₂ × Z) along ⟨a⟩ yields a non-virtually-free-by-cyclic tubular group.","Cyclic subgroup separability does not imply property (VRC): the classical rank-three example is cyclic subgroup separable but has no virtual retraction onto ⟨a⟩.","For tubular groups, 'free-by-cyclic' automatically means F_n-by-Z (Theorem 3.2), so the infinite-rank/finite-rank distinction collapses in this class."],"supporting_citations":[{"why":"Supplies the equitable-set criterion for free cubulation of tubular groups, the fact that virtually special tubular groups are CAT(0), and the parallelism-class obstruction to cocompact cubulation used in Lemma 4.7 and Corollary 4.10.","marker":"[38]"},{"why":"Provides the criterion that a tubular group is virtually special iff it acts freely on a finite-dimensional CAT(0) cube complex; this is the bridge from non-dilated walls to virtual specialness in Theorem 4.2 and Theorem 4.5.","marker":"[41]"},{"why":"Supplies the dilation criterion for finite-dimensionality of the dual cube complex; Theorem 4.2 shows every immersed wall has weight 1 to apply it.","marker":"[40]"},{"why":"Gives the criterion characterizing when a tubular group is free-by-cyclic via a homomorphism to Z that is nonzero on every edge group; underpins Theorem 2.1(2) and Theorem 3.1.","marker":"[12]"},{"why":"Provides the flat torus theorem and the CAT(0) gluing theorem; Lemma 2.3 translates CAT(0)-ness into the existence of a norm-equalizing matrix A.","marker":"[10]"},{"why":"Supplies the CAT(0) free-by-cyclic groups F_3 ⋊_Ψ Z whose cocompact cubulation question is answered negatively in Example 4.11.","marker":"[33]"},{"why":"Provides the theory of virtual retractions, the (VRC) property, and the result used to show the classical rank-three example fails (VRC) in Example 6.3 and Section 5.","marker":"[34]"},{"why":"Supplies the classical rank-three free-by-cyclic example and its non-CAT(0) companion; this example is the source group for the amalgamation counterexample and the cyclic-separable non-(VRC) example.","marker":"[19]"}],"fun_headline_variants":["CAT(0) tubular groups are virtually special, but some resist cubulation","One-vertex tubular free-by-cyclic: CAT(0) iff virtually special","Tubular groups: CAT(0) implies virtually special; cubulation often fails","Free-by-cyclic tubular groups: CAT(0) and specialness equivalence","CAT(0) tubular groups: virtually special, yet not always freely cubulable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a single common angle $\\varphi$ in Theorem 2.6's cosine equation really detects the existence of the norm-equalizing matrix $A$; as written, the identity it rests on only holds when every coordinate product is non-positive, and no argument in the paper removes that sign restriction.","fun_headline_variants_meta":{"raw":{"variants":["CAT(0) tubular groups are virtually special, but some resist cubulation","One-vertex tubular free-by-cyclic: CAT(0) iff virtually special","Tubular groups: CAT(0) implies virtually special; cubulation often fails","Free-by-cyclic tubular groups: CAT(0) and specialness equivalence","CAT(0) tubular groups: virtually special, yet not always freely cubulable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001291,"raw_usage":{"total_tokens":5307,"prompt_tokens":1016,"completion_tokens":4291,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":4185}},"tokens_in":632,"tokens_out":4291,"duration_ms":29383,"temperature":1.0,"reasoning_tokens":4185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:58:55.691911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct $G = \\langle \\mathbb{Z}^2, s_1, s_2 \\mid s_1(1,0)s_1^{-1} = (0,1),\\ s_2(2,1)s_2^{-1} = (1,-1)\\rangle$. Taking $B_1 = (1,0)$ and $B_2 = (-1/2,\\sqrt{3}/2)$ gives an invertible $A = [B_1\\ B_2]$ with $\\lVert A v_1\\rVert = \\lVert A w_1\\rVert = 1$ and $\\lVert A v_2\\rVert = \\lVert A w_2\\rVert = \\sqrt{3}$, so Lemma 2.3 makes $G$ CAT(0); yet Theorem 2.6's equation for $i=2$ reads $5 - 4\\cos\\varphi = 2 - 2\\cos\\varphi$, which has no solution in $(0,\\pi)$. This contradicts Theorem 2.6 as stated, and checking the free-by-cyclic instances of Theorems 4.5 and 4.9 would show whether the sign issue can actually arise there.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the equitable-set criterion for free cubulation of tubular groups, the fact that virtually special tubular groups are CAT(0), and the parallelism-class obstruction to cocompact cubulation used in Lemma 4.7 and Corollary 4.10."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the criterion that a tubular group is virtually special iff it acts freely on a finite-dimensional CAT(0) cube complex; this is the bridge from non-dilated walls to virtual specialness in Theorem 4.2 and Theorem 4.5."},{"cited_title":"Woodhouse, Classifying finite dimensional cubulations of tubular groups","cited_arxiv_id":null,"evidence_quote":"Supplies the dilation criterion for finite-dimensionality of the dual cube complex; Theorem 4.2 shows every immersed wall has weight 1 to apply it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the criterion characterizing when a tubular group is free-by-cyclic via a homomorphism to Z that is nonzero on every edge group; underpins Theorem 2.1(2) and Theorem 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the flat torus theorem and the CAT(0) gluing theorem; Lemma 2.3 translates CAT(0)-ness into the existence of a norm-equalizing matrix A."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CAT(0) free-by-cyclic groups F_3 ⋊_Ψ Z whose cocompact cubulation question is answered negatively in Example 4.11."},{"cited_title":"Minasyan, Virtual retraction properties in groups, Int","cited_arxiv_id":null,"evidence_quote":"Provides the theory of virtual retractions, the (VRC) property, and the result used to show the classical rank-three example fails (VRC) in Example 6.3 and Section 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical rank-three free-by-cyclic example and its non-CAT(0) companion; this example is the source group for the amalgamation counterexample and the cyclic-separable non-(VRC) example."}],"review_version":1}