{"id":"b2c50c65-c36b-46f7-bbb5-1435b14e8fef","arxiv_id":"2607.20401","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The conjugator length function is quadratic for the integral Heisenberg group and Stallings' group, and cyclic-subgroup distortion can be promoted to conjugator length.","lead":"This paper turns the shortest conjugating word between conjugate elements—the conjugator length—into a systematic, geometrically studied invariant, and proves it grows quadratically for the Heisenberg and Stallings groups. It is a compact map of a young quantitative area with consequences for algorithms, geometry, and cryptography.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.14's upper bound uses an inaccurate description of the set of all conjugators in H = F(a,b) × F(c,d): the set W misses valid conjugators when a component is a proper power, so the Diophantine step may lack an integer solution.","rationale":"The reader identified Corollary 3.7(iii) as the weakest assumption, focusing on the diagram excision lemma. Our stress-test found a more internal, algebraic flaw in the upper-bound proof of Theorem 4.14: the characterization of the set of all conjugators W in H is incorrect when the relevant components are proper powers. This is load-bearing because the proof uses W to assert that a known conjugator w_l ∈ K can be written as θ1 θ2^p φ1 φ2^q, enabling a Diophantine equation with a bound from Lemma 4.10. Without that, the O(n^2) upper bound does not follow. The flaw is easily repairable by using the maximal roots of the components instead of v's components, but as written the proof has a gap. Therefore the paper should not be accepted without modification; the verdict is CONDITIONAL on fixing this step. We do not dispute the overall novelty or significance, and the lower bound and other sections appear sound. The disagreement with the reader is partial because they flagged a different and arguably less immediate issue.","tokens_in":39231,"tokens_out":28167,"duration_ms":207238,"concrete_test":"Compute directly in F(a,b): let u_ab = a^2, v_ab = b^{-1} a^2 b, θ1 = b, θ2 = b^{-1} a^2 b. Enumerate the reduced words in {θ1 θ2^p} for p = -2,...,2 and compare with the set of all reduced words w satisfying u_ab w = w v_ab (i.e., w^{-1} u_ab w = v_ab). The latter includes a^{-1} b (since (a^{-1}b)^{-1} a^2 (a^{-1}b) = b^{-1} a^2 b), while {θ1 θ2^p} gives only a^{2p} b. This demonstrates the set W in the proof of Theorem 4.14 is incomplete. A more integrated test: embed this pair as the non-t parts of u = a^2 t, v = b^{-1} a^2 b t in Stallings' group with the presentation in (10) and verify that the K-conjugator a^{-1}b conjugates u to v in S, yet no p,q solve the paper's Diophantine equation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 4.14 (upper bound), after reducing to radial t-corridors via Corollary 3.7, the authors define θ1, φ1 as a minimal conjugator in H for u∼v and let θ2, φ2 be such that v = θ2φ2. They then assert: 'The set of all w∈H such that u w = w v in H is W = {θ1 θ2^p φ1 φ2^q | p,q ∈ Z}.' This is false in general. In H = F(a,b) × F(c,d), the centralizer of a component is cyclic, generated by its maximal root. If the a,b-component u_ab is a proper power, say u_ab = r^m with m>1, and v_ab = θ1^{-1} r^m θ1, then all conjugators from u_ab to v_ab are {r^k θ1 | k ∈ Z}, while θ1 θ2^p = θ1 (θ1^{-1} r^m θ1)^p = r^{mp} θ1, giving only k divisible by m. Thus W omits conjugators such as r θ1. This matters because the proof needs to find an integer solution to a Diophantine equation derived from the known existence of w_l ∈ K ∩ W. If w_l is an omitted conjugator, the equation may have no solution. Concretely, for H's a,b-factor take u_ab = a^2, v_ab = b^{-1} a^2 b, with θ1 = b. Then true conjugators are a^k b; the paper's set is {a^{2p} b}. The K-valuable conjugator a^{-1} b (with z(a)=z(b)=1) is not in W. Hence the assertion w_l ∈ W is unjustified and the quadratic upper-bound proof fails at this step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is a survey of conjugator length functions for finitely presented groups, with several original results: quadratic growth for the integral Heisenberg group (Theorem 4.9) and for Stallings' group (Theorem 4.14), an elementary linear bound for BS(1,m) (Theorem 4.13), and constructions promoting cyclic subgroup distortion to conjugator length (Theorems 5.2 and 5.3). The paper develops the annular-diagram toolkit in Section 3, including a reduction lemma for HNN-extensions (Corollary 3.7), and surveys the known landscape. The Heisenberg proof is clean and fully displayed; the Stallings upper bound reduces to a Diophantine estimate for a claimed description of all conjugators in F(a,b) x F(c,d), and that description is not correct as written.","tokens_in":39684,"tokens_out":17458,"duration_ms":142565,"significance":"If the Stallings proof is repaired, the paper supplies two explicit quadratic benchmarks for conjugator length, the quantitative invariant attached to the conjugacy problem, complementing the theory of Dehn functions. The survey is well organized, and the diagrammatic foundations (annular diagrams, t-corridors, excision arguments) are useful for researchers entering the area. The proofs are self-contained modulo standard lemmas (van Kampen's lemma, annular diagram characterizations, Bezout-type Diophantine bounds), and the authors are explicit about the limitations of the hypothesis in Theorem 5.3.","major_comments":[{"comment":"The assertion that W={θ1 θ2^p φ1 φ2^q | p,q∈Z} is the set of all w∈H with uw=wv in H is false in general. In the F(a,b) factor, take u_ab=a^2, v_ab=b^{-1}a^2b, and θ1=b. Then all conjugators from u_ab to v_ab are {a^k b : k∈Z}, but θ1 θ2^p = b(b^{-1}a^2b)^p = a^{2p}b, so conjugators such as a^{-1}b are omitted. Such an omitted conjugator can lie in K (e.g., it has z-length zero), so the word w_l obtained from the radial-corridor reduction need not lie in W. Consequently the Diophantine equation p z(θ2)+q z(φ2) = -z(θ1)-z(φ1) need not have a solution, and the O(n^2) upper-bound proof fails at this step. The proof should parametrize the actual set of conjugators using maximal roots in the free factors, then re-run the bounded-solution argument.","section":"§4.5, proof of Theorem 4.14 (upper bound, displayed set W)"},{"comment":"Width_M(ℓ) is defined as an infimum over all pairs of loops of total length at most ℓ. With this definition Width_M is identically zero: take ρ0=ρ1 to be a constant loop. Thus Theorem 2.1 cannot hold as stated. The proof outline uses a worst-case width, and the intended definition should be a supremum over such pairs. Please correct the definition and adjust the surrounding text.","section":"§2.4, definition of Width_M"}],"minor_comments":[{"comment":"Typo: 'jeodardises' should be 'jeopardizes'.","section":"§3.3"},{"comment":"Typo: 'seimhyperbolic' should be 'semihyperbolic'.","section":"§4.2"},{"comment":"In Corollary 3.7, 'not-corridor' should be 't-corridor'; near the end of the proof of Theorem 5.3, 's-corridor' should be 'r-corridor'.","section":"§4.5 / §5.3"},{"comment":"The notation S=H ˙∗_K is nonstandard; please state explicitly that this is the HNN-extension with stable letter t commuting with K, matching presentation (10). Also, the sentence justifying CL_H(n)≃n via CL_{A×B}≃max{CL_A,CL_B} could use a one-line explanation.","section":"§4.5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about arXiv:2607.20401. First, it is a genuinely useful survey; it will become the standard entry point for the conjugator length function. Second, the proof of the main new theorem for Stallings' group (Thm 4.14, upper bound) contains a real gap, and the stress-test note is right.\n\nWhat is good: The Heisenberg theorem (Thm 4.9) is clean and self-contained, a nice model for how Diophantine estimates control conjugator length. The survey pulls together a scattered literature — hyperbolic, CAT(0), nilpotent, metabelian, mapping class groups, etc. — and the discussion of why conjugator length is not a quasi-isometry invariant is honest and useful. Theorems 5.2 and 5.3 are original, and the authors disclose the limitation on 5.3's hypothesis.\n\nThe soft spot: In the proof of Thm 4.14, the set of all conjugators in H = F(a,b) × F(c,d) is asserted to be W = {θ1 θ2^p φ1 φ2^q | p,q ∈ Z}. That is false in general. If the a,b-component of u is a proper power, say u_ab = r^m with m>1, then the full set of conjugators from u_ab to v_ab is {r^k θ1 | k ∈ Z}, while θ1 θ2^p only gives k divisible by m. Concretely, take u_ab = a^2, v_ab = b^{-1} a^2 b, θ1 = b; then a^{-1} b is a conjugator not in W. Since the proof needs w_l ∈ K ∩ W, the Diophantine step does not follow.\n\nThis is repairable — one should use a root of u_ab rather than v_ab, and the length bounds probably survive — but as written it is not a minor terse step; it is a false parametrization. The lower bound for Stallings' group is independent and fine.\n\nThe paper deserves a serious referee. For anyone working on quantitative decision problems or filling invariants, the survey is worth reading now, and the Heisenberg proof is a nice model. But before Thm 4.14 is taken as established, the upper-bound proof needs to be fixed. If you send it to review, I'd ask the authors to patch that step or state the result as conditional.","headline":"A genuinely useful survey, but the upper-bound proof for Stallings' group (Thm 4.14) has a real gap: the set of all conjugators in H is misdescribed when a free-group component is a proper power.","tokens_in":40183,"tokens_out":5269,"would_cite":true,"duration_ms":41107,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F10","20F06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the conjugator length functions of the 3-dimensional integral Heisenberg group and of Stallings' group grow quadratically, and it promotes the systematic study of this invariant.","keywords":["conjugacy problem","conjugator length function","Heisenberg group","Stallings' group","annular diagrams","HNN extensions","Diophantine equations","geometric group theory"],"falsifier":"Produce, for every constant C, a conjugate pair u,v in Stallings' group or the Heisenberg group with |u|+|v| ≤ n and shortest conjugator length > C n^2. In the Heisenberg case the search is concrete: write elements in normal form a^α b^β c^γ and solve the single linear equation (6); the theorem predicts a solution with |x|,|x̂| bounded by the coefficients, so a counterexample pair violating that bound falsifies the upper bound. For Stallings, the corresponding check is finding a pair that cannot be trimmed by the excision lemma without lengthening the words.","tokens_in":39125,"feed_emoji":"📈","tokens_out":10530,"duration_ms":82576,"temperature":0.7,"pith_summary":"The paper aims to make the conjugator length function — the shortest word needed to conjugate one element to another when they are conjugate — a standard quantitative invariant for the conjugacy problem, on a par with the Dehn function for the word problem. It proves two new benchmark results: in the 3-dimensional integral Heisenberg group and in Stallings' group, the conjugator length function grows quadratically. The Heisenberg proof reduces the problem to a single linear Diophantine equation in the two parameters of a candidate conjugator and applies a classical bound on the size of its smallest solution; the Stallings proof uses annular diagrams and a trimming lemma for HNN extensions to reduce to the same type of arithmetic estimate. For a sympathetic reader, the payoff is a concrete toolkit — annular diagrams, stable-letter corridors, Diophantine bounds — plus explicit examples that calibrate the difficulty of the conjugacy problem.","feed_headline":"Heisenberg and Stallings' group have quadratic conjugator length","feed_subtitle":"The shortest conjugating word grows like n^2 in both, setting the benchmark for the conjugacy problem.","key_machinery":"Annular diagrams are the central objects: finite planar 2-complexes shaped like a cylinder whose boundary cycles read the two conjugate words u and v, with a path across the annulus carrying a conjugator w; Lemma 3.4 shows CL(u,v) is the least length of such a crossing. In HNN extensions, stable-letter t-corridors and t-annuli decompose these diagrams, and Corollary 3.7(iii) — the excision lemma — allows the authors to replace u,v by words of no greater length whose diagram has only radial t-corridors and no essential t-annuli. The second load-bearing tool is Lemma 4.10, a sharp bound on minimal solutions of a linear Diophantine equation Ax+By=C, which turns the search for short conjugators","core_discovery":"The paper's central new results are Theorems 4.9 and 4.14: the conjugator length function of the 3-dimensional integral Heisenberg group H_3(Z) and of Stallings' group both grow quadratically. For a finitely generated group, CL(n) is the least upper bound on the length of a shortest word w with uw = wv in G, taken over all conjugate words u,v with |u|+|v| ≤ n. In the Heisenberg group, conjugacy of two elements in normal form forces the two off-diagonal parameters to match and leaves a single linear Diophantine equation in the two unknown parameters of a conjugator; an elementary bound on the smallest solution of such an equation gives the O(n^2) upper bound. The matching lower bound is witne","pith_inferences":["The Heisenberg lower-bound witness suggests a general route for class-2 nilpotent groups: compute CL as the minimal norm of a solution to the defining system of linear Diophantine equations; the paper's survey already shows polynomial degrees of all integers are attainable, so one could test whether every class-2 nilpotent group has CL bounded by such a solution norm.","Because conjugator length is not a quasi-isometry invariant, the quadratic benchmarks imply that no coarse-geometric invariant alone can predict conjugator length; the paper's examples show index-two subgroups can differ in solvability of the conjugacy problem, so one can expect equally stark quantitative differences in finitely presented groups with identical large-scale geometry.","Theorem 2.1's equivalence with the width of free homotopies in a Riemannian manifold means numerical experiments on the Heisenberg nilmanifold could estimate CL by measuring minimal basepoint sweep; a measured quadratic width would independently corroborate the quadratic benchmark."],"forward_implications":["In the Heisenberg group, the conjugacy search problem admits conjugators of length O(n^2); by Remark 4.11 the same holds for every higher-dimensional integral Heisenberg group.","Stallings' group now has a complete quadratic conjugator length function up to the standard equivalence, adding a classic exotic group to the short list of groups for which this invariant is known exactly.","The amalgamation construction of Theorem 5.2 shows that any distortion function of an infinite cyclic subgroup yields a lower bound on conjugator length, so the quadratic floor combines with distorted subgroups to produce groups with conjugator length at least as large as any prescribed distortion.","The survey's spectrum — linear for free, hyperbolic, BS(1,m), and mapping class groups, quadratic for Heisenberg and Stallings, non-recursive in other cases — makes conjugator length a discriminating quantitative invariant for the conjugacy problem."],"fun_headline_variants":["Quadratic conjugator length for Heisenberg and Stallings' groups","Heisenberg and Stallings' groups have n^2 conjugator length","Conjugator length is quadratic in Heisenberg and Stallings' groups","Heisenberg and Stallings' groups: conjugator length grows as n^2","Quadratic conjugator length: Heisenberg and Stallings' groups"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The quadratic upper bound for Stallings' group rests on the excision lemma (Corollary 3.7(iii)): every conjugate pair of words can be replaced, without increasing lengths, by a pair whose annular diagram has only radial stable-letter corridors and no essential annuli; if that trimming step fails, the reduction to the free-product estimate and hence the O(n^2) bound collapses.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic conjugator length for Heisenberg and Stallings' groups","Heisenberg and Stallings' groups have n^2 conjugator length","Conjugator length is quadratic in Heisenberg and Stallings' groups","Heisenberg and Stallings' groups: conjugator length grows as n^2","Quadratic conjugator length: Heisenberg and Stallings' groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00101,"raw_usage":{"total_tokens":4036,"prompt_tokens":609,"completion_tokens":3427,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":353,"completion_tokens_details":{"reasoning_tokens":3331}},"tokens_in":353,"tokens_out":3427,"duration_ms":19383,"temperature":1.0,"reasoning_tokens":3331,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:56:22.361333+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Produce, for every constant C, a conjugate pair u,v in Stallings' group or the Heisenberg group with |u|+|v| ≤ n and shortest conjugator length > C n^2. In the Heisenberg case the search is concrete: write elements in normal form a^α b^β c^γ and solve the single linear equation (6); the theorem predicts a solution with |x|,|x̂| bounded by the coefficients, so a counterexample pair violating that bound falsifies the upper bound. For Stallings, the corresponding check is finding a pair that cannot be trimmed by the excision lemma without lengthening the words.","supporting_citations":[],"review_version":1}