{"id":"53dc48b9-ec28-4e90-bf09-53d4564213ab","arxiv_id":"1908.04460","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A set of algorithms detects stability and Morseness of finitely generated subgroups in mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and limit groups.","lead":"Algorithms are given that detect whether a finitely generated subgroup of a mapping class group, right-angled Artin group, or toral relatively hyperbolic group is stable or Morse. The paper assembles known structural characterizations into partial and complete decision procedures, including a new search using Dehn fillings.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A(i)'s first proof is invalid: it asserts a stability characterization involving all geodesics of Mod(S), which fails even for the trivial subgroup.","rationale":"The reader identified the same load-bearing weakness: the first proof of Theorem A(i) substitutes an overstrong ambient-geodesic condition for the actual orbit condition defining stability. My reading confirms this and sharpens it: the trivial subgroup is stable, yet geodesics to large powers of a Dehn twist have images in C(S) that are arbitrarily long closed paths, so they are not quasigeodesics. Consequently the proof's claimed equivalence is false, and the termination argument for the partial algorithm is unsupported. This gap is central because Theorem A(i) is the paper's first headline result and Theorems A(ii) and A(iii) rely on it. I do not see a reason to move the verdict beyond conditional: the theorem is likely repairable by running the same local-to-global argument on geodesics in the Cayley graph of H rather than of all Mod(S), and independent partial evidence for related claims exists (for example, the second proof for closed hyperbolic surfaces and the cube-complex proof for Theorem B(i)). The paper's other algorithmic contributions may still be sound, but the current text does not establish the general mapping-class-group statements.","tokens_in":26707,"tokens_out":28050,"duration_ms":291428,"concrete_test":"Independently re-derive the equivalence asserted in the first proof of Theorem A(i) for the stable subgroup H = {1} in Mod(S). Take a Dehn twist τ and a base curve x fixed by τ. For the geodesic w_n = τ^n, compute length(f(w_n)) and d_C(x, τ^n x). If the asserted equivalence were true, these paths would be uniform quasigeodesics for some λ,ε; but length(f(w_n)) → ∞ while the endpoint distance is 0, so no such λ,ε exists. This settles that the proof's central characterization is false; a correct algorithm must check the H-orbit in C(S), not all ambient geodesics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The first proof of Theorem A(i) (Section 2) asserts that H is stable in Mod(S) iff there exist λ ≥ 1, ε ≥ 0 such that for every geodesic w in the full Cayley graph ΓMod(S), the orbit-map image f(w) is a (λ,ε)-quasigeodesic in C(S). This equivalence is false. Let x be a curve fixed by a Dehn twist τ and take the geodesic w_n = τ^n in ΓMod(S) (with τ included in a finite generating set of Mod(S)). Since τ acts elliptically on C(S), d_C(x, τ^n x) = 0 for all n, but by construction f(w_n) has length at least n. Hence f(w_n) cannot be a (λ,ε)-quasigeodesic for any fixed λ,ε as n → ∞. The trivial subgroup H = {1} is stable, so the 'only if' direction fails. The algorithm then checks only geodesic edge-paths of length at most K starting at 1; no statement in the proof connects this finite check to the H-orbit property that actually characterizes stability. Thus Theorem A(i) is not proved as written, and the proofs of A(ii) and A(iii) inherit the gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes algorithms for detecting stable and Morse subgroups of finitely generated groups in several classes. Theorem A gives a partial algorithm for stability of subgroups of mapping class groups, a complete algorithm for stability of undistorted subgroups, and a partial algorithm for Morseness. Theorem B gives a complete algorithm for stability in right-angled Artin groups and a partial algorithm for Morseness. Theorem C gives partial and complete algorithms for stability and Morseness in toral relatively hyperbolic groups, with Corollary D deciding finite index for undistorted subgroups and Corollary E applying to groups discriminated by locally quasiconvex torsion-free hyperbolic groups. The proofs reduce the target properties to previously known characterizations—convex cocompactness in the curve graph, pure loxodromicity in right-angled Artin groups, and Tran's intersection characterizations in relatively hyperbolic groups—and then invoke existing algorithmic tools.","tokens_in":26738,"tokens_out":34114,"duration_ms":360300,"significance":"If the gaps identified below are repaired, the paper would provide the first algorithmic recognition results for stable and Morse subgroups in mapping class groups and toral relatively hyperbolic groups, and a complete recognition algorithm for stability in right-angled Artin groups. The use of prior characterizations is systematic, and the Dehn-filling strategy for Theorem C(iv) is nontrivial and plausible. The cube-complex algorithm for Theorem B(i) in Section 3.2 appears to be a genuine algorithmic contribution. However, the current proof of Theorem A(i) rests on a false equivalence, and the first algorithm for Theorem B(i) has the same defect; since Theorem A is one of the paper's headline results, the manuscript requires substantial revision.","major_comments":[{"comment":"The equivalence asserted in the fourth paragraph of this proof—that H is stable in Mod(S) iff there exist λ≥1, ε≥0 such that for every geodesic w in ΓMod(S) the image f(w) is a (λ,ε)-quasigeodesic in C(S)—is false. Theorem 2.2(2) only guarantees that the H-orbit is quasi-isometrically embedded in C(S); it does not constrain geodesics of Mod(S) outside H. To see the failure, fix a finite generating set B of Mod(S) that contains a Dehn twist τ with τ·x=x. For the stable subgroup H={1}, a geodesic representative of τ^n has length L_n→∞, and its image under f is a path in C(S) from x to x of length at least L_n, so it cannot be a (λ,ε)-quasigeodesic for fixed λ,ε as n→∞. The algorithm therefore checks a condition that fails for a stable subgroup and will never terminate on H={1}. This invalidates the proofs of Theorem A(i), A(ii), and A(iii) for the full class of surfaces; the second proof in Section 2.2 covers only closed hyperbolic surfaces. The proof should apply the local-to-global check to geodesics in the Cayley graph of H with the given generating set, not to all geodesics of Mod(S). There is also a notational ambiguity: A is introduced as a generating set for H, but ΓMod(S) denotes the Cayley graph of Mod(S); on the literal reading, one must extend A to a generating set of Mod(S), and with that reading the displayed equivalence is false.","section":"Section 2, first proof of Theorem A(i)"},{"comment":"The same type of false global condition appears here. The proof asserts that H is stable in AΓ iff there exist λ,ε such that every geodesic w in (ΓAΓ,d) is mapped to a (λ,ε)-quasigeodesic in Γ^e, and then reduces this to the condition that every geodesic starting from 1 in (ΓAΓ,d) is a quasigeodesic in (ΓAΓ,d_*). But Theorem 3.6 concerns only the H-orbit in Γ^e, and the identity map from the word metric d to the star metric d_* is not a quasi-isometry. For the connected anti-connected path graph a-b-c, the word w=a^n c a^{-n} is a geodesic in AΓ of length 2n+1, while its star length is 1 because the entire word lies in St(b). Since the trivial subgroup is stable, the claimed condition fails for a stable subgroup, so this first algorithm for Theorem B(i) is invalid. The second algorithm in Section 3.2 appears to be a valid complete algorithm for Theorem B(i), so this error does not by itself overturn Theorem B, but the first algorithm must be corrected or removed.","section":"Section 3.1, first proof of Theorem B(i)"}],"minor_comments":[{"comment":"Theorem 2.3 is missing the predicate: it should read 'A finitely generated subgroup H of Mod(S) is Morse if and only if either H is stable in Mod(S) or H has finite index in Mod(S)'.","section":"Theorem 2.3"},{"comment":"In the benign-filling criterion, the indices are mixed: since the tuple contains γ∈H^{g_j}∩P_j, the intersection to check should be π(H)^{π(g_j)}∩π(P_j) and the quotient that must be infinite should be π(P_j)=P_j/N_j; the printed 'π(P_i)' appears to be a typo.","section":"Section 4.2, proof of Theorem C(iv)"},{"comment":"There are numerous typographical errors, including 'qausigeodesics' (Definition 1.3), 'undisto rted' (Introduction), 'hyerbolic' (Section 4), 'patrial' (Questions 1.7 and 1.8), 'Moreseness' (Question 1.8), 'Cayely graph' (Section 3.1), and an extra closing bracket in the displayed quotient in Theorem 4.22. A thorough copyedit is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main issue is Theorem A: the first proof is invalid for the general case, and the second proof covers only closed hyperbolic surfaces. The RAAG section has a similar flaw in its first algorithm, though the second algorithm appears sound. The paper should not be published in its current form, but the problems are local enough that a careful revision could make the central claims stand."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know this paper before citing it. It assembles known characterizations of stability and Morseness into algorithms for mapping class groups, right-angled Artin groups, and toral relatively hyperbolic groups, and it contains a few genuinely new pieces. But the main mapping class group result, Theorem A(i), is not proved as written.\n\nWhat is new is real. Lemma 3.11 computes the star length in a RAAG from normal forms, which is a clean algorithmic fact. Lemma 3.15 gives a cube-complex criterion for detecting non-loxodromic elements in a quasiconvex subgroup, and the \"benign Dehn filling\" search behind Theorem C(iv) is a clever way to detect non-Morseness in toral relatively hyperbolic groups. The paper also does a service by turning characterizations from Durham–Taylor, Koberda–Mangahas–Taylor, Tran, Kharlampovich–Myasnikov–Weil, and Groves–Manning into explicit decision procedures.\n\nHere is the main problem. The first proof of Theorem A(i) claims H is stable in Mod(S) iff every geodesic in the Cayley graph of Mod(S) maps to a quasigeodesic in the curve graph. That equivalence is false. Take H trivial. It is stable, but a geodesic from 1 to τ^n, where τ is a Dehn twist fixing the base curve, maps to a closed path of length roughly 2n in the curve graph, which cannot be a quasigeodesic for any fixed constants as n grows. The algorithm's finite local check over geodesics of length at most K is not connected to the H-orbit property that actually characterizes stability. So Theorem A(i) is unproved as written; the second proof only covers closed hyperbolic surfaces. A similar overstrong check appears in the first proof of Theorem B(i), but Theorem B(i) survives because the cube-complex proof looks sound. I also think the proof of Theorem C(iv) skips one condition in Definition 4.27: it declares a filling benign after checking that π(H)∩π(P_i) is finite and π(P_i) is infinite, without verifying H^{g_i}∩N_i is infinite. The enumeration can likely be fixed by putting γ into N_i, so this looks repairable.\n\nCitation pattern is fine, and the paper does not manufacture constants. This is a paper for people working on algorithmic geometric group theory. The flaws are real but localized; the paper deserves a serious referee and probably a major revision. I would send it to peer review.\n\nBest,\n[Your name]","headline":"Real algorithmic contributions in the RAAG and relatively hyperbolic sections, but the mapping class group stability algorithm rests on a false characterization and needs repair.","tokens_in":27430,"tokens_out":10792,"would_cite":false,"duration_ms":112974,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","20E07","57M07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stability and Morseness of finitely generated subgroups are algorithmically detectable in mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and limit-type groups.","keywords":["stable subgroup","Morse subgroup","mapping class group","right-angled Artin group","limit group","curve graph","relatively hyperbolic group","algorithmic group theory"],"falsifier":"Run the first mapping-class-group stability algorithm on the trivial subgroup, using a base curve fixed by a nontrivial mapping class element: the trivial subgroup is stable, so a correct partial algorithm must eventually halt, whereas the proof's stated equivalence between stability and quasigeodesic images of every full-Cayley geodesic would predict failure on a long geodesic ending at that element; the observed behavior tests the termination claim.","tokens_in":26314,"feed_emoji":"🧩","tokens_out":15039,"duration_ms":145776,"temperature":0.7,"pith_summary":"This paper aims to turn stability and Morseness of finitely generated subgroups—two generalizations of quasiconvexity from word-hyperbolic groups—into properties that algorithms can certify for four families of groups. For mapping class groups it claims a partial algorithm that halts exactly on stable subgroups, a complete stability decision for undistorted subgroups, and a partial Morseness detector. For right-angled Artin groups it claims a complete stability decision and a partial Morseness detector, while for toral relatively hyperbolic groups it claims partial detectors plus a complete Morseness decision for undistorted subgroups. The same results specialize to finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group, including ordinary limit groups, where the algorithms are complete because every finitely generated subgroup is undistorted. A sympathetic reader would care because these are algorithmic results of a kind previously known only for quasiconvexity in word-hyperbolic groups.","feed_headline":"Algorithms detect stable and Morse subgroups in four group families","feed_subtitle":"Mapping class groups, right-angled Artin groups, and toral relatively hyperbolic groups get termination guarantees.","key_machinery":"The load-bearing object is the orbit map into a δ-hyperbolic space with an algorithmic distance oracle: the curve graph for mapping class groups (vertices are isotopy classes of essential simple closed curves, edges record disjoint realizations), and the extension graph for right-angled Artin groups (vertices are conjugates of standard generators, edges record commutation), with the star metric providing a computable quasi-isometric model. The local-to-global principle for quasigeodesics in δ-hyperbolic spaces is the mechanism that lets an algorithm certify a global quasigeodesic by checking only paths of length at most a computable constant. For toral relatively hyperbolic groups the mechanism shifts to the induced peripheral structure of a relatively quasiconvex subgroup and its intersections with conjugates of peripheral subgroups; benign Dehn fillings are used to certify failures of peripheral finite index.","core_discovery":"On the paper's own terms, the central discovery is that stability of a finitely generated subgroup can be recognized by a uniform local-to-global check in a computable hyperbolic test space. In the mapping class group, the test space is the curve graph: the subgroup is stable exactly when its orbit is quasi-isometrically embedded, and the algorithm enumerates candidate quasigeodesic constants, checks all geodesic segments of length up to a computable bound starting at the identity, and terminates when the local checks pass. The same template works in right-angled Artin groups through the extension graph and the star metric. In toral relatively hyperbolic groups, stability and Morseness are instead characterized by intersections with peripheral subgroups, and the paper combines partial algorithms for computing those intersections with a Dehn-filling search that certifies failure of Morseness. Corollary E then yields complete algorithms for stability and Morseness in limit-type groups because every finitely generated subgroup there is undistorted.","pith_inferences":["The local-to-global template is not tied to the four classes in the paper; any group with a computable hyperbolic test space and a stability-orbit characterization could in principle receive the same partial algorithm, though the paper does not claim this.","For the mapping class group proof, the algorithm's dependence on enumerating geodesics from the identity suggests that a practical implementation would first need an efficient curve-graph distance oracle and explicit hyperbolicity constants.","The right-angled Artin cube-complex algorithm connects stability to the absence of simple loops labeled by join words, pointing toward an automata-theoretic or regular-language description of stable subgroups that could extend to other subgroup properties.","If a complete algorithm for detecting infinite index in mapping class groups or right-angled Artin groups were found, the partial Morseness algorithms would become complete; the paper explicitly leaves this as an open problem."],"forward_implications":["In a mapping class group, every undistorted subgroup can be certified either stable or non-stable in finite time; the only gap for a complete Morseness decision is an algorithm for detecting infinite index.","In a right-angled Artin group, stability of any finitely generated subgroup is decidable in full; Morseness has a partial algorithm that halts on every Morse subgroup and runs forever only on non-Morse infinite-index subgroups.","In a toral relatively hyperbolic group, an undistorted subgroup can be certified Morse or non-Morse, and as a corollary the finite-index property for undistorted subgroups is decidable.","For ordinary limit groups, stability and Morseness of every finitely generated subgroup are completely decidable without an undistortedness hypothesis.","When these algorithms halt they also produce certificates: a quasiconvexity constant, a non-loxodromic witness, a nontrivial peripheral intersection, or a benign Dehn filling."],"supporting_citations":[{"why":"Establishes that stability of a subgroup of Mod(S) is equivalent to convex cocompactness, so the curve-graph orbit test can detect stability.","marker":"[DT15]"},{"why":"Shows convex cocompactness in Mod(S) is equivalent to quasi-isometric embeddedness of an orbit in the curve complex, the geometric criterion the algorithm checks.","marker":"[KL08]"},{"why":"Characterizes convex cocompactness as undistorted and purely pseudo-Anosov, used in the complete algorithm for undistorted subgroups in Theorem A(ii).","marker":"[BBKL18]"},{"why":"Proves a finitely generated subgroup of Mod(S) is Morse exactly when it is stable or has finite index, which Theorem A(iii) turns into an algorithm.","marker":"[K19]"},{"why":"Characterizes stable subgroups of a right-angled Artin group as purely loxodromic with quasi-isometrically embedded extension-graph orbits, the criterion behind Theorem B(i).","marker":"[KMT17]"},{"why":"Introduces the star metric and shows it is quasi-isometric to the extension graph, making the quasigeodesic checks in the first proof of Theorem B(i) computable.","marker":"[KK14]"},{"why":"Supplies the peripheral-intersection characterization of stability and Morseness for undistorted subgroups of relatively hyperbolic groups, used in Theorem C.","marker":"[T17]"},{"why":"Supplies the partial algorithms for relatively quasiconvex subgroups with peripherally finite index and for computing intersections, which drive the toral relatively hyperbolic algorithms.","marker":"[KMW17]"},{"why":"Provides the relatively hyperbolic Dehn-filling results that let the algorithm detect failure of peripheral finite index and hence non-Morseness in Theorem C(iv).","marker":"[GM17]"},{"why":"Provides algorithmic computations of distances and geodesics in the curve graph and its hyperbolicity, needed for the finite geodesic checks in Theorem A(i).","marker":"[B06]"}],"fun_headline_variants":["Algorithmic stability tests for mapping class and RAAG groups","Stable subgroup detection now algorithms for four group families","Detect stability and Morseness algorithmically in four group classes","Terminating algorithms spot stable and Morse subgroups in limit groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithms assume that stability of a subgroup is exactly the geometric or peripheral condition they check—quasi-isometrically embedded orbit in a curve or extension graph, purely loxodromic behavior, or trivial intersections with peripheral subgroups—and that this characterization is effective enough that finitely many geodesic checks or intersection computations can certify it.","fun_headline_variants_meta":{"raw":{"variants":["Algorithmic stability tests for mapping class and RAAG groups","Stable subgroup detection now algorithms for four group families","Detect stability and Morseness algorithmically in four group classes","Terminating algorithms spot stable and Morse subgroups in limit groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1349,"prompt_tokens":855,"completion_tokens":494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":426}},"tokens_in":471,"tokens_out":494,"duration_ms":5969,"temperature":1.0,"reasoning_tokens":426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:45:58.082691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the first mapping-class-group stability algorithm on the trivial subgroup, using a base curve fixed by a nontrivial mapping class element: the trivial subgroup is stable, so a correct partial algorithm must eventually halt, whereas the proof's stated equivalence between stability and quasigeodesic images of every full-Cayley geodesic would predict failure on a long geodesic ending at that element; the observed behavior tests the termination claim.","supporting_citations":[],"review_version":1}