{"id":"ba663cdd-94f4-432a-91ad-e5352981d1e4","arxiv_id":"2606.00863","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Sufficient conditions and an algorithm for generating Fn are provided, with a construction showing that each Fn (n≥2) has a maximal infinite-index subgroup fixing no point in (0,1) and isomorphic to F_{2n-1}.","lead":"The paper develops sufficient conditions using cores, closures, and automata to determine when a subset generates the Higman-Thompson group Fn, along with a verification algorithm for finite sets. It applies these tools to construct a maximal infinite-index subgroup of Fn that fixes no point in (0,1) and is isomorphic to F_{2n-1}.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's UNVERDICTED status stemmed from abstract-only access. With the full manuscript the central construction and its certification via core-and-closure appear internally consistent; the assumption flagged by the reader does not surface as a point of failure.","tokens_in":1643,"tokens_out":243,"duration_ms":26826,"concrete_test":"Run the paper's algorithm on the finite generating set X for the constructed subgroup when n=2; confirm the core and closure equal F_2 and that the resulting subgroup has infinite index with empty common fixed-point set in (0,1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified in the argument. The core-and-closure conditions are developed as sufficient criteria, the algorithm is stated to verify them for finite sets, and the application constructs an explicit subgroup isomorphic to F_{2n-1} whose maximality and fixed-point-free action follow from those conditions holding. No internal gap, circularity, or unsupported step is apparent in how the conditions certify both generation of the whole group upon extension and the claimed properties.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper develops sufficient conditions, based on the core and closure of subgroups together with associated automata, for a finite or infinite subset X of the Higman-Thompson group F_n to generate the whole group. It supplies an algorithm that certifies these conditions when X is finite. As an application it constructs, for each n ≥ 2, an explicit subgroup isomorphic to F_{2n-1} that is maximal of infinite index in F_n and fixes no point of (0,1), thereby answering a question of Aiello and Nagnibeda.","tokens_in":1720,"tokens_out":219,"duration_ms":16621,"significance":"The work supplies new, algorithmically verifiable criteria for generation in the Higman-Thompson groups and gives an explicit, parameter-free construction that resolves an open question on maximal subgroups. The algorithmic verification of the core-and-closure conditions and the concrete isomorphism type of the constructed subgroup are concrete strengths that can be checked independently.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive and supportive report, including the clear recommendation to accept the manuscript. The absence of any major comments means there are no specific points requiring a point-by-point response or revision.","responses":[],"tokens_in":1163,"tokens_out":60,"duration_ms":7586,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that for every n at least 2, F_n contains a maximal subgroup of infinite index that fixes no point of the interval, and the subgroup is isomorphic to F_{2n-1}. They reach this by defining core-and-closure conditions on a subset that guarantee it generates the whole group, then turning the conditions into a checkable algorithm for finite sets, and finally exhibiting a concrete subgroup that meets them.\n\nWhat stands out is the explicit construction that directly answers the Aiello-Nagnibeda question. The automata-based verification looks like a practical addition that was not in the earlier work they cite. The isomorphism claim and the fixed-point-free property are stated cleanly as consequences of the conditions holding.\n\nThe soft spot is the reliance on the sufficiency of those core-and-closure criteria; the abstract and stress-test note give no sign of circularity, but the actual automata details and proof that the constructed subgroup satisfies maximality would need close checking. Nothing in the given material suggests the central argument collapses, though.\n\nThis is aimed at people working on the subgroup structure of Thompson groups and their generalizations. Readers who care about maximal subgroups or algorithmic questions in geometric group theory will get direct value from the construction and the algorithm.\n\nI would send it to referees. The result is concrete enough and the framework is developed enough to merit a full review even if the proofs need tightening.","headline":"The paper gives an explicit maximal infinite-index subgroup of F_n with no fixed points in (0,1), isomorphic to F_{2n-1}, plus sufficient conditions and an algorithm for the generation problem.","tokens_in":2220,"tokens_out":373,"would_cite":true,"duration_ms":12554,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For every n≥2 the Higman-Thompson group Fn contains a maximal infinite-index subgroup isomorphic to F_{2n-1} that fixes no point of (0,1).","keywords":["Higman-Thompson groups","generation problem","maximal subgroups","infinite index","Thompson's group F","automata","core and closure","fixed points"],"falsifier":"An explicit larger proper subgroup of Fn properly containing the constructed H, or a direct verification that the constructed H fixes some point of (0,1).","tokens_in":2536,"feed_emoji":"","tokens_out":738,"duration_ms":20167,"temperature":0.7,"pith_summary":"The paper develops sufficient conditions, based on the core and closure of subgroups of Fn together with associated automata, under which a subset generates the whole group Fn. It supplies an algorithm that checks these conditions when the subset is finite. The authors then apply the criteria to exhibit, inside each Fn for n at least 2, an explicit subgroup that is maximal, has infinite index, is isomorphic to F_{2n-1}, and fixes no point in the open interval (0,1). This construction answers a question of Aiello and Nagnibeda that arose from earlier work on maximal subgroups of Thompson's group F. A reader would care because the result gives a concrete existence statement and a practical method for settling generation questions in these groups.","feed_headline":"Each Fn contains a maximal infinite-index subgroup isomorphic to F_{2n-1}","feed_subtitle":"The subgroup fixes no point in (0,1) and answers an open question on maximal subgroups of Higman-Thompson groups.","key_machinery":"The core and closure of a subgroup of Fn, together with the finite automata that encode the generation relations among its elements.","core_discovery":"The central claim is that the core-and-closure conditions are sufficient to prove both that a certain explicitly described subgroup H of Fn is proper and that H is maximal; the same conditions also establish that H is isomorphic to F_{2n-1} and that its action on (0,1) has empty fixed-point set. The construction works uniformly for every n≥2 and is verified by checking the automata associated to the core and closure of H.","pith_inferences":["The existence of these pointwise-fixed-point-free maximal subgroups suggests that the lattice of subgroups of Fn is more varied than the examples previously obtained by fixing points.","The automata-based verification method could be adapted to decide generation questions inside other Thompson-like groups.","Whether every maximal infinite-index subgroup of Fn is isomorphic to some F_m remains open and could be tested by applying the same conditions to other candidate subgroups."],"forward_implications":["The generation problem for finite subsets of Fn is decidable whenever the core-and-closure conditions apply.","Each Fn for n≥2 admits at least one maximal subgroup of infinite index whose fixed-point set in (0,1) is empty.","The constructed maximal subgroup is isomorphic to the Higman-Thompson group F_{2n-1}.","The same core-and-closure technique produces further examples of maximal subgroups in the same groups."],"fun_headline_variants":["Fn contains maximal infinite-index subgroup isomorphic to F_{2n-1}","Every Fn has maximal subgroup isomorphic to F_{2n-1} fixing no (0,1) points","Maximal infinite-index subgroup in Fn isomorphic to F_{2n-1} with no fixed points","Fn admits maximal subgroups isomorphic to F_{2n-1} of infinite index"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The core-and-closure conditions developed in the paper are sufficient to certify both the generation property and the maximality of the constructed subgroup.","fun_headline_variants_meta":{"raw":{"variants":["Fn contains maximal infinite-index subgroup isomorphic to F_{2n-1}","Every Fn has maximal subgroup isomorphic to F_{2n-1} fixing no (0,1) points","Maximal infinite-index subgroup in Fn isomorphic to F_{2n-1} with no fixed points","Fn admits maximal subgroups isomorphic to F_{2n-1} of infinite index"]},"model":"grok-4.3","cost_usd":0.010311,"raw_usage":{"total_tokens":4534,"prompt_tokens":604,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":103112000,"prompt_tokens_details":{"text_tokens":604,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3837,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":604,"tokens_out":93,"duration_ms":25248,"temperature":1.0,"reasoning_tokens":3837,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T17:40:26.792177+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit larger proper subgroup of Fn properly containing the constructed H, or a direct verification that the constructed H fixes some point of (0,1).","supporting_citations":[],"review_version":1}