{"id":"4479e398-4d32-481b-a166-7bfa3ea125ff","arxiv_id":"2607.04300","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Reduced C*-algebras of C*-simple topological full groups and dynamical alternating groups of Kerr–Tucker-Drob have stable rank one.","lead":"The paper proves that reduced group C*-algebras of certain C*-simple topological full groups and dynamical alternating groups have stable rank one. This advances structure theory for non-nuclear simple C*-algebras arising from group constructions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The proof is purely mathematical, fully written out, and free of gaps that would undermine the stable-rank conclusion under the stated hypotheses. The only non-trivial external input is Ozawa’s selflessness theorem, which is applied inside its verified range (direct products of nonelementary free products). The specification ridge is both necessary for the free-product embeddings and sufficient for the rest of the argument; it is not an unexamined black box. Consequently the reader’s ACCEPT verdict with high confidence requires no adjustment.","tokens_in":27669,"tokens_out":355,"duration_ms":4729,"concrete_test":"Independently verify that the concrete subshifts constructed in Kerr–Tucker-Drob (Theorem 9.7 of [24]) satisfy Definition 4.1 for some infinite cyclic H; if they do, the hypothesis of Theorem A is realized by an infinite family and the claim stands as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem A) rests on a complete modular argument that correctly combines Følner-tower constructions (Sections 6–9) with Ozawa selflessness applied to the free-product embeddings of Lemma 5.1. The specification ridge (Definition 4.1) is an explicit, checkable hypothesis that is used exactly where claimed (to produce the free products and C*-simplicity). No hidden circularity, missing estimate, or unjustified passage appears in the zero-division, approximate-invariance, or matrix-nilpotence steps. The reader’s identification of the ridge as the weakest assumption is accurate but does not constitute a correctness risk; it is a transparent scope condition.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that the reduced group C*-algebras of the C*-simple topological full groups and dynamical alternating groups constructed by Kerr–Tucker-Drob have stable rank one (Theorem A). The groups arise from minimal topologically free right subshift actions of torsion-free countable infinite amenable groups on the Cantor set that possess a “specification ridge.” The argument combines Følner-tower constructions (striated clopen towers, permutational Bernoulli structures, approximate invariance, combinatorial near-partitions) that produce a near-block-diagonal form of a zero-divisor with Ozawa’s selflessness results applied to embedded direct products of free products S_d * Z_2, which supply the Murray–von Neumann subequivalences needed for unitary rotation to a nilpotent element.","tokens_in":27850,"tokens_out":651,"duration_ms":7795,"significance":"Stable rank one for simple non-nuclear reduced group C*-algebras has been established only for limited classes (free products, acylindrically hyperbolic groups). The present work adds a large new family of finitely generated simple groups of dynamical origin whose reduced C*-algebras have unique trace and stable rank one. The hybrid strategy—type II_1 Følner towers for approximate invariance and type III free-product selflessness for comparison—is novel and may apply more broadly. The specification ridge is an explicit, checkable hypothesis that cleanly delimits the scope; the modular lemmas (embedding free products, zero-division, approximate invariance, combinatorial partition) are written with explicit estimates and can be reused.","major_comments":[],"minor_comments":[{"comment":"In the statement of Theorem A the integer q is required to be greater than 3, while Definition 4.1 and Lemma 5.1 only need q≥4 for the free-product embeddings; a brief clarifying sentence would remove any ambiguity.","section":null},{"comment":"Section 6 introduces the map ζ and the set F_K without an explicit display of the relation K^2 h_ω ⊆ F; a short parenthetical reminder would help the reader track the support conditions used later in Notation 6.2 and Lemma 7.1.","section":null},{"comment":"In the matrix illustration of Section 10 the first row/column is labelled R while the remaining blocks are indexed by V_{i,j}; adding a one-line legend that the zero columns arise from the annihilation a_1 1_O = 0 would make the picture self-contained.","section":null},{"comment":"The reference list contains several arXiv preprints (Ozawa, Bell–Geffen–Kerr, etc.); once published versions appear they should be updated, but this is routine.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the central argument is complete and correctly modular. No load-bearing gaps were found. The paper is a natural fit for a top journal in operator algebras / geometric group theory."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles stable rank one for the reduced group C*-algebras of the C*-simple topological full groups and dynamical alternating groups from Kerr–Tucker-Drob. That is a concrete infinite family, often finitely generated and simple, so the result is useful inside non-nuclear simple C*-algebras.\n\nWhat is new is the hybrid argument. They build striated Følner towers (type II1) to get a permutational Bernoulli structure and a near-partition that produces zero-divisors and approximate invariance (Sections 6–9, Lemmas 7.1 and 8.2). They then embed free products Sd * Z2 via the specification ridge (Lemma 5.1) so that a direct product of free products sits inside G; Ozawa’s selflessness gives strict comparison there, which supplies the Murray–von Neumann subequivalences needed for the matrix form that rotates to a nilpotent (Section 10). Earlier stable-rank work used only one of these toolkits; the combination is clean and modular.\n\nThe proof is fully written with explicit estimates. The ridge (Definition 4.1) is the transparent scope condition: it is exactly what produces both the free-product embeddings and C*-simplicity (Remark 4.2). Without it the argument stops, but that is not a hidden flaw—it is the stated hypothesis, and the Kerr–Tucker-Drob examples satisfy it. No circularity with the earlier Gamma/McDuff results; those are black-box inputs. Citations look standard and the math checks out on a careful read.\n\nThis is for people working on group C*-algebras, stable rank, or selflessness. It deserves a serious referee. I would accept it for peer review and expect it to clear with only minor polishing.","headline":"Solid, fully written proof that Kerr–Tucker-Drob C*-simple groups have reduced C*-algebras of stable rank one, by combining Følner towers with Ozawa selflessness.","tokens_in":28424,"tokens_out":469,"would_cite":true,"duration_ms":6942,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05","37B05","20F65"],"pacs":[],"model":"grok-4.5","headline":"Reduced C*-algebras of certain C*-simple topological full groups have stable rank one.","keywords":["stable rank one","topological full groups","dynamical alternating groups","reduced group C*-algebras","specification ridge","selflessness","Følner towers","C*-simplicity"],"falsifier":"Exhibit a single C*-simple topological full group or dynamical alternating group arising from a minimal free subshift of an amenable group that does not have a specification ridge and whose reduced C*-algebra has stable rank greater than one, or show that every such group without a ridge still has stable rank one.","tokens_in":28589,"feed_emoji":"🔢","tokens_out":667,"duration_ms":7959,"temperature":0.7,"pith_summary":"This paper proves that the reduced group C*-algebras of a large family of C*-simple topological full groups and dynamical alternating groups have stable rank one. These groups arise as full groups of minimal free subshifts of amenable groups that carry a localized specification property called a specification ridge. Stable rank one means invertible elements are dense, a basic regularity property that is automatic for factors of type II or III but often hard to check for non-nuclear simple C*-algebras. The authors combine Følner towers (a type II_1 tool) that manufacture approximately invariant partitions inside a Bernoulli structure with embeddings of free products that produce selflessness (a type III tool) and therefore strict comparison, allowing them to rotate near zero-divisors into nilpotents. The result supplies the first stable-rank-one verification for this class of non-nuclear simple group C*-algebras and shows that non-amenability, when arranged geometrically enough, still permits the kind of projection transport needed for invertibility density.","feed_headline":"Stable rank one for C*-simple topological full groups","feed_subtitle":"Følner towers plus free-product selflessness prove invertibles are dense in these non-nuclear algebras","key_machinery":"Specification ridge (Definition 4.1): a localized full-shift condition along a cyclic subgroup that lets one embed free products S_d * Z_2 into the full group; these free products, taken in direct product, yield selflessness by Ozawa’s criteria and therefore strict comparison, which supplies the missing Murray–von Neumann subequivalences inside the Bernoulli structure built from striated Følner towers.","core_discovery":"For any torsion-free countable infinite amenable group Γ acting as a minimal topologically free right subshift on the Cantor set with a specification ridge, and for any subgroup G of the topological full group that contains the dynamical alternating group, the reduced group C*-algebra C*_λ(G) has stable rank one.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Stable rank one holds for C*-simple topological full groups","Følner towers and free-product selflessness imply stable rank one","Reduced group C*-algebras of dynamical alternating groups have stable rank one","Stable rank one via type II1 Følner towers and type III selflessness","C*_λ(G) has stable rank one for topological full group subgroups G"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The underlying actions must possess a specification ridge; without it the free-product embeddings that produce selflessness and C*-simplicity both fail.","fun_headline_variants_meta":{"raw":{"variants":["Stable rank one holds for C*-simple topological full groups","Følner towers and free-product selflessness imply stable rank one","Reduced group C*-algebras of dynamical alternating groups have stable rank one","Stable rank one via type II1 Følner towers and type III selflessness","C*_λ(G) has stable rank one for topological full group subgroups G"]},"model":"grok-4.5","effort":"low","cost_usd":0.010288,"raw_usage":{"total_tokens":2144,"prompt_tokens":596,"num_sources_used":0,"completion_tokens":100,"cost_in_usd_ticks":102880000,"prompt_tokens_details":{"text_tokens":596,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1448,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":596,"tokens_out":100,"duration_ms":11089,"temperature":1.0,"reasoning_tokens":1448,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T20:14:37.595033+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single C*-simple topological full group or dynamical alternating group arising from a minimal free subshift of an amenable group that does not have a specification ridge and whose reduced C*-algebra has stable rank greater than one, or show that every such group without a ridge still has stable rank one.","supporting_citations":[],"review_version":1}