{"id":"b04c86ca-4d20-4427-9996-13bf8d2e5831","arxiv_id":"2501.00908","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local decomposability characterises when a piecewise full group admits a unique non-discrete t.d.l.c. topology, and the resulting alternating full group is an open, compactly generated, simple derived subgroup that is universal within the class SLD.","lead":"This paper identifies a clean condition, local decomposability, under which a piecewise full group of homeomorphisms of the Cantor space carries a non-discrete locally compact topology, and the topology is unique when it exists. A generalist might read it because it unifies the Neretin, Röver and Lederle constructions and shows alternating full groups are universal simple building blocks inside a large family of simple locally compact groups.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7 and the universality claim (Theorem 9.1) depend on unverified companion results [20] and on a step in the proof of Theorem 9.1 that appears to overstate Lemma 5.11.","rationale":"The reader's weakest assumption correctly identifies the unverified import from [20] as the main load-bearing point: Theorem 1.7 and the universality theorem 9.1 rest on Theorem 5.12 and Proposition 5.13, and the reviewer cannot check them without the companion article. I agree that this warrants a conditional verdict. However, I also found a separate potential gap in the proof of Theorem 9.1: in the implication (iii)=>(vi), the text claims H is expansive on the basis of Lemma 5.11, which as stated only gives regional expansivity. If regionally expansive is defined as in Definition 5.8 (a compactly generated subgroup, not necessarily open), the inference is not valid. This is an internal proof issue that is independent of [20] and would need to be addressed, but it may be resolvable if the definition in [11] is more precise or if faithful local decomposability provides the extra argument. Because the main verdict remains conditional on external and internal verification, I recommend keeping the reader's verdict unchanged rather than moving to reject or accept.","tokens_in":61560,"tokens_out":22474,"duration_ms":205823,"concrete_test":"Obtain [20] and verify Theorem 1.13 and Corollary 1.10 in the form quoted as Theorem 5.12 and Proposition 5.13, and confirm that the action of A(G) on X is compressible in the setting of Corollary 5.14. Additionally, inspect the proof of Theorem 9.1 (iii)=>(vi): check whether Lemma 5.11(i) is sufficient to conclude that a compactly generated H in RLD is expansive, or whether the definition of regionally expansive in the cited [11, Proposition 5.1.2] uses an open subgroup. If the latter holds, the gap is only terminological; if not, an alternative argument is needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is Theorem 1.7: under faithful minimal locally decomposable action with A(G) compactly generated, A(G) is open, abstractly simple, equals D(F(G)), and the action is fully compressible. The proof in Section 5 invokes two results imported from the companion article [20] without proof: Theorem 5.12 (normal subgroups with compressible action are open) and Proposition 5.13 (compressible piecewise full actions have simple monolith M(G)=D(H)). These are then used repeatedly: Corollary 5.14, Theorem 8.3, and the whole of Section 9 (Theorem 9.1) rely on them. If either companion theorem is false, or if its hypotheses (notably compressibility of the A(G)-action on X) are not verified in this setting, the paper's main structural conclusions collapse. This is an external dependency, not an internal inconsistency, but it is load-bearing because no proof or proof sketch is given in the preprint. A second, internal concern arises in the proof of Theorem 9.1, implication (iii)=>(vi): the text states that by Lemma 5.11, a compactly generated open subgroup H in RLD is 'an expansive t.d.l.c. group'. Lemma 5.11(i) only asserts regional expansivity, and Definition 5.8 defines regionally expansive via a compactly generated subgroup (not necessarily open). If that subgroup is not open, containing it does not make H itself expansive. The conclusion (vi) requires an open expansive subgroup of L, so this step is not justified as written unless the intended definition of regionally expansive in [11] is 'has an open expansive subgroup' (in which case the text is merely imprecise) or there is an additional argument that faithful locally decomposable actions force expansivity. The paper's own Remark 9.5(1) flags the RLD-to-SLD implication as new, so this proof step is not a routine citation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of locally compact piecewise full groups of homeomorphisms of the Cantor space, working partly through topological Boolean inverse monoids. The main results are: local decomposability is necessary and sufficient for extending a group topology to the piecewise full group; a compact generation criterion for Nekrashevych's alternating full group A(G); and a structural theorem (Theorem 1.7) saying that under faithful, minimal, locally decomposable action with A(G) compactly generated, A(G) is open, abstractly simple, equal to D(F(G)), and has fully compressible action. The paper then introduces the class A of t.d.l.c. groups containing such alternating full groups, proves Neretin-type properties for A (free action on the Furstenberg boundary, absence of general-type hyperbolic actions), and derives a universality result for A within the local structure theory of simple t.d.l.c. groups with locally decomposable actions. The final section lists open questions.","tokens_in":61662,"tokens_out":4821,"duration_ms":45021,"significance":"If the main results are correct, the paper gives a substantial unifying framework for non-discrete piecewise full groups, placing Neretin, Röver and Lederle constructions in a common setting, and it yields a structural characterisation of local isomorphism classes of a large family of simple t.d.l.c. groups. The inverse-monoid topology extension result (Theorem 1.3, Corollary 3.34) is a genuinely useful technical contribution, and the applications to Furstenberg boundaries and hyperbolic actions give strong new constraints on groups in A. The paper is carefully written and the proofs inspected are detailed; the authors also clearly separate their own open questions.","major_comments":[{"comment":"The proof of Theorem 1.7 and the subsequent structural results (Corollary 5.14, Theorem 8.3, Theorem 9.1) rest on two results imported from the companion article [20] without proof: Theorem 5.12 ([20, Theorem 1.13]) and Proposition 5.13 ([20, Corollary 1.10]). These results carry the claims that A(G) is open in F(G) and equals D(F(G)), and that compressible piecewise full actions have simple monolith M(G)=D(H). Because the hypotheses of these results must be verified in the present setting (notably compressibility of the A(G)-action on X), and because no proof or proof sketch is supplied, the central claims cannot be fully checked from this manuscript alone. Please state the full hypotheses of these results and either reproduce their proofs in an appendix or indicate clearly where they can be found if [20] is publicly available.","section":"Section 5, Theorem 1.7 and Corollary 5.14"},{"comment":"The text asserts that, by Lemma 5.11, a compactly generated open subgroup H in RLD is 'an expansive t.d.l.c. group'. Lemma 5.11(i) only gives regional expansivity, i.e. the existence of a compactly generated subgroup (not necessarily open) that is expansive. Containing such a subgroup does not imply that H itself is expansive, and conclusion (vi) explicitly requires an open expansive subgroup A of L. This step is therefore not justified as written unless the definition of 'regionally expansive' (Definition 5.8) is strengthened to require an open witness, or an additional argument is supplied.","section":"Theorem 9.1, proof of (iii)⇒(vi)"}],"minor_comments":[{"comment":"The statement contains the typo 'acts mimimally' for 'acts minimally'; the same typo appears in the introduction where Theorem 1.7 is stated.","section":"Theorem 1.7"},{"comment":"The definition of 'regionally expansive' says 'it has a compactly generated subgroup H that is expansive' without specifying whether H must be open. Since Theorem 9.1(vi) needs an open expansive subgroup, please clarify this point or adjust the proof accordingly.","section":"Definition 5.8"},{"comment":"The phrase 'faithul continuous action' should read 'faithful continuous action'.","section":"Definition 6.5"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript is a serious contribution and the concerns above are about verifiability rather than internal inconsistency. The companion-article dependency in Section 5 is the main issue; if [20] is available or its proofs are included, and the regional-expansivity step in Theorem 9.1 is repaired, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this is a serious piece of work that, if the companion results hold, gives the first systematic framework for non-discrete piecewise full groups and proves a genuine universality statement: the local isomorphism classes of SLD and RLD coincide with those of the alternating full group class A (Theorem 9.1). The machinery in Sections 3–4—topological Boolean inverse monoids, local decomposability as the right topology-extension criterion, and the compact generation criterion for A(G) generalizing Nekrashevych—is new and well built. Theorem 1.7 (A(G) open, simple, equal to D(F(G)) under compact generation) is a substantial payoff, and the applications to Furstenberg boundaries and hyperbolic actions (Theorems 8.5, 8.8) are genuine.\n\nThe soft spots are real but not fatal. The main structural conclusions (Theorem 1.7 and everything after) rest on two imports from the companion [20]: Theorem 5.12 (compressible normal subgroups of locally decomposable actions are open) and Proposition 5.13 (simple monolith from compressible piecewise full actions). These are stated but not proved. That is a load-bearing external dependency: a referee cannot fully verify the paper without [20]. The good news is that the statements are precise and the hypotheses are clearly checked in the present setting; the companion is by the same authors, so this is a conscious division of labor rather than a hidden assumption. Still, the paper would be stronger with at least proof sketches.\n\nA smaller issue: Definition 5.8 defines regionally expansive via a compactly generated subgroup, without requiring it to be open, while the introduction and the proof of Theorem 9.1 use the open version (page 7 and the (iii)=>(vi) step). If the non-open definition is literal, the step ‘H is expansive’ does not follow from Lemma 5.11. I suspect this is a typo—the standard definition and the one used in the argument require an open subgroup—but it should be fixed. Also note the 5-point orbit condition in Theorem 4.14 excludes the Juschenko–Monod 3-point case; that is a scoping caveat, not an error.\n\nOverall: the paper deserves a serious referee. It is long, careful, and honest about its open questions. I would accept it for peer review, with the companion article supplied to the referee, and I'd want the definitional slip fixed.","headline":"Major, likely-correct reorganization of simple t.d.l.c. group theory via piecewise full groups, but the main theorems lean on two unproved companion results and one definitional slip.","tokens_in":62538,"tokens_out":6034,"would_cite":true,"duration_ms":52686,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22D99","22D05","22F50","20E32","22A15","37C85","20E08","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that alternating full groups of locally decomposable piecewise full actions are compactly generated simple open subgroups realizing every local isomorphism class of simple locally decomposable t.d.l.c. groups.","keywords":["totally disconnected locally compact groups","piecewise full groups","alternating full groups","Boolean inverse monoids","locally decomposable actions","compact generation","fully compressible actions","local isomorphism"],"falsifier":"Exhibit a faithful minimal locally decomposable t.d.l.c. action on the Cantor space whose alternating full group is compactly generated but is not open in the full group, or whose quotient by the alternating full group is not discrete abelian. A more targeted check of Theorem 9.1: construct a compactly generated group in the robustly monolithic locally decomposable class whose group of germs has no compactly generated open subgroup that is expansive.","tokens_in":61116,"feed_emoji":"🔄","tokens_out":10171,"duration_ms":88168,"temperature":0.7,"pith_summary":"The paper sets out to show that piecewise full groups of homeomorphisms of the Cantor space—groups built by gluing together restrictions of a smaller group—can carry a canonical non-discrete locally compact topology, and that the alternating full subgroup inside them is the clean object to study. Working through inverse monoids of partial homeomorphisms, it proves a compact-generation criterion: if the Boolean completion of the monoid is compactly generated and every orbit has at least five points, then the alternating full group is compactly generated. Under minimality and local decomposability, that implies the alternating full group is abstractly simple, open, of countable index in the full group, and equal to its commutator subgroup. The payoff is a classification statement: every compactly generated simple t.d.l.c. group with a faithful locally decomposable action occurs as an open subgroup of such an alternating full group, and every robustly monolithic locally decomposable group is locally isomorphic to one.","feed_headline":"Alternating full groups cover all simple locally compact classes","feed_subtitle":"A single group construction realizes every local isomorphism class of locally decomposable simple groups.","key_machinery":"The central object is the Boolean inverse monoid BI(M) of clopen partial homeomorphisms of the Cantor space, whose group of units is the piecewise full group; it is equipped with a locally decomposable topology in which restriction to clopen sets and compatible joins are continuous. Inside it lives the alternating full group A(M), generated by the '3-cycles' of the action—homeomorphisms supported on three disjoint clopen pieces cyclically permuted. The load-bearing mechanism is the equivalence between compact generation of BI(M) and the existence of an expansive, piecewise dense, compactly generated open subgroup of the acting group; this converts the expansive dynamics of the action into a compact generating set for A(M), and then compressibility forces A(M) to be simple and open.","core_discovery":"The central claim is that local decomposability of the action is exactly the condition that lets the topology of a t.d.l.c. group extend to its piecewise full group, and that once the alternating full group A(G) is compactly generated, structure follows almost for free: A(G) is simple, open in F(G), of countable index, equal to D(F(G)), and acts fully compressibly on the Cantor space. The universality theorem then asserts that the class A of groups between D(F) and F built from such actions is the natural ambient space for the local isomorphism classes in SLD and RLD: every group in SLD is an open subgroup of a group in A∩SLD, and every group in RLD is locally isomorphic to a group in SLD. If true, this means that alternating full groups are the canonical representatives of the local isomorphism classes of all simple t.d.l.c. groups admitting faithful locally decomposable actions.","pith_inferences":["Editorial inference: The five-point orbit condition in the compact-generation theorem looks like a real boundary, not a convenience: because the proof requires the alternating group on five letters to be perfect, actions whose orbits have only three points—the range of the minimal homeomorphism constructions—may fail compact generation even when the Boolean inverse monoid is compactly generated.","Editorial inference: Theorem 9.1 turns the problem of constructing new simple locally compact groups into a search for compactly generated expansive t.d.l.c. groups with a faithful minimal micro-supported action; any such group can be completed to a group in A and then stripped down to a simple alternating full group, which should be a productive source of new examples.","Editorial inference: The paper's open question about adapting a homology theory to non-discrete full groups suggests a testable programme: if such homology is built, the quotient F(G)/A(G) could be computed by an index map that detects the modular function, giving new invariants to distinguish groups within the same local isomorphism class."],"forward_implications":["The classical almost automorphism groups of regular trees and earlier tree-based constructions become special cases of a uniform criterion: any locally decomposable t.d.l.c. group with a compactly generated expansive piecewise dense open subgroup has a compactly generated simple alternating full group.","Groups in A act freely on their Furstenberg boundary, so every conjugacy class of closed relatively amenable subgroups accumulates at the trivial subgroup in the space of closed subgroups.","Groups in A admit no general-type actions on hyperbolic spaces; their perfect alternating full subgroups have no loxodromic elements at all and are one-ended.","Every group in SLD is an open subgroup of a group in A∩SLD, and every group in RLD is locally isomorphic to a group in SLD, so a local isomorphism class is controlled by one compactly generated expansive open subgroup together with a faithful minimal micro-supported action.","For profinite branch groups, being locally isomorphic to a group in SLD is equivalent to some compactly generated open subgroup of the group of germs being expansive; otherwise, in the just-infinite case, all compactly generated subgroups have arbitrarily small invariant identity neighbourhoods."],"supporting_citations":[{"why":"Imported as Theorem 5.12: a normal subgroup with compressible action in a faithful minimal locally decomposable t.d.l.c. action is open, making A(G) open.","marker":"[20, Theorem 1.13]"},{"why":"Imported as Proposition 5.13: compressible piecewise full actions have simple monolith D(H), giving A(G) = D(F(G)).","marker":"[20, Corollary 1.10]"},{"why":"The alternating-group simplicity theorem that makes A(G) the monolith of the full group.","marker":"[39, Theorem 4.1]"},{"why":"The finite-generation theorem whose proof is adapted to produce compact generation of A(M) in the inverse-monoid setting.","marker":"[39, Theorem 5.10]"},{"why":"Establishes that faithful locally decomposable actions correspond to invariant subalgebras of the decomposition lattice, the bridge from dynamics to local isomorphism classes.","marker":"[12, Theorem 5.18]"},{"why":"Source of the class S structure theory and the fully compressible action criterion that turns compact generation into simplicity and openness.","marker":"[13]"},{"why":"Source of robustly monolithic groups and the regional expansivity and compressible monolith results used in Theorem 9.1.","marker":"[11]"}],"fun_headline_variants":["Alternating full groups realize all simple local classes","Simple t.d.l.c. groups from local decomposability","Canonical representatives for simple t.d.l.c. groups","Unifying Neretin, Roever, and Lederle constructions","Compact generation in alternating full groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The structural conclusions of Sections 5–9 depend on two theorems imported from the companion paper: a normal subgroup with compressible action is open, and a compressible piecewise full action has simple monolith (the intersection of all nontrivial normal subgroups) equal to its derived subgroup; if either theorem is false, or its hypotheses are not met in a particular action, the openness and universality claims collapse.","fun_headline_variants_meta":{"raw":{"variants":["Alternating full groups realize all simple local classes","Simple t.d.l.c. groups from local decomposability","Canonical representatives for simple t.d.l.c. groups","Unifying Neretin, Roever, and Lederle constructions","Compact generation in alternating full groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1297,"prompt_tokens":977,"completion_tokens":320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":242}},"tokens_in":593,"tokens_out":320,"duration_ms":3917,"temperature":1.0,"reasoning_tokens":242,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:41:29.063663+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a faithful minimal locally decomposable t.d.l.c. action on the Cantor space whose alternating full group is compactly generated but is not open in the full group, or whose quotient by the alternating full group is not discrete abelian. A more targeted check of Theorem 9.1: construct a compactly generated group in the robustly monolithic locally decomposable class whose group of germs has no compactly generated open subgroup that is expansive.","supporting_citations":[{"cited_title":"Caprace, C","cited_arxiv_id":null,"evidence_quote":"Source of the class S structure theory and the fully compressible action criterion that turns compact generation into simplicity and openness."},{"cited_title":"Caprace, C","cited_arxiv_id":null,"evidence_quote":"Source of robustly monolithic groups and the regional expansivity and compressible monolith results used in Theorem 9.1."}],"review_version":1}