{"id":"5abfddc0-3527-4e75-9e2d-f578c73ecac2","arxiv_id":"2601.15185","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Thompson groups F and T have Zariski topology equal to the compact-open topology, while V and Homeo(2^ω) have irreducible Zariski topology; among connected manifolds, exactly those of dimension ≤ 1 have homeomorphism groups with Hausdorff Zariski topology.","lead":"This paper shows that the Zariski topology on Thompson's groups F and T recovers the usual compact-open topology, while for Thompson's group V it is irreducible and therefore not a Hausdorff group topology. It also classifies the connected manifolds whose homeomorphism groups have a Hausdorff Zariski topology.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's central claims are supported by detailed proofs. The only place I would have expected trouble is the infinite-support hypothesis, which the reader also identified; but it holds in the intended applications, so it is not a live threat. The remaining unproved assertions (e.g., high transitivity of V_n on its orbits) are standard and can be verified by prefix-replacement constructions. Thus the reader's ACCEPT at moderate confidence stands.","tokens_in":11634,"tokens_out":54364,"duration_ms":530433,"concrete_test":"As a check on the core of Lemma 3.1, independently re-derive the final-stage case d=l_i−1 for k=+1 and k=−1, verifying explicitly that β^k is defined on (p_i)f_{i,σ}, that (p_i)f_{i,σ}β^k is the unique element of (domβ∪imβ)\\C_σ, and that the subsequent application of a_{i,l_i} lands outside C_σ. If either sign fails, the induction is broken.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined Lemma 3.1, Theorem 3.2/3.3, and Lemma 4.2. The reader's flagged assumption—that every non-identity element has infinite support—is indeed the load-bearing hypothesis, but it is satisfied in every target case: for a nontrivial homeomorphism of a Hausdorff space with no isolated points, the support is a nonempty open set and hence infinite; for V_n on an orbit this follows from faithfulness and density of the orbit. The inductive construction of the finite partial permutation is intricate but internally coherent: the counter d_σ increments by exactly one on each step, conditions (1)–(3) are preserved for all indices, and the final extension exists by high transitivity. Lemma 4.2's commutator argument correctly transfers non-commutation to non-disjointness of supports. I found no step that would invalidate Theorem 1.3 or the compact-open recovery theorems.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the group Zariski topology (verbal topology) on homeomorphism groups. It proves that for Thompson's groups F and T (and their generalizations F_n, T_n), the Zariski topology on any containing subgroup of Homeo([0,1]) or Homeo(S^1) coincides with the compact-open topology, hence is a Hausdorff group topology. In contrast, for Thompson's group V (and V_n) and for Homeo(2^ω), the Zariski topology is irreducible, hence neither Hausdorff nor a group topology. The main technical tool is Lemma 3.1, an inductive construction showing that for a highly transitive permutation group with no nontrivial finite-support elements, every finite intersection of basic Zariski-open sets is nonempty. Lemma 4.2 uses commutator conditions to show that under a support-density hypothesis on subgroups of Homeo(S^1), the Zariski topology contains every uniform ε-ball around the identity. These results are combined to classify connected manifolds whose homeomorphism groups admit a Hausdorff (or group) Zariski topology: exactly those of dimension at most 1.","tokens_in":11800,"tokens_out":27410,"duration_ms":216176,"significance":"If correct, the paper establishes striking new phenomena: the Zariski topology can coincide with a natural analytic topology for F and T, while for V it is irreducible despite V being a highly transitive group, in contrast to the Hausdorff property proved for permutation groups containing all finite-support elements. The manifold classification is a clean application of the main theorems. The proofs are long and self-contained, with the central inductive argument in Lemma 3.1 providing a strong tool applicable to many highly transitive groups. The paper also gives concrete evidence that the Zariski and Markov topologies differ from the compact-open topology in the V case. The results are likely to be of interest to researchers in topological group theory and Thompson groups.","major_comments":[],"minor_comments":[{"comment":"The definition of p_{i,l}, p_{i,r} contains the phrase 'for all 0≤j < l', where l is undefined; it should presumably be 'for all 0≤j < k' or simply 'with p_{i,r}-p_{i,l} < ε/64k'. As written, the quantifier over j is meaningless.","section":"Lemma 4.2, p.7"},{"comment":"After the sentence about the images of the intervals (p_j,p_{j+1}), the displayed sum is written as Σ_{0≤j<k-1} λ((p_j,p_{j+1})g)=1, but the list (p_0,p_1),...,(p_{k-1},p_k) has k intervals; the sum should be over 0≤j<k. The next line uses the correct range, so this is a typo.","section":"Lemma 4.2, p.8"},{"comment":"The normalization step ('By applying conjugation, grouping terms, and omitting trivial cases...') is terse. In particular, eliminating a leading constant a0 by conjugating by a0^{-1} changes the trailing constant; a short explanation would help readability. This is not a correctness issue.","section":"Lemma 3.1, p.4"},{"comment":"The sentence 'The group V_n does not act highly transitively on the Cantor set' could be misunderstood, since V_n is highly transitive on each of its orbits. Briefly noting that the Cantor set splits into many orbits (e.g., eventually-constant sequences vs. others) would clarify why the orbit argument is needed.","section":"Theorem 3.3, p.6"},{"comment":"There is a typo: 'Zarsiki' should be 'Zariski' in the sentence about Rubin's theorem. Also, the comment about the semigroup Zariski topology differing from the group Zariski topology is cryptic; consider expanding it briefly.","section":"Introduction, p.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the central claims are well supported. The few typographical issues and unclear quantifiers are easily fixed. I see no need for further technical verification. The author's self-citations are numerous but appropriate to the contextual material. This is a strong contribution suitable for publication after minor revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. The headline result — the Zariski topology on Thompson's V is irreducible, hence not Hausdorff — is real and new. So are the F/T compact-open recovery theorems and the connected-manifold classification. The paper is honest, the main proofs are self-contained, and the contrast with Banakh–Guran–Protasov is correctly drawn: V lacks finite-support elements, so the BGP theorem simply does not apply.\n\nWhat is genuinely new: Lemma 3.1 is a useful general tool — a highly transitive permutation group with no nontrivial finite-support element has irreducible Zariski topology. The application to V and Homeo(2^ω) is clean and surprising. The F/T results, while less shocking, are properly proven via the Lemma 4.2 commutator-and-support measure argument. The manifold classification falls out nicely from the earlier theorems plus standard facts.\n\nSoft spots: the proofs are long and dense. The normalization of subbasic open sets in Lemma 3.1 is stated tersely; I think it is correct (conjugate, regroup, drop trivial pieces), but a referee will want that spelled out. The inductive construction of σ is intricate — the counter d_σ increments by exactly one at each stage, and conditions (1)–(3) are preserved; I do not see a gap, but this is where a careful referee earns their keep. Minor typos exist (e.g., 'Zarsiki' in the introduction), and the measure estimate in Lemma 4.2 is delicate, though the bounds do add up. The assumption that every non-identity element has infinite support is load-bearing but satisfied in every target case: for a homeomorphism of a Hausdorff space without isolated points, the support is open and nonempty, hence infinite.\n\nI disagree with any suggestion that this is just an application of known results. The Banakh et al. theorem requires all finite-support permutations; V and Homeo(2^ω) fail that, and the irreducibility is a genuinely new phenomenon. The self-citations are for context and related work; the main theorems do not depend on them.\n\nWho this is for: people working on topological group theory, homeomorphism groups, and Thompson groups. It is a solid paper, not flashy, but the V result will be cited. Send it to a serious referee; it deserves a full review.","headline":"The V-irreducibility result is real and new; this paper deserves a serious referee.","tokens_in":12285,"tokens_out":1901,"would_cite":true,"duration_ms":18615,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F38","22A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Zariski topology on Thompson's group V is irreducible, so it is neither Hausdorff nor a group topology.","keywords":["Zariski topology","Markov topology","Thompson groups","homeomorphism groups","irreducible topology","highly transitive action","compact-open topology","Cantor space"],"falsifier":"Find a finite list of words w_1,...,w_n such that for every g in V (or Homeo(2^ω)), at least one w_i(g) equals the identity; the corresponding intersection of nonempty Zariski-open sets would be empty, contradicting irreducibility. Since the proof of Lemma 3.1 shows such a finite conjunction always has a solution, a counterexample would pinpoint a flaw in the induction.","tokens_in":11498,"feed_emoji":"🌀","tokens_out":9356,"duration_ms":79657,"temperature":0.7,"pith_summary":"This paper pins down the Zariski topology — the coarsest topology defined by non-solutions to word equations — on several important homeomorphism groups. For Thompson's groups F and T, it equals the classical compact-open topology, so it is Hausdorff and compatible with the group operations. For Thompson's group V and the full homeomorphism group of the Cantor space, it is irreducible: every two nonempty open sets intersect, so the topology is neither Hausdorff nor a group topology. The paper proves a general criterion: a highly transitive action by homeomorphisms on a space without isolated points forces the Zariski topology to be irreducible. It also classifies connected manifolds M for which Homeo(M) has a Hausdorff Zariski topology: exactly the 0- and 1-dimensional ones.","feed_headline":"Thompson's V has an irreducible Zariski topology","feed_subtitle":"Unlike F and T, V and Homeo(Cantor) have non-Hausdorff Zariski topologies: every nonempty open set is dense.","key_machinery":"The engine for the negative result is Lemma 3.1, a construction that takes any finite list of Zariski-subbasic conditions (inequalities of the form 1 ≠ g0 x^{k0} g1 ... g_{l-1} x^{k_{l-1}} g_l) and builds a group element x satisfying all of them at once. The construction inductively grows a finite partial permutation, choosing new points outside a finite forbidden set; it works because each non-identity group element has infinite support, so there are always new points whose images also avoid the forbidden set. For homeomorphism groups, infinite support follows from the fact that supports are open and the space has no isolated points. For the positive results on F and T, the key tool is Lemm","core_discovery":"The central claim is Theorem 1.3: the group Zariski topology on Thompson's group V (and the V_n for n≥2) and on Homeo(2^ω) is irreducible. Irreducibility means any two nonempty open sets have nonempty intersection, so the topology has no disjoint open sets; in particular it is not Hausdorff and not compatible with the group operation. The supporting Theorem 1.4 states that if a Hausdorff space X has no isolated points and a group G acts highly transitively on X by homeomorphisms, then the Zariski topology on G is irreducible. A separate pair of results shows the opposite behavior for F and T: the Zariski topology on these groups (and on any subgroup of Homeo([0,1]) or Homeo(S^1) containing t","pith_inferences":["The irreducibility of the Zariski topology on V suggests that no algebraic equation in one variable can distinguish elements of V; this is a strong obstruction to reconstructing the Cantor set from the group structure via algebraic equations.","The contrast between F/T and V indicates that the boundary between Hausdorff and irreducible Zariski topology may be tied to the degree of transitivity: order-preserving actions on 1-manifolds are only finitely transitive, while V's high transitivity kills separation. One could test whether intermediate transitivity (e.g., n-transitive for all n but not highly transitive) still yields irreducibili","For manifolds of dimension at least 2, the result implies that any Hausdorff group topology on Homeo(M) must use data beyond word equations; this may connect to rigidity phenomena for homeomorphism groups.","The proof of Lemma 3.1 is constructive and could potentially be turned into an algorithm that, given a finite list of inequalities, finds a group element satisfying them; this would give a computational check of irreducibility for finitely presented subgroups."],"forward_implications":["For Thompson's V, the Markov topology — the intersection of all Hausdorff group topologies — is non-Hausdorff, so no Zariski-definable set can separate points of V.","For F and T, every Hausdorff group topology must contain the compact-open topology, because the Zariski topology is coarser than every Hausdorff group topology and here coincides with compact-open.","For any connected manifold of dimension at least 2, the Zariski topology on Homeo(M) is irreducible, hence not Hausdorff and not a group topology.","The irreducibility result extends to the full homeomorphism group of the Cantor space, not just Thompson's V, and to every Thompson group V_n with n≥2."],"fun_headline_variants":["Thompson's V breaks Hausdorff: Zariski topology is irreducible","V's Zariski topology is irreducible, unlike F and T's","Irreducible Zariski topology on Thompson's V and Homeo(Cantor)","F and T match compact-open; V does not: irreducible Zariski"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction that builds an element satisfying every given Zariski inequality assumes that every non-identity group element moves infinitely many points; if the group contains an element with finite support, the inductive argument can fail, which is exactly why symmetric groups on infinite sets escape the conclusion.","fun_headline_variants_meta":{"raw":{"variants":["Thompson's V breaks Hausdorff: Zariski topology is irreducible","V's Zariski topology is irreducible, unlike F and T's","Irreducible Zariski topology on Thompson's V and Homeo(Cantor)","F and T match compact-open; V does not: irreducible Zariski"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1558,"prompt_tokens":962,"completion_tokens":596,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":706,"completion_tokens_details":{"reasoning_tokens":514}},"tokens_in":706,"tokens_out":596,"duration_ms":5709,"temperature":1.0,"reasoning_tokens":514,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:18:54.952742+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finite list of words w_1,...,w_n such that for every g in V (or Homeo(2^ω)), at least one w_i(g) equals the identity; the corresponding intersection of nonempty Zariski-open sets would be empty, contradicting irreducibility. Since the proof of Lemma 3.1 shows such a finite conjunction always has a solution, a counterexample would pinpoint a flaw in the induction.","supporting_citations":[],"review_version":2}