{"id":"2aebb280-29c0-474b-ac0f-6f36e27927f6","arxiv_id":"2506.11500","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs a countable abelian group whose bounded Zariski topologies are all distinct, shows that groups with no algebraicity have hyperconnected semigroup Zariski topology, and proves that on symmetric groups the semigroup Hausdorff-Markov topology equals the pointwise convergence…","lead":"This mathematical note studies topologies on groups that arise purely from the group operation, such as Zariski topologies, and answers two open questions about when these topologies coincide or differ. A generalist might read it to see how abstract algebra can force the existence of many distinct intrinsic topologies on the same group.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.20's proof has an unjustified openness step: from Ux,x∩Uy,y closed only in the subspace Ux,x, it infers W\\(Ux,x∩Uy,y) ∈ τ; the density contradiction therefore lacks support.","rationale":"The paper's main results are Theorem 1.6 (answering Dikranjan–Toller), Theorem 1.12 (hyperconnectedness for no-algebraicity groups), and Theorem 1.20 on symmetric groups. I checked the proofs of Theorems 1.6 and 1.12 carefully. Theorem 1.6's construction is sound: the group G is a direct sum of quotients of a free abelian group; Claim 2.1 correctly shows that polynomials of degree at most n have finitely many zeroes on Tm, and the infinite closed set {x^m = 1} ∩ Tm witnesses the failure of cofiniteness in Zm. The notation clash in Claim 2.1 (using n for both the degree bound and the generator index) is confusing but repairable. Theorem 1.12's proof is intricate but the induction on partial bijections works: the choice of q' outside (T)P ensures the extension preserves condition (4), and the p-values strictly increase, bounding the induction length. The missing label 'condition (c)' is a typo, not a mathematical gap. Lemma 3.2 is correct: cancellation allows the reductions. So the central constructions hold up. The only genuine gap is in Theorem 1.20, as the reader noted. The step from closedness in the subspace Ux,x to openness of V = W \\ (Ux,x ∩ Uy,y) is unjustified because Ux,x ∩ Uy,y is not known to be closed in the whole space. This is precisely where the density argument needs V to be open. The theorem may well be true, but the written proof does not establish it. Therefore the reader's CONDITIONAL verdict is appropriate; no change is needed.","tokens_in":13867,"tokens_out":26468,"duration_ms":230505,"concrete_test":"Analytically verify whether Ux,x ∩ Uy,y is closed in (Sym(X), τ) for every Hausdorff semigroup topology τ. If it is not, the assertion V = W \\ (Ux,x ∩ Uy,y) ∈ τ in Theorem 1.20 fails and the proof needs a new argument. A minimal test: attempt to strengthen Lemma 4.2 from closedness of Sym(X){x,y} to closedness of the two-point stabilizer Ux,x ∩ Uy,y; if this cannot be done, the gap is real.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.20 (Section 4), after applying Lemma 4.1 to the subspace Ux,x, the paper obtains W ∈ τ with W ∩ Ux,x = Ux,x ∩ Uy,y. It then asserts that V = W \\ (Ux,x ∩ Uy,y) ∈ τ and uses density of Ux,x to conclude V = ∅. However, closedness of Ux,x ∩ Uy,y has only been established in the subspace Ux,x, not in the whole space (Sym(X), τ). Since Ux,x is not closed (otherwise it would be open by Lemma 4.1, contradicting the assumption), the complement of Ux,x ∩ Uy,y need not be open, so W \\ (Ux,x ∩ Uy,y) is not necessarily open. Without V ∈ τ, the density argument fails: a nonempty set W\\A can exist without intersecting Ux,x if it is not open. Thus the proof of Theorem 1.20 is incomplete. This does not affect Theorems 1.6 and 1.12, whose proofs appear sound (modulo the notation clash in Claim 2.1 and a missing label in Theorem 1.12's induction, both cosmetic). The gap is real but localized; the theorem may still be true, but the argument as written does not establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies intrinsic topologies on groups and semigroups: the group and semigroup Zariski topologies, their bounded-degree versions, and the Hausdorff-Markov and Fréchet-Markov topologies. The main results are: (1) a countable abelian group G is constructed for which the bounded Zariski topologies Z_n^{±}(G)=Z_n(G) form a strictly increasing chain, so no bounded version equals the full Zariski topology (answering a question of Dikranjan and Toller); (2) for any subgroup G of Sym(X) with no algebraicity, the semigroup Zariski topology Z(G) is hyperconnected, giving a broad class of groups with Z(G) ≠ Z^{±}(G) and sharpening the dichotomy for groups between Sym_ω(X) and Sym(X); (3) the semigroup Hausdorff-Markov topology on Sym(X) coincides with the topology of pointwise convergence, complementing results of Gaughan and of Banakh–Guran–Protasov. The paper also proves, using the recent Poór–Rinot group, that the Fréchet-Markov topology can be discrete while the group Zariski topology is not, settling a question of Elliott et al. in ZFC.","tokens_in":14149,"tokens_out":16898,"duration_ms":150908,"significance":"If the results hold, the paper answers two open questions and adds substantial new structure theory for Zariski and Markov topologies. The abelian example of Theorem 1.6 is explicit and elementary, and its separation of bounded Zariski topologies is a clean construction. The hyperconnectedness theorem for groups with no algebraicity is broad, covering many natural permutation groups, and the resulting dichotomy for Sym_ω(X)⊆G⊆Sym(X) is elegant. The final theorem on Sym(X) is a strong complement to classical results. The proofs of Theorems 1.6, 1.12, and 1.19 are detailed and appear structurally sound. However, the proof of Theorem 1.20 contains a gap in an openness step; the gap is localized and appears repairable, but it affects a main theorem as written.","major_comments":[{"comment":"The assertion 'Then V = W \\ (Ux,x ∩ Uy,y) ∈ τ' is not justified. The set Ux,x ∩ Uy,y is shown to be closed only in the subspace Ux,x, and since Ux,x is not closed in (Sym(X),τ) (otherwise Lemma 4.1 would make it open, contrary to the standing assumption), the set need not be closed in the whole space. Hence W minus this set need not be open, and the density argument concluding V = ∅ does not go through. This is a genuine gap in the proof of a main theorem. The gap is localized and repairable: because Ux,x is dense in (Sym(X),τ) and W is open, one has W ⊆ Cl(W∩Ux,x) = Cl(Ux,x∩Uy,y) ⊆ Sym(X)_{x,y}, where the last inclusion uses that Sym(X)_{x,y} is closed in τ and contains Ux,x∩Uy,y. Thus W is a nonempty open subset of the subgroup Sym(X)_{x,y}, so Sym(X)_{x,y} is open, and the remainder of the contradiction proceeds as written. The proof should be revised to replace the flawed V-step with this (or an equivalent) argument.","section":"Section 4, proof of Theorem 1.20"}],"minor_comments":[{"comment":"The symbol n is used both for the degree bound (as in Z_n(G)) and for the index of the generator x_n; for instance 'p ≤ n < m' mixes the two roles. Please use a different letter, e.g. k, for the generator index throughout the claim and its proof.","section":"Section 2, Claim 2.1"},{"comment":"'The formulating of the second key theorem' should read 'The formulation of the second key theorem'.","section":"Section 1, before Theorem 1.8"},{"comment":"The text 'Fr´echet-Markovtopology' is missing a space; it should be 'Fr´echet-Markov topology'.","section":"Section 1, Definition 1.17"},{"comment":"In the final paragraph, 'and (mi)bi,0xe · · ·xeai,dA,i' should presumably be 'and (mi)bi,0xe · · ·xebi,dB,i'; the current expression repeats the degree of the a-row.","section":"Section 3, proof of Theorem 1.12"},{"comment":"In the (⇒) direction, it should be stated explicitly that in a semigroup topology the left and right shifts are homeomorphisms, so that cosets of an open subgroup are open; this justifies the claim that the complement of an open subgroup is open.","section":"Section 4, proof of Lemma 4.1"},{"comment":"The phrase 'closed subset of Ux,x' should be clarified as 'closed in the subspace topology on Ux,x' to avoid ambiguity.","section":"Section 4, proof of Theorem 1.20"}],"recommendation":"major_revision","confidential_remarks":"The only substantive issue is the gap in the proof of Theorem 1.20; it is localized and the repair I outline is short. The rest of the manuscript is mathematically sound in its main lines and the results are of good interest for the topology-of-groups community. I recommend major revision rather than rejection, with the expectation that the authors can fix the step cleanly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper answers two named open questions and gives a sharp degree-4 dichotomy for Zariski topologies on large permutation groups. Theorem 1.6 is the first example where all bounded Zariski topologies are pairwise distinct, resolving Dikranjan-Toller's Question 1.3. Theorem 1.12 replaces Goffer-Greenfeld's small-cancellation example with a broad structural class: any group with no algebraicity has hyperconnected semigroup Zariski topology. Theorem 1.19, using Poor-Rinot's ZFC Shelah group, settles Elliott et al.'s Question 1.18 in ZFC. These are real results.\n\nWhat's good: the proofs of 1.6 and 1.12 are detailed and appear sound. The paper is honest about what comes from where; the load-bearing steps are external theorems, and the citation pattern looks right. The main gap is in the proof of Theorem 1.20. After applying Lemma 4.1 to the subspace Ux,x, they get W in tau with W cap Ux,x = Ux,x cap Uy,y. They then claim V = W \\ (Ux,x cap Uy,y) is open. But Ux,x cap Uy,y is only known to be closed in the subspace Ux,x, not in the whole space, so V need not be open. Without that, the density argument for V = empty does not follow. This is a real gap, but it's localized and repairable: since Ux,x is dense and Sym(X)_{x,y} is closed, W cap Ux,x is dense in W and lies in Sym(X)_{x,y}, so W is contained in Sym(X)_{x,y} directly, and the rest of the proof goes through. I'd expect the theorem to hold.\n\nMinor issue: in Claim 2.1, n is used both for the degree bound and as the index of the generator x_n. Confusing but recoverable.\n\nBottom line: this is a paper for people in topological algebra and permutation groups. It deserves a serious referee; the proof gap should be fixed in revision, not used to kill the paper.","headline":"Answers two open questions and gives a sharp degree-4 dichotomy for Zariski topologies on symmetric groups; the main results look right, but the proof of Theorem 1.20 has a repairable gap.","tokens_in":14710,"tokens_out":4643,"would_cite":true,"duration_ms":41846,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B30","20B35","20K45","22A05","54H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"There is a countable abelian group on which every bounded Zariski topology is strictly contained in the next, so the bounded hierarchy never reaches the full Zariski topology.","keywords":["group Zariski topology","semigroup Zariski topology","bounded Zariski topologies","symmetric group","Hausdorff-Markov topology","Fréchet-Markov topology","no algebraicity","hyperconnected"],"falsifier":"For the group $G$ built in the proof of Theorem 1.6, examine the set of solutions of $x^m=1$ inside the distinguished subset $T_m$: the theorem says this set is closed in the subspace topology inherited from $Z_m(G)$ but not in that inherited from $Z_n(G)$ for any $n<m$. If it were closed already in $Z_n(G)$ for some $n<m$, then $Z_n(G)=Z_m(G)$ at that stage, falsifying the claim.","tokens_in":13663,"feed_emoji":"♾️","tokens_out":18769,"duration_ms":182922,"temperature":0.7,"pith_summary":"The paper studies intrinsic topologies on groups—topologies defined purely from the algebraic structure, without any external metric or order. Its main result is a countable abelian group $G$ for which the bounded Zariski topologies $Z_n(G)$ form a strictly increasing chain $Z_0(G)\\subsetneq Z_1(G)\\subsetneq\\cdots$ that never reaches the full Zariski topology $Z(G)$. This answers an open question in the literature and shows that no finite bound on polynomial degree can reproduce the full Zariski topology in general. The paper also proves that permutation groups with no algebraicity have hyperconnected semigroup Zariski topology, making the semigroup and group Zariski topologies distinct for a large class, and that on the full symmetric group the semigroup Hausdorff-Markov topology coincides with the topology of pointwise convergence.","feed_headline":"Bounded Zariski topologies never stabilize on some abelian group","feed_subtitle":"A countable abelian group makes the bounded Zariski hierarchy climb forever; permutation groups add two more results.","key_machinery":"The separating construction works with the direct sum $G=\\bigoplus_{k\\in\\mathbb{N}}G_k$ of quotients of the free abelian group $F$; the subgroups $N_k$ are chosen so that, inside the distinguished subset $T_m=\\{\\phi\\in G:(k)\\phi=1\\text{ for }k\\neq m,\\ (m)\\phi\\in\\{x_nN_m:n\\in\\mathbb{N}\\}\\}$, equations of bounded degree have finite solution sets while $x^m=1$ has infinitely many. For the permutation-group results, the key mechanism is the condition of no algebraicity—the orbit of any point outside a finite set under its pointwise stabilizer is infinite—together with a representation of basic open sets of $Z(G)$ by pairs of ragged matrices; an inductive extension of partial bijections, avoiding a finite set of forbidden images at each step, produces an element lying in any two prescribed basic open sets, proving hyperconnectedness. For the symmetric-group theorem, the machinery is the pointwise stabilizer $U_{x,x}=\\{g:(x)g=x\\}$: it is a maximal subsemigroup, and in any semigroup topology it is open exactly when it is closed, which forces every Hausdorff semigroup topology to contain all the subbasic sets of the pointwise-convergence topology.","core_discovery":"The paper's central claim is that the bounded Zariski topologies on a group need not stabilize. Concretely, it constructs a countable commutative group $G=\\bigoplus_{k\\in\\mathbb{N}}G_k$ with $G_k=F/N_k$, where $F$ is the free abelian group on generators $\\{x_n:n\\in\\mathbb{N}\\}$ and $N_k$ is a subgroup generated by suitable powers, so that for every $n<m$ the topology $Z_n(G)$ is properly contained in $Z_m(G)$. Inside a carefully chosen subset $T_m$ of $G$, every polynomial equation of degree $p\\le n<m$ has only finitely many solutions, while $x^m=1$ has infinitely many solutions, so the degree-$m$ basic open sets cannot be produced by lower-degree polynomials. The paper further claims that for any subgroup $G$ of $\\mathrm{Sym}(X)$ with no algebraicity, the semigroup Zariski topology $Z(G)$ is hyperconnected, and hence distinct from the group Zariski topology in many cases; and that on the full symmetric group $\\mathrm{Sym}(X)$, the semigroup Hausdorff-Markov topology is exactly the topology of pointwise convergence.","pith_inferences":["The strict infinite chain in Theorem 1.6 suggests that, for abelian groups with unbounded exponent-like structure, the full Zariski topology is an essential infinite union: no finite set of bounded-degree equations can define all open sets. One could test whether even longer or ordinal-indexed chains occur for uncountable groups, an extension the paper does not address.","The hyperconnectedness argument for no-algebraicity groups likely applies beyond permutation groups; any faithful action with sufficiently transitive pointwise stabilizers might make the semigroup Zariski topology hyperconnected, so the phenomenon is probably not confined to the examples listed in the paper.","If the stronger closedness property needed in the symmetric-group proof can be established, the same maximal-subsemigroup strategy may show that the semigroup Hausdorff-Markov topology on the full transformation monoid on $X$ is also the pointwise-convergence topology, a statement the paper does not make."],"forward_implications":["The open question about stabilization of bounded Zariski topologies is answered negatively: in general no finite bound on polynomial degree reproduces the full Zariski topology of a group.","For the constructed group, each increase in degree adds genuinely new open sets, so the hierarchy $Z_0(G)\\subsetneq Z_1(G)\\subsetneq\\cdots\\subsetneq Z(G)$ is an infinite strict chain rather than a finite stabilization.","For any group with no algebraicity acting on an infinite set, the semigroup Zariski topology is hyperconnected, hence not Hausdorff and not compatible with multiplication; this gives a large class where $Z(G)\\neq Z^\\pm(G)$, complementing the earlier single counterexample.","For groups sandwiched between the subgroup of finitely supported permutations and the full symmetric group, the bounded group Zariski topologies have a sharp phase transition: degree at most $3$ is hyperconnected, while degree at least $4$ is the totally disconnected pointwise-convergence topology.","On the full symmetric group, every Hausdorff semigroup topology contains the topology of pointwise convergence, so pointwise convergence is intrinsic to the algebra of the symmetric group even when inversion is not assumed continuous."],"supporting_citations":[{"why":"It poses the stabilization question that Theorem 1.6 answers negatively.","marker":"[8]"},{"why":"It introduces the bounded versions of the group Zariski topology whose hierarchy the paper studies.","marker":"[2]"},{"why":"It gives the classical link between the group Zariski topology and the existence of Hausdorff group topologies.","marker":"[24]"},{"why":"It supplies the basic facts on semigroup Zariski topologies, including the inclusion of the semigroup Zariski topology in the Hausdorff-Markov topology.","marker":"[14]"},{"why":"It provides the theorem that the degree-four group Zariski topology on permutation groups is the pointwise-convergence topology.","marker":"[1]"},{"why":"It gives the earlier example where the semigroup and group Zariski topologies differ, which Theorem 1.8 generalizes.","marker":"[18]"},{"why":"It proves that the Hausdorff-Markov group topology on the symmetric group is the pointwise-convergence topology.","marker":"[16]"},{"why":"It supplies the ZFC construction of a Shelah-like group used in Theorem 1.19.","marker":"[25]"}],"fun_headline_variants":["Countable abelian group with Zariski topologies climbing indefinitely","No stabilization: bounded Zariski topologies strictly increase on abelian group","Semigroup Zariski hyperconnected on no-algebraicity groups","Symmetric groups: Hausdorff-Markov topology equals pointwise convergence","Infinite Zariski chain on countable abelian group"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The symmetric-group theorem relies on the set of permutations fixing both $x$ and $y$ being closed in the whole symmetric group for every Hausdorff semigroup topology; the proof establishes closedness only inside the pointwise stabilizer of $x$, and the density step needs the stronger statement.","fun_headline_variants_meta":{"raw":{"variants":["Countable abelian group with Zariski topologies climbing indefinitely","No stabilization: bounded Zariski topologies strictly increase on abelian group","Semigroup Zariski hyperconnected on no-algebraicity groups","Symmetric groups: Hausdorff-Markov topology equals pointwise convergence","Infinite Zariski chain on countable abelian group"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001113,"raw_usage":{"total_tokens":4627,"prompt_tokens":930,"completion_tokens":3697,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":3604}},"tokens_in":546,"tokens_out":3697,"duration_ms":29689,"temperature":1.0,"reasoning_tokens":3604,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:04:58.934258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the group $G$ built in the proof of Theorem 1.6, examine the set of solutions of $x^m=1$ inside the distinguished subset $T_m$: the theorem says this set is closed in the subspace topology inherited from $Z_m(G)$ but not in that inherited from $Z_n(G)$ for any $n<m$. If it were closed already in $Z_n(G)$ for some $n<m$, then $Z_n(G)=Z_m(G)$ at that stage, falsifying the claim.","supporting_citations":[{"cited_title":"Dikranjan, D","cited_arxiv_id":null,"evidence_quote":"It poses the stabilization question that Theorem 1.6 answers negatively."},{"cited_title":"Zariski topologies on groups","cited_arxiv_id":"1001.0601","evidence_quote":"It introduces the bounded versions of the group Zariski topology whose hierarchy the paper studies."},{"cited_title":"Markov, On unconditionally closed sets , Mat","cited_arxiv_id":null,"evidence_quote":"It gives the classical link between the group Zariski topology and the existence of Hausdorff group topologies."},{"cited_title":"Elliott, J","cited_arxiv_id":null,"evidence_quote":"It supplies the basic facts on semigroup Zariski topologies, including the inclusion of the semigroup Zariski topology in the Hausdorff-Markov topology."},{"cited_title":"Banakh, I","cited_arxiv_id":null,"evidence_quote":"It provides the theorem that the degree-four group Zariski topology on permutation groups is the pointwise-convergence topology."},{"cited_title":"Goffer, B","cited_arxiv_id":null,"evidence_quote":"It gives the earlier example where the semigroup and group Zariski topologies differ, which Theorem 1.8 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It proves that the Hausdorff-Markov group topology on the symmetric group is the pointwise-convergence topology."},{"cited_title":"A Shelah group in ZFC","cited_arxiv_id":"2305.11155","evidence_quote":"It supplies the ZFC construction of a Shelah-like group used in Theorem 1.19."}],"review_version":1}