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A note on intrinsic topologies of groups

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.11500 v2 pith:CQGSZFGA submitted 2025-06-13 math.GR math.GN

S. Bardyla , L. Elliott , J. D. Mitchell , Y. Péresse This is my paper · ORCID
classification math.GRmath.GN MSC 20B3020B3520K4522A0554H15
keywords groupZariskitopologysemigroupboundedtopologiessymmetricHausdorff-MarkovFréchet-Markovnoalgebraicityhyperconnected
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 6 minor

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.

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 (1)
  1. [Section 4, proof of Theorem 1.20] 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.
minor comments (6)
  1. [Section 2, Claim 2.1] 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.
  2. [Section 1, before Theorem 1.8] 'The formulating of the second key theorem' should read 'The formulation of the second key theorem'.
  3. [Section 1, Definition 1.17] The text 'Fr´echet-Markovtopology' is missing a space; it should be 'Fr´echet-Markov topology'.
  4. [Section 3, proof of Theorem 1.12] 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.
  5. [Section 4, proof of Lemma 4.1] 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.
  6. [Section 4, proof of Theorem 1.20] The phrase 'closed subset of Ux,x' should be clarified as 'closed in the subspace topology on Ux,x' to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stated theorems are proved from explicit constructions and external cited theorems, with no load-bearing definitional or self-citational reduction.

full rationale

The paper's main results are derived from self-contained arguments rather than from their own conclusions. Theorem 1.6 is established by constructing the abelian group G as the direct sum of quotients F/N_k and explicitly comparing Zn(G) and Zm(G) via the subset T_m: Claim 2.1 proves the subspace topology on T_m is cofinite in (G, Zn(G)), while the set {f : (m)f = x^{2n}N_m} is closed in (G, Zm(G)), showing the two topologies differ. No fitted parameter or predicted quantity appears anywhere in this argument. Theorem 1.12 is proved by a direct induction constructing partial bijections satisfying conditions (1)-(5), with no reliance on the theorem itself. Theorem 1.20 is proved from Lemmas 4.1-4.3 together with the standard fact that any Hausdorff semigroup topology contains the semigroup Zariski topology; this fact is used as an input premise, not as an unverified self-citation that defines the conclusion. The Banakh-Guran-Protasov theorem [1] and the Poor-Rinot construction [25] are external results, and the paper does not rename or refit them. No equation is defined in terms of the quantity being derived, no fitted input is called a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. A possible localized proof gap in Theorem 1.20 concerning whether W \ (U_{x,x} ∩ U_{y,y}) is open would be a correctness issue, not a circularity issue, and does not affect this assessment.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters and no invented entities. The paper relies on standard mathematics and previously published theorems (Banakh-Guran-Protasov, Poór-Rinot, Elliott et al.), which are cited and not re-derived.

assumptions (4)
  • standard math ZFC set theory plus standard group theory and topology
    The paper works in ordinary ZFC and uses standard facts about group topologies, semigroups, and permutation groups.
  • domain assumption Theorem 1.9 (Banakh-Guran-Protasov): for Sym_ω(X) ⊆ G ⊆ Sym(X), Z±_4(G) = Z±(G) is the pointwise convergence topology
    Used directly to prove Theorem 1.8 and the dichotomy Corollary 1.15; cited from [1] without proof.
  • domain assumption Existence of a 10^120-Shelah group of size ℵ1 in ZFC (Poór-Rinot [25])
    The proof of Theorem 1.19 builds on this group construction and [25, Lemma 5.11].
  • domain assumption Z(S) ⊆ HM(S) for every semigroup S
    Used in the proof of Theorem 1.20; this result is from [14].

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Pith. "Pith review of A note on intrinsic topologies of groups." pith.science (2026). https://pith.science/paper/CQGSZFGA

@misc{pith2026250611500,
  author       = {Pith},
  title        = {Pith review of: A note on intrinsic topologies of groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQGSZFGA}},
  note         = {Machine review of arXiv:2506.11500}
}
read the original abstract

We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that there exists a countable abelian group in which no bounded version of the Zariski topology coincides with the full Zariski topology. Complementing a recent result by Goffer and Greenfeld, we show that on any group with no algebraicity the semigroup Zariski topology is hyperconnected and hence, in many cases, is distinct from the group Zariski topology. Finally, we show that on the symmetric groups, the semigroup Hausdorff-Markov topology coincides with the topology of pointwise convergence.

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Forward citations

Cited by 2 Pith papers

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Reference graph

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