REVIEW 5 minor 1 cited by
The Zariski Topology on Homeomorphism groups
T0 review · 0 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read The Zariski topology on Thompson's group V is irreducible, so it is neither Hausdorff nor a group topology.
desk verdict The V-irreducibility result is real and new; this paper deserves a serious referee. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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
What would settle it
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.
Extended reading notes
Core claim
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
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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.
minor comments (5)
- [Lemma 4.2, p.7] 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.
- [Lemma 4.2, p.8] 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.
- [Lemma 3.1, p.4] 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.
- [Theorem 3.3, p.6] 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.
- [Introduction, p.2] 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.
Circularity Check
No significant circularity: the main theorems are proved from self-contained Lemmas 3.1 and 4.2.
full rationale
The paper's central results are not circular. Theorem 1.4 (and hence Theorem 1.3 for V and Homeo(2^ω)) rests on Lemma 3.1, whose proof is an explicit inductive construction of a finite partial permutation, using only high transitivity and the hypothesis of infinite support. That hypothesis is verified independently for the target groups: for a nontrivial homeomorphism of a Hausdorff space with no isolated points, the support is a nonempty open set, hence infinite; for V_n one passes to a faithful highly transitive orbit with no isolated points. No fitted parameter is renamed as a prediction, and no group or topology is defined in terms of the conclusion. Theorems 1.1 and 1.2 follow from Lemma 4.2, which is also proved in the paper with direct commutator and measure estimates and is applied to explicitly described Thompson group elements. Theorem 1.5 assembles these results with standard facts (high transitivity of Homeo(M) for n≥2, classification of 1-manifolds). Citations such as [BEM25], [BEMP25], and [BGP12] are used for context, comparison, or standard background, not as load-bearing support for the present theorems. The remark that the main lemma 'closely resembles Theorem 1.12 of [BEMP25]' but needs stronger hypotheses is a comparison, not a reduction to that cited result. I found no step where an equation or construction reduces by definition to an input, and no self-citation chain that forces the main conclusions. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (7)
- standard math Zariski topology is coarser than every Hausdorff group topology, and coincides with the Markov topology for countable or abelian groups
- standard math For a compact metric space X, Homeo(X) with the compact-open topology is a topological group induced by the sup-metric
- standard math Every T1 topological group is Tychonoff and hence Hausdorff
- domain assumption Connected n-manifolds with n ≥ 2 have highly transitive homeomorphism groups
- standard math Connected 1-manifolds are homeomorphic to R or S^1
- standard math The support of a homeomorphism of a Hausdorff space is open, because the fixed-point set is closed
- domain assumption Thompson groups F_n and T_n contain elements with support exactly an arbitrary interval with endpoints in Z[1/n]
Cite this review
Pith. "Pith review of The Zariski Topology on Homeomorphism groups." pith.science (2026). https://pith.science/paper/FUCAVHQC
@misc{pith2026260115185,
author = {Pith},
title = {Pith review of: The Zariski Topology on Homeomorphism groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/FUCAVHQC}},
note = {Machine review of arXiv:2601.15185}
}
abstract
The Zariski topology on a group G is the coarsest topology such that all sets of the form $\{x \in G | 1_G \neq g_0 x^{k_0} g_1 ... g_{l-1} x^{k_{l-1}} g_l\}$ are open. Originally introduced by Bryant as the verbal topology, it serves as a fundamental tool for investigating the topological structure of infinite groups and is always a $T_1$ topology with continuous shifts and inversion. Since the Zariski topology is coarser than every Hausdorff group topology on G, it provides a natural starting point for topologizing groups; specifically, for countable or abelian groups, it is known that the Zariski topology coincides with the Markov topology-the intersection of all Hausdorff group topologies on G. In this paper, we analyze the Zariski topology on various homeomorphism groups. We demonstrate that for the Thompson groups F and T, the Zariski (and thus Markov) topology coincides with the standard compact-open topology derived from their respective actions on $[0,1]$ and $S^1$. In contrast, we show that the Zariski (and thus Markov) topology on Thompson's group V is irreducible, and therefore neither Hausdorff nor a group topology. As V acts highly transitively on each of its orbits, this result stands in notable opposition to a theorem by Banakh et al, which establishes that the Zariski topology on any permutation group containing all finitely supported elements is a Hausdorff group topology. Our results for the Zariski topologies on $F,T$ and $V$ also apply to the full homeomorphism groups $\operatorname{Homeo}([0,1])$, $\operatorname{Homeo}(S^1)$, and $\operatorname{Homeo}(2^\omega)$ respectively. We conclude by providing a classification of the connected manifolds $M$ for which the homeomorphism group $\mathrm{Homeo}(M)$ admits a Hausdorff Zariski topology.
Forward citations
Cited by 1 Pith paper
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All mixed identities are singular in groups with no algebraicity
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Reference graph
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Reviewed August 4, 2026 · model on record in the stance chip above.
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