The Zariski Topology on Homeomorphism groups

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 $S<sup>1$. 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. We also extend these investigations to Homeo$([0,1])$, Homeo$(S<sup>1)$, and Homeo$(2<sup>ω)$. We conclude by providing a classification of the topological manifolds $M$ for which the homeomorphism group Homeo$(M)$ admits a Hausdorff Zariski topology.

• The paper identifies a definitive distinction: For Thompson's groups F and T, the Zariski topology matches the compact-open topology, while for Thompson's group V, it is non-Hausdorff and irreducible.
• The study demonstrates that for any Hausdorff space with no isolated points and a highly transitive homeomorphism group, the Zariski topology is irreducible and non-Hausdorff.,In contrast, for compact metrics space, this topological group is always Hausdorff.
• There are precisely two connected homogeneous groups where the Zariski topology is Hausdorff, which matches neatly with the dimensions of their underlying action.

The Zariski topology $\mathfrak{Z}_G$ on a group $G$—introduced by Bryant as the "verbal topology"—is defined as the coarsest topology whose subbasis consists of the non-solution sets of one-variable equations over $G$, i.e., sets of the form $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$. It is always a $T_1$ topology admitting continuous shifts and inversion, and it is contained in every Hausdorff group topology on $G$. For countable or abelian groups, work of Markov and Dikranjan–Shakhmatov shows that $\mathfrak{Z}_G$ coincides with the Markov topology, the intersection of all Hausdorff group topologies. The paper under review computes this topology for several prominent homeomorphism groups, obtaining a sharp dichotomy: for Thompson's groups $F$ and $T$ (and their higher-rank analogues) the Zariski topology is exactly the compact-open topology, while for Thompson's group $V$ it is irreducible and hence not even Hausdorff. These results culminate in a complete classification of connected manifolds whose homeomorphism groups carry a Hausdorff Zariski topology.

Zariski versus compact-open topologies on $G$0 and $G$1

The first main result establishes that the abstractly defined Zariski topology recovers the classical analytic topology on interval and circle homeomorphism groups. Specifically, if $G$2 contains Thompson's group $G$3 (or any $G$4, $G$5), then $G$6 equals the compact-open topology; the analogous statement holds for any $G$7 containing $G$8 or some $G$9. Since the compact-open topology on $G$0 of a compact metric space is a Hausdorff group topology induced by the uniform metric $G$1, these Zariski topologies are Hausdorff group topologies. Consequently, every Hausdorff group topology on $G$2 or $G$3 must contain the compact-open topology.

The proof rests on a lemma of independent interest: if $G$4 contains elements $G$5 supported exactly on $G$6 for all pairs from a dense subset $G$7, then $G$8 coincides with the compact-open topology. The argument proceeds by showing that each uniform ball $G$9 contains a finite intersection of non-commutation conditions of the form $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$0, which are Zariski-open since they are negations of equations. A measure-theoretic bookkeeping argument using Lebesgue measure on $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$1 then forces any $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$2 satisfying these conditions to move every point by less than $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$3. The hypothesis is verified by taking $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$4, for which $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$5 and $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$6 supply the required interval-supported elements.

This outcome is consistent with Rubin's reconstruction theorem, which recovers the underlying space algebraically from sufficiently rich homeomorphism groups [rubin1989reconstruction]. However, the paper notes that the techniques behind Rubin's theorem do not transfer to the Zariski setting, as the case of $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$7 demonstrates decisively.

Irreducibility for highly transitive actions

The second main theorem goes in the opposite direction. If $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$8 is a Hausdorff space with no isolated points and $$\{x \in G \mid 1_G \neq g_0 x^{k_0} g_1 \dots g_{l-1} x^{k_{l-1}} g_l\}$$9 acts on $T_1$0 by homeomorphisms highly transitively (every bijection between finite subsets extends to an element of $T_1$1), then $T_1$2 is irreducible: all non-empty open sets are dense, so no two non-empty open sets are disjoint. In particular, such a Zariski topology is neither Hausdorff nor a group topology (a $T_1$3 topological group is necessarily Tychonoff).

The key lemma requires that $T_1$4 be a highly transitive permutation group of an infinite set $T_1$5 in which every non-identity element has infinite support. Its proof is combinatorial: given a basic Zariski-open set described by finitely many inequations $T_1$6, one inductively constructs a finite partial permutation $T_1$7 of $T_1$8 that realizes all the required inequations at prescribed points $T_1$9, carefully avoiding collisions between the partial words evaluated at distinct basepoints. High transitivity then extends $G$0 to a global element witnessing that the basic set is non-empty; since the basis is closed under finite intersections, every basis element meets every other, giving irreducibility. The passage to topological spaces uses the observation that supports of homeomorphisms of a Hausdorff space are open, hence infinite when there are no isolated points.

Applying this to Thompson's group $G$1 requires a small adjustment: $G$2 does not act highly transitively on the whole Cantor space $G$3, but it does act faithfully and highly transitively on each of its orbits, and those orbits have no isolated points. Hence the group Zariski topologies on $G$4, all $G$5, and the full group $G$6 are irreducible. This stands in direct opposition to a theorem of Banakh, Guran, and Protasov stating that the Zariski topology on any permutation group containing all finitely supported permutations is a Hausdorff group topology—the contrast arises because every non-identity element of $G$7 has infinite support. It also yields, via Bardyla–Elliott–Mitchell–Péresse, a new infinite family of groups whose semigroup Zariski topology differs from the group Zariski topology.

Classification for manifolds

Combining the two strands, the paper classifies connected manifolds by the Hausdorffness of the Zariski topology on their homeomorphism groups. For an $G$8-manifold $G$9 with exactly one connected component, the following are equivalent:

Condition	Status
$\mathfrak{Z}_G$0 is Hausdorff	equivalent to $\mathfrak{Z}_G$1
$\mathfrak{Z}_G$2 is a group topology	equivalent to $\mathfrak{Z}_G$3
$\mathfrak{Z}_G$4	—

Here $\mathfrak{Z}_G$5 fails because $\mathfrak{Z}_G$6 is highly transitive by a result attributed to Mann's survey, triggering irreducibility; $\mathfrak{Z}_G$7 succeeds because the only connected manifolds of dimension at most one are a point, $\mathfrak{Z}_G$8, and $\mathfrak{Z}_G$9, handled respectively by triviality, the $F$0-result via $F$1, and the $F$2-result. Note that the classification covers boundaryless manifolds only, and relies on the high transitivity theorem for $F$3 in dimension $F$4.

Limitations and open questions

Several questions remain open. Since the Markov topology on $F$5 is not Hausdorff (by Markov's theorem, as $F$6 is countable), yet $F$7 admits many Hausdorff group topologies beyond the pointwise, compact-open, discrete, and join topologies, the paper asks whether any remaining Hausdorff group topologies on $F$8 are metrizable, and how many Hausdorff group topologies $F$9 admits in total. For $T$0, which is neither countable nor abelian, the equality of the Markov and Zariski topologies—and even the Hausdorffness of the Markov topology—are unresolved. Finally, the results suggest a dichotomy on $T$1: circular-order-preserving subgroups large enough to contain a copy of $T$2 have Hausdorff Zariski topology, while all highly transitive subgroups do not; classifying "sufficiently large" subgroups with Hausdorff Zariski topology remains open, as does the parallel question for $T$3 and $T$4 of whether the compact-open, pointwise, and discrete topologies exhaust the Hausdorff group topologies.

Conclusion

This paper gives a precise account of how the Zariski topology interacts with natural analytic topologies on homeomorphism groups. The dichotomy is clean: algebraic richness of the action in the order-preserving setting ($T$5, $T$6) forces the Zariski topology to be the finest possible lower bound, namely the compact-open topology, whereas high transitivity ($T$7, $T$8, and homeomorphism groups of manifolds of dimension at least two) collapses it to an irreducible, non-Hausdorff topology. The manifold classification distills this into a dimension-one threshold, and the accompanying questions delineate precisely where current techniques end.