The Zariski Topology on Homeomorphism groups
Luna Elliott
TL;DR
This paper investigates the Zariski topology $ abla_G$ on homeomorphism groups and its relation to the Markov and compact-open topologies. It shows that for Thompson groups, $ abla_G$ coincides with the compact-open topology for $F$ and $T$, yielding Hausdorff group topologies, while for $V$ the topology is irreducible and non-Hausdorff, challenging expectations from related reconstruction results. The authors extend the analysis to $ ext{Homeo}([0,1])$, $ ext{Homeo}(S^1)$, and $ ext{Homeo}(2^{\omega})$, and classify manifolds $M$ for which $ ext{Homeo}(M)$ admits a Hausdorff $ abla_G$. A key irreducibility lemma for highly transitive actions with infinite support underpins these results, highlighting distinct behavior between highly transitive groups and permutation groups containing finitely supported elements. The findings illuminate the interplay between algebraic equations defining the Zariski topology and the geometric/topological actions of these groups, with implications for the landscape of Hausdorff topologies on large transformation 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 $S^1$. 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$(S^1)$, and Homeo$(2^ω)$. We conclude by providing a classification of the topological manifolds $M$ for which the homeomorphism group Homeo$(M)$ admits a Hausdorff Zariski topology.