The Zariski Topology on Homeomorphism groups

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.