The Zariski Topology on Homeomorphism groups
Luna Elliott

TL;DR
This paper investigates the Zariski topology on various homeomorphism groups, revealing that for some groups it coincides with standard topologies, while for others it is non-Hausdorff and irreducible, providing new insights into their topological structures.
Contribution
It analyzes the Zariski topology on homeomorphism groups like Thompson groups and classifies manifolds with Hausdorff Zariski topologies.
Findings
Zariski topology on F and T matches the compact-open topology.
Zariski topology on V is irreducible and non-Hausdorff.
Classification of manifolds with Hausdorff Zariski topology on their homeomorphism groups.
Abstract
The Zariski topology on a group G is the coarsest topology such that all sets of the form 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 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…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Topology and Set Theory · Advanced Operator Algebra Research
