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Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

Abstract : Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup. This produces vast families of kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their ho-mological properties, such as acyclicity or contrariwise large Schur mul-tipliers; their topological properties, such as unique polishability. Our construction is carried out within the framework of homeomor-phism groups of topological dendrites.
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Bruno Duchesne, Nicolas Monod, Phillip Wesolek. Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms. Fundamenta Mathematicae, Instytut Matematyczny, Polskiej Akademii Nauk,, 2019, 247 (3), pp.229-274. ⟨10.4064/fm702-4-2019⟩. ⟨hal-01695772v2⟩

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