Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms
Volume 247 / 2019
Fundamenta Mathematicae 247 (2019), 229-274
MSC: 03E15, 22F50.
DOI: 10.4064/fm702-4-2019
Published online: 10 October 2019
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 homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique Polishability.
Our construction is carried out within the framework of homeomorphism groups of topological dendrites.
