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 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.