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.

References in corpus (2)

• Group actions on dendrites and curves
• Indicability, residual finiteness, and simple subquotients of groups acting on trees

Cited by in corpus (3)

• Topological properties of Wazewski dendrite groups
• A group with Property (T) acting on the circle
• On Thompson Groups for Ważewski Dendrites