Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

TL;DR

This paper introduces a universal construction of infinite permutation groups called kaleidoscopic groups, derived from dendrite homeomorphisms, exploring their algebraic, homological, and topological properties.

Contribution

It presents a novel method to construct vast families of permutation groups from dendrite homeomorphisms, expanding understanding of their structural and topological characteristics.

Findings

Proved conditions for simplicity and oligomorphy of kaleidoscopic groups.

Analyzed homological properties, including acyclicity and Schur multipliers.

Established topological features like unique polishability.

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.