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Kaleidoscopic groups: permutation groups constructed from dendrite homeomorphisms

Volume 247 / 2019

Bruno Duchesne, Nicolas Monod, Phillip Wesolek 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.

Authors

  • Bruno DuchesneInstitut Élie Cartan de Lorraine
    CNRS & Université de Lorraine
    Nancy, France
    e-mail
  • Nicolas MonodEPFL
    Lausanne, Switzerland
    e-mail
  • Phillip WesolekDepartment of Mathematical Sciences
    Binghamton University
    Binghamton, NY 13902, U.S.A.
    e-mail

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