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Chief factors in Polish groups

Part of: Connections with other structures, applications Topological and differentiable algebraic systems

Published online by Cambridge University Press:  30 June 2021

COLIN D. REID ,
PHILLIP R. WESOLEK  and
FRANÇOIS LE MAÎTRE
COLIN D. REID
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, University Drive, CallaghanNSW 2308, Australia. e-mail: colinreid29@gmail.com
PHILLIP R. WESOLEK
Affiliation:
Zendesk, Boston, MA 02101, U.S.A. e-mail: prwesolek@gmail.com
FRANÇOIS LE MAÎTRE
Affiliation:
Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris cedex 13, France. e-mail: francois.le-maitre@imj-prg.fr
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Abstract

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups.

The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

MSC classification

Primary: 22A05: Structure of general topological groups
Secondary: 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 239 - 296
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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