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Chief factors in Polish groups

Published online by Cambridge University Press:  30 June 2021

University of Newcastle, School of Mathematical and Physical Sciences, University Drive, CallaghanNSW 2308, Australia. e-mail:
Zendesk, Boston, MA 02101, U.S.A. e-mail:
Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris cedex 13, France. e-mail:


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.

Research Article
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Ando, H. and Matsuzawa, Y.. On Polish groups of finite type. Publ. Res. Inst. Math. Sci. 48 (2012)(2), 389408.10.2977/PRIMS/73CrossRefGoogle Scholar
Bourbaki, N.. General topology. Chapters 1–4. Elements of Mathematics (Berlin). (Springer-Verlag, Berlin 1989), translated from the French. Reprint of the 1966 edition.10.1007/978-3-642-61703-4CrossRefGoogle Scholar
Bourbaki, N.. General topology. Chapters 5–10. Elements of Mathematics (Berlin). (Springer-Verlag, Berlin 1989), translated from the French. Reprint of the 1966 edition.10.1007/978-3-642-61703-4CrossRefGoogle Scholar
Bourbaki, N.. Lie Groups and Lie Algebras. Chapters 1–3. Elements of Mathematics (Berlin). (Springer-Verlag, Berlin 1998), translated from the French. Reprint of the 1989 English translation.Google Scholar
Broise, M.. Commutateurs dans le groupe unitaire d’un facteur. J. Math. Pures Appl. (9), 46 (1967), 299312.Google Scholar
Caprace, P.-E. and Monod, N.. Decomposing locally compact groups into simple pieces. Math. Proc. Cam. Philos. Soc. 150 (2011)(1), 97128.10.1017/S0305004110000368CrossRefGoogle Scholar
Cornulier, Y.. Commutators in an unrestricted infinite wreath product. MathOverflow, (version: 2020-06-11).Google Scholar
Danilenko, A. I. and Silva, C. E.. Ergodic theory : Nonsingular transformations. Mathematics of Complexity and Dynamical Systems, pages 329356, (Springer 2011).Google Scholar
Ding, L.. On surjectively universal Polish groups. Adv. Math. 231 (2012)(5), 25572572.10.1016/j.aim.2012.06.029CrossRefGoogle Scholar
Eigen, S. J.. On the simplicity of the full group of ergodic transformations. Israel J. Math. 40 (1981)(3-4), 345349 (1982).10.1007/BF02761375CrossRefGoogle Scholar
Farah, I. and Solecki, S.. Borel subgroups of Polish groups. Adv. Math. 199 (2006)(2), 499541.10.1016/j.aim.2005.07.009CrossRefGoogle Scholar
Friedman, N. A.. Introduction to ergodic theory, (Van Nostrand Reinhold Co., New York-Toronto, Ont.-London 1970), van Nostrand Reinhold Mathematical Studies, no. 29.Google Scholar
Gao, S.. Invariant descriptive set theory, vol. 293 Pure and Appl. Math . (Boca Raton). (CRC Press, Boca Raton, FL 2009).Google Scholar
Isaacs, M.. Algebra, a Graduate Course. (Brooks/Cole 2009), 2nd edition.Google Scholar
Kechris, A. S.. Classical descriptive set theory, vol. 156 Graduate Texts in Mathematics, (Springer-Verlag, New York 1995).10.1007/978-1-4612-4190-4CrossRefGoogle Scholar
Kechris, A. S.. Global aspects of ergodic group actions. (American Mathematical Society, Providence, RI 2010).10.1090/surv/160CrossRefGoogle Scholar
Lang, S.. Algebraic number theory. (Springer 2013), 2nd edition.Google Scholar
Reid, C. D. and Wesolek, P. R.. Dense normal subgroups and chief factors in locally compact groups. Proc. London Math Soc. 116 (2018)(4), 760812.10.1112/plms.12088CrossRefGoogle Scholar
Reid, C. D. and Wesolek, P. R.. The essentially chief series of a compactly generated locally compact group. Math. Ann. 370 (2018)(1–2), 841861.10.1007/s00208-017-1597-0CrossRefGoogle Scholar