Chief factors in Polish groups

TL;DR

This paper extends the concept of chief factors from finite group theory to Polish groups, establishing foundational theorems and classifications for their structure.

Contribution

It develops a theory of chief factors for Polish groups, including a Schreier refinement theorem, a classification of simple groups, and new notions of semisimplicity.

Findings

Proved a version of the Schreier refinement theorem for Polish groups.

Established a trichotomy for topologically characteristically simple Polish groups.

Introduced new concepts of semisimplicity for Polish groups.

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 factorization via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalizations of direct products and use these to isolate a notion of semisimplicity for Polish groups.