Universal subgroups of Polish groups

TL;DR

This paper investigates the existence of universal subgroups within Polish groups, demonstrating their presence in locally compact cases and many Banach spaces, and exploring their properties and relationships.

Contribution

It establishes the existence of universal K-sigma and compactly generated subgroups in locally compact Polish groups and many Banach spaces, and analyzes their relationships.

Findings

Existence of universal subgroups in locally compact Polish groups

Universal subgroups in many Banach spaces

Conditions where K-sigma and compactly generated subgroups coincide

Abstract

Given a class C of subgroups of a topological group G, we say that a subgroup H in C is a universal C subgroup of G if every subgroup K in C is a continuous homomorphic preimage of H. Such subgroups may be regarded as complete members of C with respect to a natural pre-order on the set of subgroups of G. We show that for any locally compact Polish group G, the countable power of G has a universal K-sigma subgroup and a universal compactly generated subgroup. We prove a weaker version of this in the non-locally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universal K-sigma and universal compactly generated subgroups. As an aside, we explore the relationship between the classes of K-sigma and compactly generated subgroups and give conditions under which the two coincide.