Complexity classes of Polishable subgroups

TL;DR

This paper characterizes the Borel complexity classes of Polishable subgroups of Polish groups, providing a comprehensive classification of their possible complexities and applications to continuous homomorphisms in various contexts.

Contribution

It offers a complete classification of Borel complexity classes for Polishable subgroups and their ranges in multiple mathematical settings, extending previous work.

Findings

Complete list of Borel complexity classes for Polishable subgroups

Classification of complexity classes of ranges of continuous homomorphisms

Application to non-Archimedean Polish groups and linear maps in functional analysis

Abstract

In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical approximations in terms of their Borel complexity class. As an application we provide a complete list of all the possible Borel complexity classes of Polishable subgroups of Polish groups or, equivalently, of the ranges of continuous group homomorphisms between Polish groups. We also provide a complete list of all the possible Borel complexity classes of the ranges of: continuous group homomorphisms between non-Archimedean Polish groups; continuous linear maps between separable Fr\'{e}chet spaces; continuous linear maps between separable Banach spaces.