Complexity classes of Polishable subgroups

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 complete characterization of the Borel complexity class of a Polishable subgroup in terms of its canonical approximation. 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 Fréchet spaces; and continuous linear maps between separable Banach spaces.