Complexity classes of Polishable subgroups

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 Fréchet spaces; continuous linear maps between separable Banach spaces.

(Looking For) The Heart of Abelian Polish Groups

“…In this section, we consider the complexity of Polishable subgroups of groups with a Polish cover. We reformulate in this context some results from [HKL98,FS06,Lup22]. Recall that a Borel complexity class Γ is an assignment X → Γ (X) from Polish spaces to classes of Borel sets such that for every continuous function f : X → Y between Polish spaces X, Y and for every A ∈ Γ (X), A ⊆ X and f −1 (A) ∈ Γ (Y ).…”

“…The main result here is that images and preimages of Polishable subgroups of abelian groups with a Polish cover are Polishable; see Proposition 4.5. In Section 5 we reformulate in this context some results concerning the Borel complexity of Polishable subgroups from [Lup22]. In Section 6 we describe a canonical chain of Polishable subgroups of a given abelian group with a Polish cover, which we call Solecki subgroups.…”