Complexity classes of Polishable subgroups

The goal of this paper is to exactly pin down the possible Borel complexity classes of Polishable subgroups of Polish groups. Most of the equivalence relations studied in the context of Borel complexity theory (and mathematics in general) arise as orbit equivalence relations associated with continuous actions of Polish groups on Polish spaces. Many of these actions can be seen as the (left) translation action associated with a continuous group homomorphism $\varphi:H\rightarrow G$ between Polish groups. In such a case, the image $\varphi\left(H\right)$ of $\varphi$ inside of $G$ is Borel, and the potential complexity class (in the sense of Louveau [11]) of the orbit equivalence relation associated with the translation action of $H$ on $G$ is essentially the same as the Borel complexity class of $\varphi\left(H\right)$ of $\varphi$ inside of $G$; see Theorem 3.3 for a precise statement. It is thus an interesting problem to determine what are the possible values for the Borel complexity class of such a subgroup.

The study of Polishable subgroups of Polish groups, which are precisely the ranges of continuous homomorphisms between Polish groups, has been undertaken by several authors over a number of years. The problem of determining their complexity has been considered as early as the 1970s, when Saint-Raymond proved that there exist Polishable subgroups of $\mathbb{R}^{\mathbb{N}}$ that are arbitrarily high in the Borel hierarchy [18]. A construction of arbitrarily complex non-Archimedean Polishable subgroups of $\mathbb{Z}_{2}^{\mathbb{N}}$ was presented by Hjorth, Kechris, and Louveau in [8]. Hjorth constructed in [7] arbitrarily complex Polishable subgroups of any uncountable abelian Polish groups. Farah and Solecki in [5], building on previous work of Sain-Raymond in the context of separable Fréchet spaces [18], related the least multiplicative Borel class containing a given Polishable subgroup to the length of the canonical approximation of that Polishable subgroup as in [16, 17].

In this paper, we refine the analysis from [5] by considering not only the multiplicative classes in the Borel hierarchy, but also the additive and difference classes. By relating the Borel complexity class of a Polishable subgroup to its canonical approximation, we completely characterize the possible Borel complexity classes of Polishable subgroups of Polish groups.

Theorem 4.1 from [8, Section 5] shows that the complexity class $D(\boldsymbol{\Pi}_{1+\lambda+1}^{0})$ where $\lambda$ is either zero or a countable limit ordinal cannot arise in the context of Theorem 1.1 if one demands $H$ to be non-Archimedean. In this case, we have the following characterization.

The existence assertions in Theorem 1.1 and Theorem 1.2 are proved by providing a unified approach to the constructions in [18] and [8, Section 5] of arbitrarily complex Polishable subgroups, together with a careful analysis of their canonical approximations in the sense of Solecki [16, 17].

Theorem 1.1 entails in particular a negative answer to a Question 6.3(1) from [4]. Let $X$ be a separable Banach space with a Schauder basis $\left(x_{n}\right)_{n\in\mathbb{N}}$. Then the collection $\mathrm{coef}(X,(x_{n}))$ of $(\lambda_{n})_{n\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}$ such that $\sum_{n\in\mathbb{N}}\lambda_{n}x_{n}$ converges in $X$ is a Polishable subgroup of $\mathbb{R}^{\mathbb{N}}$. Question 6.3(1) asks whether there is an example of such a Polishable subgroup that is $\boldsymbol{\Delta}_{3}^{0}$ and not $D(\boldsymbol{\Sigma}_{2}^{0})$. By Theorem 1.1, a $\boldsymbol{\Delta}_{3}^{0}$ Polishable subgroup of a Polish group must be $D(\boldsymbol{\Sigma}_{2}^{0})$.

We also apply the techniques of this paper to provide a complete characterization of the Borel complexity classes of the ranges of continuous homomorphisms between separable Fréchet spaces and between separable Banach spaces.

Continuous linear maps with arbitrarily complex range, with a fixed separable Banach space or separable Frechet space as target, were constructed in [3, 12].

The rest of this paper is organized as follows. In Section 2 and Section 3 we recall some definitions and known results concerning Polishable subgroups and their Borel complexity class. In Section 4 we recall the definition of the canonical approximation of a Polishable subgroup, whose elements we call Solecki subgroups as they were originally described by Solecki in [16]. In Section 5, building on the work of Farah and Solecki, we refine the analysis from [5] to characterize the Solecki subgroups in terms of their Borel complexity class. This is then applied in Section 6 to obtain the characterization of complexity classes of Polishable subgroups as in Theorem 1.1. Section 7 shows that the length of the canonical approximation, called the Polishable rank in [5], coincides with a notion of rank originally considered by Saint-Raymond in [18] in the context of separable Fréchet spaces. The existence assertions in Theorem 1.1 and Theorem 1.2 are proved in Section 8. Finally, Section 9 and Section 10 contain a proof of Theorem 1.3 and Theorem 1.4, respectively.

A Polish space is a second countable topological space whose topology is induced by a complete metric. A Polish group is a group in the category of Polish spaces, namely a Polish space that is endowed with a continuous group operation such that the function that maps each element to its inverse is also continuous (in fact, the latter requirement holds automatically; see the remark after [9, Corollary 9.15]). A subgroup $H$ of a Polish group $G$ is Polishable if it is Borel and there exists a Polish group topology on $H$ whose open sets are Borel in $G$. Notice that such a Polish topology on $H$, if it exists, it is unique by [9, Theorem 9.10]. In the following, we will regard $H$ as a Polish group with respect to its unique Polish group topology, which is in general finer than the subspace topology induced from $G$. Equivalently, $H$ is a Polishable subgroup of $G$ if and only if there exists a Polish group $\tilde{H}$ and a continuous group homomorphism $\varphi:\tilde{H}\rightarrow G$ with image equal to $H$. Noticing that one can assume without loss of generality that $\varphi$ is an injection, the equivalence of the two definitions follows from [9, Theorem 9.10] and the fact that if $f:X\rightarrow Y$ is an injective Borel function between standard Borel spaces, then $f(A)$ is a Borel subset of $Y$ and $f|_{A}$ is a Borel isomorphism between $A$ and $f(A)$ [9, Theorem 15.1]. If $G$ is a Polish group and $H\$is a Polishable subgroup of $G$, then $G$ is a Polish $H$-space with respect to the left translation action of $H$ on $G$ [2, Section 2.2]. We will denote by $E_{H}^{G}$ the corresponding orbit equivalence relation. Recall that a Polish group $G$ is non-Archimedean if it admits a basis of neighborhoods of the identity consisting of open subgroups; see [6, Theorem 2.4.1] for equivalent characterizations.

A complexity class $\Gamma$ is a function $X\mapsto\Gamma\left(X\right)$ that assigns to each Polish space $X$ a collection $\Gamma\left(X\right)$ of Borel subsets, such that if $X,Y$ are Polish spaces and $f:X\rightarrow Y$ is a continuous function, then $f^{-1}(A)\in\Gamma\left(X\right)$ for every $A\in\Gamma\left(Y\right)$. For a complexity class $\Gamma$, we let $D\left(\Gamma\right)$ be the complexity class consisting of differences between sets in $\Gamma$; see [9, Section 22.E] where it is denoted by $D_{2}\left(\Gamma\right)$. We let $\check{\Gamma}$ be the dual complexity class of $\Gamma$, such that $\check{\Gamma}\left(X\right)$ comprises the complements of the elements of $\Gamma\left(X\right)$. We say that $\Gamma$ is self-dual if $\Gamma=\check{\Gamma}$. If $\Gamma$ is a complexity class that is not self-dual, then we say that $\Gamma$ is the complexity class of $A\subseteq X$ if $A\in\Gamma\left(X\right)$ and $A\notin\check{\Gamma}\left(X\right)$. We will be mainly interested in the complexity classes $\boldsymbol{\Sigma}_{\alpha}^{0}$, $\boldsymbol{\Pi}_{\alpha}^{0}$, $\boldsymbol{\Delta}_{\alpha}^{0}$, and $D(\boldsymbol{\Pi}_{\alpha}^{0})$ for $\alpha\in\omega_{1}$; see [9, Section 11.B].

If $X$ is a standard Borel space and $E$ is an equivalence relation on $X$, then $E$ has potential complexity $\Gamma$ if there exists a Polish topology $\tau$ on $X$ that induces the Borel structure of $X$ such that $E\in\Gamma\left(\tau\times\tau\right)$ [11]. This is equivalent to the assertion that there exists a Borel equivalence relation $F$ on a Polish space $Y$ such that $F\in\Gamma\left(Y\times Y\right)$ and $E$ is Borel reducible to $F$; see [6, Lemma 12.5.4]. The following result is essentially proved in [8, Section 5].

A similar proof as Proposition 3.1 gives the following.

Applying Proposition 3.1 to the left translation action associated with a Polishable subgroup of a Polish group, we obtain Items (1) and (3) of the following result. The proof of Item (2) is postponed to Section 6.

We now recall some results concerning the possible complexity classes of Polishable subgroups. The following proposition is a reformulation of [5, Corollary 3.4].

The following proposition is a consequence of [8, Theorem 4.1] and Proposition 3.1.

Suppose that $G$ is a Polish group, and $H$ is a Polishable subgroup of $G$. Then $H$ admits a canonical approximation by Polishable subgroups indexed by countable ordinals. As these were originally described by Solecki in [16], we call them Solecki subgroups of $G$ associated with $H$. They have also been considered in [17, 5].

Lemma 2.3 from [16, Lemma 2.3] implies that $G$ has a smallest $\boldsymbol{\Pi}_{3}^{0}$ Polishable subgroup containing $H$, which we denote by $s_{1}^{H}(G)$. One can explicitly describe $s_{1}^{H}(G)$ as the $\boldsymbol{\Pi}_{3}^{0}$ subgroup of $G$ defined by

where $V$ ranges among the open neighborhoods of the identity in $H$, and $\overline{V}^{G}$ is the closure of $V$ inside of $G$. It is proved in [16, Lemma 2.3] that $s_{1}^{H}(G)$ satisfies the following properties—see also [19, Lemma 4.5] and [5, Section 3]:

• •

  $H$ is dense in $s_{1}^{H}(G)$;

• •

  a neighborhood basis of $x\in s_{1}^{H}(G)$ consists of sets of the form $\overline{Wx}^{G}\cap s_{1}^{H}(G)$ where $W$ is an open neighborhood of the identity in $H$;

• •

  if $A\subseteq G$ is $\boldsymbol{\Pi}_{3}^{0}$ and contains $H$, then $A\cap s_{1}^{H}(G)$ is comeager in the Polish group topology of $s_{1}^{H}(G)$.

A similar argument as in the proof of [16, Lemma 2.3] gives the following.

The sequence of Solecki subgroups $s_{\alpha}^{H}(G)$ for $\alpha<\omega_{1}$ of $G$ associated with $H$ is defined recursively by setting:

• •

  $s_{0}^{H}(G)=\overline{H}^{G}$;

• •

  $s_{\alpha+1}^{H}(G)=s_{1}^{H}\left(s_{\alpha}^{H}(G)\right)$ for $\alpha<\omega_{1}$;

• •

  $s_{\lambda}^{H}(G)=\bigcap_{\beta<\lambda}s_{\beta}^{H}(G)$ for a limit ordinal $\lambda<\omega_{1}$.

Using Lemma 2.1 at the limit stage, one can prove by induction on $\alpha<\omega_{1}$ that $s_{\alpha}^{H}(G)$ is a Polishable subgroup of $G$, and $H$ is dense in $s_{\alpha}^{H}(G)$. Furthermore, by Lemma 4.1, if $H$ is non-Archimedean, then $s_{\alpha}^{H}(G)$ is non-Archimedean for every $1\leq\alpha<\omega_{1}$. It is proved in [16, Theorem 2.1] that there exists $\alpha<\omega_{1}$ such that $s_{\alpha}^{H}(G)=H$. We call the least countable ordinal $\alpha$ such that $s_{\alpha}^{H}(G)=H$ the Solecki rank of $H$ in $G$.

One can define the Polish groups $s_{\alpha}^{H}(G)$ solely in terms of $H$ endowed with the subspace topology inherited from $G$. Indeed, $s_{0}^{H}(G)$ can be seen as the completion of $H$ with respect to a suitable metric that induces the subspace topology inherited from $G$; see [16, Section 2.1]. Using Lemma 4.2 one can describe the Solecki subgroups of products, as follows.

Suppose that $G$ is a Polish group, and $H$ is a Polishable subgroup of $G$. For a complexity class $\Gamma$, we define $\Gamma(G)|_{H}$ to be the collection of sets of the form $A\cap H$ for $A\in\Gamma(G)$. The following results are essentially established in [5]. In the statements and proofs, we adopt the Vaught transform notation in reference to the action of $H$ on $G$ by left translation; see [6, Section 3.2].

Recall that, by Proposition 3.4, if $\alpha$ is either zero or a countable limit ordinal, and $H$ is a $\boldsymbol{\Pi}_{1+\alpha+1}^{0}$ Polishable subgroup of a Polish group, then $H$ is $\boldsymbol{\Pi}_{1+\alpha}^{0}$.

The goal of this section is to establish the following theorem, characterizing the possible values of the complexity class of a Polishable subgroup of a Polish group.

Theorem 1.1 and Theorem 1.2 are immediate consequences of Theorem 6.1. We will obtain Theorem 6.1 as a consequence of a number of complexity reduction lemmas. We fix a Polish group $G$ and a Polishable subgroup $H$ of $G$. We adopt the Vaught transform notation, in reference to the left translation action of $H$ on $G$.