3.1 Basic properties

Definition 3.1 [Reference Ding and Zheng8].

Let G be a Polish group. We define an equivalence relation $E(G)$ on $G^\omega $ as

$$ \begin{align*}xE(G)y\iff\lim_nx(n)y(n)^{-1}\text{ converges in }G\end{align*} $$

for $x,y\in G^\omega $ .

We say that $E(G)$ is the equivalence relation induced by G. They are ${\Pi }_3^0$ equivalence relations. Moreover, $E(G)$ is an orbit equivalence relation. To see this, let d be a compatible metric on G and let $c(G)$ be the group consisting of all convergent sequences in G. We define the supremum metric as $d_u(x,y)=\sup \{d(x(n),y(n)):n\in \omega \}$ for $x,y\in G^\omega $ . Then $(c(G),d_u)$ is also a Polish group. Therefore, $E(G)$ is the orbit equivalence relation induced by $c(G)\curvearrowright G^\omega $ , the action of left multiplication of $c(G)$ on $G^\omega $ . If G is CLI (TSI, or non-archimedean), so is $c(G)$ .

Based on the definition of $E(G)$ , we got the following proposition.

Focusing on several groups with relatively simple structures such as countable discrete group, we established the following theorem.

As an application of Theorem 3.3, the Borel complexity of $E(G)$ can be used to characterize countable discrete groups: G is a countable discrete group containing at least two elements iff $E(G)\sim _BE_0$ . To see this, if G is a countable discrete group containing at least two elements, then by Theorem 3.3(1), we have $E(G)\sim _BE_0$ . On the other hand, suppose $E(G)\sim _BE_0$ , from that $E_0^\omega \nleq _B E_0$ (cf. [Reference Gao13, Theorem 8.5.2]), by Theorem 3.3(2), we know that G is countable. Therefore, G is a countable discrete group containing at least two elements.

This also indicates that the Borel complexity of $E(G)$ can well describe the structure of the G.

In addition, we considered equivalence relations induced by some quotient groups. We need the following definition.

Definition 3.4. Let G be a Polish group. We define equivalence relation $E_*(G)$ on $G^\omega $ as, for $x,y\in G^\omega $ ,

$$ \begin{align*}xE_*(G)y\iff\lim_nx(0)x(1)\dots x(n)y(n)^{-1}\dots y(1)^{-1}y(0)^{-1}\text{ converges}.\end{align*} $$

It is easy to check that $E(G)\sim _B E_*(G)$ (see [Reference Ding and Zheng9]). It turns out that, for TSI Polish groups, $E_*(G)$ is a more convenient research object than $E(G)$ . Let G be a TSI Polish group and d a compatible complete two-sided invariant metric on G, then

(*) $$ \begin{align}xE_*(G)y\iff\lim_n\sup_{n\le m}d(x(n)\dots x(m),y(n)\dots y(m))=0,\end{align} $$

for $x,y\in G^\omega $ . The relation $(*)$ is crucial in the proof of many theorems.

Ding [Reference Ding6] defined the equivalence relations $E(M;0)$ . For any metric space $(M,d)$ , the $E(M;0)$ is an equivalence relation on $M^\omega $ with

$$ \begin{align*}xE(M;0)y\iff\lim_nd(x(n),y(n))=0\end{align*} $$

for $x,y\in M^\omega $ . If G is a Polish group, then the definition of $E(G;0)$ is independent of the choice of left-invariant compatible metric d on G, since $d(x(n),y(n))\to 0$ iff $x(n)^{-1}y(n)\to 1_G$ . If G is a TSI Polish group, then

$$ \begin{align*}E(G)\times E(G;0)\le_BE(G),\end{align*} $$

the corresponding Borel reduction is $\theta :G^\omega \times G^\omega \to G^\omega $ with $\theta (x,x')(n)=x(k)$ for $ n=2k$ ; and $\theta (x,x')(n)=x(k)x'(k)$ for $n=2k+1$ .

Definition 3.5. Given two sets $X,Y$ and a map $S:X\to Y$ , we define two maps $S^\omega :X^\omega \to Y^\omega $ and $S^\#:X^\omega \to Y^\omega \times X^\omega $ as, for $x\in X^\omega $ and $n\in \omega $ ,

$$ \begin{align*}S^\omega(x)(n)=S(x(n)),\end{align*} $$

$$ \begin{align*}S^\#(x)=(S^\omega(x),x).\end{align*} $$

We can directly verify the equivalence of (1) and (2) of the proposition. Using the above results, we got the following theorem.

The key idea of the proof is as follows. First, we verify that $\psi ^\#$ witnesses that $E_*(G)\le _BE_*(H)\times E(G;0)$ . This can be done by the condition that $\ker (\varphi \circ \psi )$ is non-archimedean. Secondly, we check that $E(G;0)\leq _BE(H;0)$ . It follows from [Reference Ding6, Theorem 3.4(ii)] that $E(G;0)\le _BE([0,1];0)$ . Let $f:[0,1]\to H$ be an embedding. Then $E([0,1];0)\sim _BE(f([0,1]);0)\le _BE(H;0)$ . So $E(G;0)\leq _BE(H;0)$ and thus $E(G)\leq _BE(H)\times E(H;0)\leq _BE(H)$ .

A similar argument can deduce that the next statement is true.

Let ${\mathbb {T}}$ be the multiplicative group of all complex numbers with modulus $1$ . Then ${\mathbb {T}}={\mathbb {R}}/{\mathbb {Z}}$ and Theorem 3.7 tells us that $E({\mathbb {R}})\leq _B E({\mathbb {T}})$ .

3.2 CLI, non-archimedean, and TSI Polish groups

In this section, we focus on three class of Polish groups: CLI, non-archimedean, and TSI.

3.2.2 Non-Archimedean Polish groups

Hjorth founded the theory of turbulence for proving that an orbit equivalence relation is not Borel reducible to $E_{S_\infty }^Y$ for any Polish $S_\infty $ -space Y. We omit the definition of turbulent actions since it is very complex and is not used in the article, one can find it in [Reference Gao13, Section 10].

Farah introduced the $c_0$ -equalities relations in [Reference Farah12].

Definition 3.12 (Farah [Reference Farah12]).

If $(X_n,d_n),\,n\in \omega $ is a sequence of finite metric spaces, let $D=D(\langle X_n,d_n\rangle )$ be the equivalence relation on $\prod _nX_n$ defined as

$$ \begin{align*}xDy\iff\lim_nd_n(x(n),y(n))=0.\end{align*} $$

The equivalence relations of this form are called $c_0$ -equalities.

For a finite metric space $(X,d)$ and $\varepsilon \in {\mathbb {R}}_+$ , we define

$$ \begin{align*}\delta(\varepsilon,X)=\min\{\delta:\exists x_0,x_1,\ldots,x_n\,(d(x_0,x_n)\ge\varepsilon\wedge\forall i<n\,(d(x_i,x_{i+1})<\delta))\}.\end{align*} $$

Using the notions of turbulence and $c_0$ -equality, we proved the following result.

Theorem 3.13 [Reference Ding and Zheng8].

Let G and H be two Polish groups. If H is non-archimedean but G is not, then $E(G)\not \le _BE_H^Y$ for any Polish H-space Y. In particular, we have $E(G)\not \le _BE(S_\infty )$ .

The key idea of the proof is as follows. Recall the Hjorth turbulence theorem: Let G be a Polish group and X a Polish G-space. If the action of G on X is turbulent, then $E_G^X\not \le _BE_{S_\infty }^Y$ for any Polish $S_\infty $ -space Y (see [Reference Gao13, Corollary 10.4.3]). Suppose G is not non-archimedean. In order to finish the proof, we only need to show that there is a turbulent orbit equivalence relation D with $D\leq _B E(G)$ . Fix an open set $1_G\in V$ such that no open subgroup of G is contained in V. Let d be a left-invariant compatible metric on G. For each $n\in \omega $ , we can find a finite set $X_n=\{g_{n,i}:0\le i\le m_n\}\subseteq G$ so that $1_G=g_{n,0}$ and $d(g_{n,i},g_{n,i+1})<2^{-n}$ for each $i<m_n$ and $g_{n,m_n}\notin V$ . It is easy to see that $D(X_n,d)\le _BE(G)$ and there is some $\varepsilon $ with $\delta (\varepsilon ,X_n)\le 2^{-n}\to 0$ as $n\to \infty $ . By [Reference Farah12, Theorem 3.7(a)], there exists a turbulent $c_0$ -equality D such that $D\le _BD(X_n,d)$ and thus $D\le _BE(G)$ . This finishes the proof.

Here is an important corollary. The Borel complexity of $E(G)$ can characterize non-archimedean Polish groups.

Corollary 3.14. Let G be a Polish group, then $E(G)\le _B=^+$ iff G is non-archimedean.

3.2.3 TSI Polish groups

Farah [Reference Farah12] introduced the notion of additive reduction, which is a powerful tool to prove Borel irreducibility.

Definition 3.15 (Farah [Reference Farah12]).

1. (1) A map $\psi :\prod _nX_n\to \prod _nX_n'$ is additive if there exist $0=l_0<l_1<\cdots <l_j<\cdots $ and maps $T_j:X_j\to \prod _{n\in [l_j,l_{j+1})}X_n'$ such that, for $x\in \prod _nX_n$ ,

   $$ \begin{align*}\psi(x)=T_0(x(0))^\smallfrown T_1(x(1))^\smallfrown T_2(x(2))^\smallfrown\dots\end{align*} $$

2. (2) Let E and F be two equivalence relations on $\prod _nX_n$ and $\prod _nX_n'$ , respectively, we say that E is additive reducible to F, denote by $E\le _AF$ , if there exists an additive reduction of E to F.

Let E be an equivalence relation on $\prod _nX_n$ , and let $I\subseteq \omega $ be infinite. Fix an element $w\in \prod _{n\notin I}X_n$ . For $x\in \prod _{n\in I}X_n$ , define $x\oplus w\in \prod _nX_n$ as $(x\oplus w)(n)=x(n)$ for $n\in I$ and $(x\oplus w)(n)=w(n)$ for $n\notin I$ . We define $E|_I^w$ on $\prod _{n\in I}X_n$ as, for $x,y\in \prod _{n\in I}X_n$ ,

$$ \begin{align*}xE|_I^wy\iff(x\oplus w)E(y\oplus w).\end{align*} $$

Let $(F_n)$ be a sequence of finite sets. The equivalence relation $E_0(\prod _nF_n)$ is defined as, for $x,y\in \prod _nF_n$ ,

$$ \begin{align*}xE_0(\prod_nF_n)y\iff\exists m\,\forall n>m\,(x(n)=y(n)).\end{align*} $$

Focusing on equivalence relations induced by TSI Polish groups, we got the following important lemma, which converts a Borel reduction to an additive reduction.

Lemma 3.16 [Reference Ding and Zheng8].

Suppose G is a TSI Polish group. Let $(F_n)$ be a sequence of finite sets, E a Borel equivalence relation on $\prod _nF_n$ with $E_0(\prod _nF_n)\subseteq E$ . If $E\le _BE_*(G)$ , then there exist an infinite $I\subseteq \omega $ and a $w\in \prod _{n\notin I}F_n$ such that $E|_I^w\le _AE_*(G)$ .

Now we explain how to use Lemma 3.16. Let G be a CLI Polish group, H a TSI Polish group. Suppose that $d_G$ is a compatible complete right-invariant metric on G, and $d_H$ is a compatible complete two-sided invariant metric on H. Assume that $E_*(G)\le _BE_*(H)$ . Let $(F_n)$ be a sequence of finite subsets of G such that

1. (i) $1_G\in F_n=F_n^{-1}$ ;

2. (ii) $F_{n-1}^{n}\subseteq F_n$ ;

3. (iii) $\bigcup _nF_n$ is dense in G.

Denote by E the restriction of $E_*(G)$ on $\prod _nF_n$ . By Lemma 3.16, there exist natural numbers $0=n_0<n_1<n_2<\cdots $ with $I=\{n_j:j\in \omega \}$ and a $w\in \prod _{n\notin I}F_n$ , $0=l_0<l_1<l_2<\cdots $ , maps $T_{n_j}:F_{n_j}\to H^{l_{j+1}-l_j}$ , and $\psi :\prod _{n\in I}F_n\to H^\omega $ with

$$ \begin{align*}\psi(x)=T_{n_0}(x(n_0))^\smallfrown T_{n_1}(x(n_1))^\smallfrown T_{n_2}(x(n_2))^\smallfrown\cdots,\end{align*} $$

such that $\psi $ is an additive reduction of $E|_I^w$ to $E_*(H)$ . Put $u_{n_0}=1_G$ , and for $j>0$ , let

$$ \begin{align*}u_{n_j}=w(n_{j-1}+1)\dots w(n_j-1).\end{align*} $$

By (i) and (ii), $u_{n_j}^{-1}\in F_{n_j}$ for each $j\in \omega $ .

Define $x_0\in \prod _{n\in I}F_n$ as

$$ \begin{align*}x_0(n_j)=u_{n_j}^{-1}\quad(\forall j\in\omega).\end{align*} $$

We may assume that $\psi (x_ 0)=1_{H^\omega }$ , otherwise we can replace $\psi $ with the following $\psi '$ : for $x\in \prod _{n\in I}F_n$ and $k\in \omega $ ,

$$ \begin{align*}\psi'(x)(k)=\psi(x_0)(0)\ldots\psi(x_0)(k-1)\psi(x)(k)\psi(x_0)(k)^{-1}\ldots\psi(x_0)(0)^{-1}.\end{align*} $$

For $s=(h_0,\ldots ,h_{l-1})$ and $t=(h_0',\ldots ,h_{l-1}')$ in $H^l$ , we define

$$ \begin{align*}d_H^\infty(s,t)=\max_{0\le i\le m<l}d_H(h_i\ldots h_m,h_i'\ldots h_m').\end{align*} $$

For any integer $j>0$ and $g\in F_{n_j-1}$ , it is worth noting that

$$ \begin{align*}u_{n_j}^{-1}g=w(n_j-1)^{-1}\ldots w(n_{j-1}+1)^{-1}g\in F_{n_j-1}^{n_j}\subseteq F_{n_j}.\end{align*} $$

We obtained the following lemmas.

Lemma 3.17 [Reference Ding and Zheng10].

For any $q\in \omega $ , there exists a $\delta _q>0$ such that

$$ \begin{align*}\forall^\infty n\in I\,\forall g,g'\in F_{n-1}\,(d_G(g,g')<\delta_q\Rightarrow d_H^\infty(T_n(u_n^{-1}g),T_n(u_n^{-1}g'))<2^{-q}).\end{align*} $$

We only sketch the proof of Lemma 3.17 here. Assume that the lemma is not true. Then there exist an $\varepsilon _0>0$ , a strictly increasing sequence $(j_k)$ , and $g_k,g_k'\in F_{n_{j_k}-1}$ for each k, such that

$$ \begin{align*}d_G(g_k,g_k')<2^{-(k+1)},\quad d_H^\infty(T_{n_{j_k}}(u_{n_{j_k}}^{-1}g_k),T_{n_{j_k}}(u_{n_{j_k}}^{-1}g_k'))\ge\varepsilon_0.\end{align*} $$

Since $d_G$ is right-invariant, we can find two subsequences $(g_{k_p}),(g_{k_p}')$ with

$$ \begin{align*}\lim_pg_{k_0}'\ldots g_{k_p}'g_{k_p}^{-1}\dots g_{k_0}^{-1}\text{ converges.}\end{align*} $$

For each $n\in I$ , put

$$ \begin{align*}x(n)=\left\{\begin{array}{@{}ll}u_n^{-1}g_{k_p}', & n=n_{j_{k_p}},\cr u_n^{-1}, & \text{otherwise,}\end{array}\right.\quad y(n)=\left\{\begin{array}{@{}ll}u_n^{-1}g_{k_p}, & n=n_{j_{k_p}},\cr u_n^{-1}, & \text{otherwise.}\end{array}\right.\end{align*} $$

Then $(x\oplus w)E_*(G)(y\oplus w)$ but $\psi (x)E_*(H)\psi (y)$ fails, contradicting that $\psi $ is a reduction.

Definition 3.18. For any $g\in \bigcup _nF_n$ , if $g\in F_n$ for some $n<n_j$ , then $u_{n_j}^{-1}g\in F_{n_j}$ , so $T_{n_j}(u_{n_j}^{-1}g)\in H^{l_{j+1}-l_j}$ . This allows us to define

$$ \begin{align*}S_{n_j}(g)=T_{n_j}(u_{n_j}^{-1}g)(0)\dots T_{n_j}(u_{n_j}^{-1}g)(l_{j+1}-l_j-1)\in H.\end{align*} $$

Recall that $\psi (x_ 0)=1_{H^\omega }$ , this implies that

$$ \begin{align*}S_{n_j}(1_G)=T_{n_j}(u_{n_j}^{-1})(0)\dots T_{n_j}(u_{n_j}^{-1})(l_{j+1}-l_j-1)=1_H\end{align*} $$

for all $j\in \omega $ .

The following proposition is well known.

Proposition 3.19 (Folklore).

Let G be a Polish group. Then G is TSI iff for all sequences $(g_p)$ and $(g_p')$ in G, we have

$$ \begin{align*}\lim_pg_pg_p'=1_G\iff\lim_pg_p'g_p=1_G.\end{align*} $$

Combining Proposition 3.19, we got the following result.

Theorem 3.20 [Reference Ding and Zheng10].

Let G and H be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$ .

We only sketch the proof of Theorem 3.20. By Theorem 3.11, we can assume that G is CLI. Let d be a compatible complete left-invariant metric on G. Assume that $E_*(G)\le _BE_*(H)$ . Since G is not TSI, there are two sequences $(g_p),(g_p')$ of elements of G such that $\lim _pg^{\prime }_pg_p=1_G$ , but $g_pg_p'\nrightarrow 1_G$ . We use notation before Lemma 3.17 and may assume that $\{g_p,g_p':p\in \omega \}\subseteq \bigcup _n F_n$ .

We can find two sequences $(p(i))$ and $(q(i))$ of natural numbers so that

1. (i) $g_{p(i)},g_{p(i)}'\in F_{n_{q(i)}-1}$ ;

2. (ii) $d_H(S_{n_{q(i)+1}}(g^{\prime }_{p(i)}),S_{n_{q(i)+1}}(g_{p(i)}^{-1}))<2^{-i}$ ;

3. (iii) for each $i\in \omega $ , $0<p(i)<q(i)<q(i)+1<p(i+1)$ .

For each $n\in I$ , define

$$ \begin{align*}x(n)=\left\{\begin{array}{@{}ll} u_n^{-1} g_{p(i)}, & n=n_{q(i)}, \cr u_n^{-1}g^{\prime}_{p(i)}, & n=n_{q(i)+1},\cr u_n^{-1},& \text{otherwise}, \end{array}\right. \quad y(n)=\left\{\begin{array}{@{}ll} u_n^{-1}g_{p(i)}, & n=n_{q(i)}, \cr u_n^{-1}g_{p(i)}^{-1}, & n=n_{q(i)+1},\cr u_n^{-1},& \text{otherwise}. \end{array}\right.\end{align*} $$

Then $\psi (x) E_*(H)\psi (y)$ but $(x\oplus w) E_*(G)(y\oplus w)$ fails, which is a contradiction.

3.3 A characterization of TSI non-archimedean Polish groups

We showed that the Borel complexity of $E(G)$ can characterize non-archimedean TSI Polish groups and there is NO Polish group G satisfying that

$$ \begin{align*}E_0^\omega<_B E(G)\leq_B {\mathbb{R}}^\omega/c_0.\end{align*} $$

To do so, we need the following notions. The author [Reference Ding7] defined equivalence relations $E(X,(x_n))$ and ${\mathbb {R}}^\omega /\mathrm {cs}$ . Let $(x_n)$ be a sequence in a Banach space X. We define an equivalence relation $E(X,(x_n ))$ on ${\mathbb {R}}^\omega $ as, for any $a,b\in {\mathbb {R}}^\omega $ ,

$$ \begin{align*}(a,b)\in E(X,(x_n ))\iff\sum_{n}(a(n)-b(n))x_n \text{ converges in } X.\end{align*} $$

The equivalence relation ${\mathbb {R}}^\omega /\mathrm {cs}$ is defined as, for any $a,b\in {\mathbb {R}}^\omega $ ,

$$ \begin{align*}(a,b)\in{\mathbb{R}}^\omega/\mathrm{cs}\iff\sum_{n}(a(n)-b(n)) \text{ converges in } {\mathbb{R}} .\end{align*} $$

The following lemma is an easy corollary of [Reference Ding7, Lemma 4.2].

We only sketch the proof of Theorem 3.22. By Theorem 3.20, G is TSI. If G is not non-archimedean, there is a sequence $(h_j)$ in G with $\lim _j h_j=1_G$ and $\lim _jh_0\ldots h_j\text { do not converges}.$ Let $(F_n)$ be a sequence of finite subsets of G such that

1. (i) $1_G,h_0\in F_n=F_n^{-1}$ ;

2. (ii) $F_{n-1}^{n}\subseteq F_n$ ;

3. (iii) for $n>0,h_n\in F_{n-1}$ .

By Lemma 3.21, there are natural numbers $0=n_0<n_1<n_2<\cdots $ with $I=\{n_j:j\in \omega \}$ , $0=l_0<l_1<l_2<\cdots $ , maps $T_{n_j}:F_{n_j}\to {\mathbb {R}}^{l_{j+1}-l_j}$ , and $\psi :\prod _{n\in I}F_n\to {\mathbb {R}}^\omega $ with

$$ \begin{align*}\psi(x)=T_{n_0}(x(n_0))^\smallfrown T_{n_1}(x(n_1))^\smallfrown T_{n_2}(x(n_2))^\smallfrown\cdots,\end{align*} $$

such that $\psi $ is an additive reduction of $E|_I^w$ to ${\mathbb {R}}^\omega /c_0$ . Put $u_{n_0}=1_G$ , and for $j>0$ , let $u_{n_j}=w(n_{j-1}+1)\dots w(n_j-1).$

For each $n\in I$ , put $y_0(n)=u^{-1}_nh_j$ and $x_0(n)=u^{-1}_{n}$ , where $n=n_j$ . Then it is not hard to show that $\psi (y_0){\mathbb {R}}^\omega /c_0\psi (x_0),$ but $(y_0\oplus w) E_*(G)(x_0\oplus w)$ fails, which is a contradiction.

Theorem 3.22 together with Theorem 3.3 proves the following theorem. The Borel complexity of $E(G)$ can characterize non-archimedean TSI Polish groups. The equivalence relation $E_0^\omega $ is precisely the Borel complexity of equivalence relations induced by TSI uncountable non-archimedean Polish groups.

5.1 Pre-rigid Theorem

Using the tool of additive reduction, we proved a highly technical theorem, namely the Pre-rigid Theorem. The existence of Borel reduction between $E(G)$ and $E(H)$ is transformed into the existence of a well-behaved continuous mapping between G and H.

We say that a topological group G has no small subgroups, or is NSS, if there exists an open subset $V\ni 1_G$ in G such that no nontrivial subgroup of G is contained in V.

Definition 5.1. A Polish group G is called strongly NSS if there exists an open set $V\ni 1_G$ in G such that

$$ \begin{align*}\forall(g_n)\in G^\omega\,( g_n\nrightarrow1_G\Rightarrow\exists n_0<\cdots<n_k\,(g_{n_0}\ldots g_{n_k}\notin V)),\end{align*} $$

where the set V is called an unenclosed set of G.

If G is a locally compact Polish group, then G is strongly NSS iff G is NSS (see [Reference Ding and Zheng10, Proposition 5.2]). Recall that a Lie group is a group which is also a smooth manifold such that the group operations are smooth functions. It is well known that a topological group is a Lie group iff it is locally compact NSS. So all Lie groups are strongly NSS. There are also some non-strongly NSS Polish groups, such as ${\mathbb {R}}^\omega $ .

We proved the Pre-rigid Theorem in [Reference Ding and Zheng10].

We only sketch the proof of Theorem 5.2. Suppose $E(G)\leq _B E(H)$ . We use notation before Lemma 3.17. Recall Definition 3.18 in which we let

$$ \begin{align*}S_{n_j}(g)=T_{n_j}(u_{n_j}^{-1}g)(0)\dots T_{n_j}(u_{n_j}^{-1}g)(l_{j+1}-l_j-1)\in H,\end{align*} $$

where $g\in F_n$ for some $n<n_j$ . And $S_{n_j}(1_G)=1_H$ .

For $m\in \omega $ , let $\pi _m:\prod _n H_n\to H_m$ be the canonical projection map, so $\pi _m(h)=h(m)$ for each h. All $\pi _m$ are continuous homomorphisms. For $m\in \omega $ , let $X_m$ be the set of all $g\in \bigcup _n F_n$ such that

$$ \begin{align*}\forall\delta>0\, \exists j_\delta\,(g\in F_{n_{j_\delta}-1}\wedge \forall j'> j\geq j_\delta\,(d_{H_m}(1_{H_m},\pi_m(S_{n_j}(g)S_{n_{j'}}(g^{-1})))<\delta)).\end{align*} $$

First step, we show, for any $m\in \omega $ , there is an $r_m>0$ such that

$$ \begin{align*}\forall g,g'\in\bigcup_n F_n\,((d_G(g,g')<r_m\wedge g\in X_m)\Rightarrow g'\in X_m).\end{align*} $$

This conclusion is Claim 1 in the proof of Theorem 5.3 in [Reference Ding and Zheng10]. This can be done by the property of strongly NSS. This property is crucial in the proof. At present, we don’t know how to remove it.

Second step, we let $V_m=\{g\in G:d_G(1_G,g)<r_m\}$ and $W_m=\bigcup _iV_m^i$ . Then $W_m$ is open and the above discussions tell us that $W_m\cap \bigcup _n F_n\subseteq X_m$ . So $\lim _{j}\pi _m(S_{n_{j}}(g))$ exists in $H_{m}$ for $g\in W_m\cap \bigcup _n F_n\subseteq X_m$ , which is denoted $\pi ^S_m(g)$ . By Lemma 3.17, $\pi ^S_m$ is uniformly continuous on $W_m\cap \bigcup _n F_n$ , which can be uniquely extended to a uniformly continuous map from $W_m$ to $H_m$ , still denoted by $\pi ^S_m$ . Put $W=\bigcap _m W_m$ , for any $g\in W$ , we let

$$ \begin{align*}S(g)=(\pi^S_0(g),\pi^S_1(g),\ldots,\pi^S_m(g),\ldots).\end{align*} $$

Finally, we can see that $W_m,\pi ^S_m,W$ , and S are as desired. By Lemma 3.8, we can conclude that the converse is also true if $G=W$ and if the interval $[0,1]$ can be embedded into H.

5.2 Rigid theorem

Using the Pre-rigid Theorem, we proved several rigid theorems for various classes of TSI Polish groups. The following lemmas characterize local continuous homomorphism and their kernel.

Consider a Polish group G and a sequence $(g_n)$ in G. Recall that $(g_n)$ is $\iota $ -Cauchy if it is d-cauchy for some compatible left-invariant metric d on G. This definition is independent of the choice of d.

Let G and H be two Polish groups and $\varphi :G\rightarrow H$ a continuous homomorphism. We define $\mathrm {IPC(\varphi )}$ as the set of all $x\in G^\omega $ that satisfies

$$ \begin{align*}(x(0)\ldots x(p))\text{ is } \iota-\text{Cauchy}\iff(\varphi(x(0))\ldots \varphi(x(p)))\text{ is } \iota-\text{Cauchy}.\end{align*} $$

Under the following assumptions, the condition of Theorem 3.7 can be weakened, and we do not require that the group homomorphism $\varphi \circ \psi $ is also surjective.

A completely metrizable topological group G is called a pro-Lie group if every open neighborhood of $1_G$ contains a normal subgroup N such that $G/N$ is a Lie group. Every locally compact TSI Polish group is a pro-Lie group (see [Reference Hofmann, Morris and Stroppel18, Theorem 3.6]). For any non-singleton connected pro-Lie Polish group G, there exists a nontrivial continuous homomorphism $\gamma :{\mathbb {R}}\rightarrow G$ (see Proposition 19 and Definition 2.6 of [Reference Hofmann and Morris17]). The interval $[0,1]$ embeds into G. For more details on pro-Lie groups, we refer to [Reference Hofmann and Morris17].

We only sketch the proof of Theorem 5.7. Clearly, every connected Polish group does not contain a nontrivial open subgroup. By Proposition 5.6, H is a closed subgroup of a countable product of separable TSI Lie groups. All Lie groups are strongly NSS. If $E(G)\leq _B E(H)$ , then it follows from Theorem 5.2, Lemmas 5.3, and 5.4 that there exists a continuous homomorphism $S:G\rightarrow H$ as required. On the other hand, let $S:G\rightarrow H$ be a continuous homomorphism such that $\ker (S)$ is non-archimedean. The interval $[0,1]$ can embed into $G/\ker (S)$ and H. Theorem 3.7 shows that $E(G)\leq _BE(G/\ker (S))$ . Let $S^*:G/\ker (S)\rightarrow H$ be the continuous injective homomorphism defined as $S^*(\ker (S)g)=S(g)$ for $g\in G$ . Note that G is locally compact, so $S^*$ is a local homeomorphism. Lemmas 3.8 and 5.5 tell us that $E(G/\ker (S))\leq _BE(H)$ . Then $E(G)\leq _BE(H)$ .

The following is a direct consequence of Theorem 5.7. Let G and H be two separable TSI Lie groups such that G is connected. Then $E(G)\leq _B E(H)$ iff there exists a continuous locally injective homomorphism $S:G\rightarrow H$ . In particular, we obtained a complete answer for abelian connected Lie groups. Recall that ${\mathbb {T}}$ is the multiplicative group of all complex numbers with modulus $1$ .

All separable Fréchet spaces, that is, separable, completely metrizable topological vector spaces (see [Reference Wilansky29, Chapter 5]), can be viewed as abelian Polish groups under the addition operation. All vector spaces are assumed to be real, that is, over the field ${\mathbb {R}}$ .

We sketch the proof of Theorem 5.9. We only need to consider the nontrivial direction. Suppose $E(X)\leq _B E(Y)$ . Then there exists a continuous homomorphism $S:X\to Y$ such that $\ker (S)$ is non-archimedean, and for $x\in X^\omega $ , if $\lim _px(p)=0$ , then

(E) $$ \begin{align}\sum_px(p)\text{ converges}\iff\sum_pS(x(p))\text{ converges}.\end{align} $$

Then $\ker (S)=\{0\}$ , since X is a linear space. And for any sequence $(x_n)$ , we can show, $\lim _nS(x_n)=0$ implies $\lim _nx_n=0$ by the relation $(E)$ and the property that for any open set $0\in W$ in Y, there is an open set $0\in V$ in Y such that

$$ \begin{align*}\forall r\in[0,1]\,\forall h\in V\, (rh\in W).\end{align*} $$

Let $(x_n)$ be a sequence in X with $\lim _nS(x_n)=0$ . Then there are natural numbers $(j_n)$ so that $\lim _n\frac {x_n}{j_n}=0$ . We suppose $j_n=2$ for all n (other situations are similar.) By the above property, we can find a subsequence $(x_{n_i})$ of $(x_n)$ so that $\frac { S(x_{n_0})}{2}+\frac { S(x_{n_0})}{2}+\frac { S(x_{n_1})}{2}+\frac { S(x_{n_1})}{2}+\cdots \text { converges}$ . The relation $(E)$ implies that $\lim _ix_{n_i}=0$ , and thus $\lim _nx_{n}=0$ . So S is an isomorphism.

It is not hard to see that a separable Banach space X is not strongly NSS iff it has a closed linear subspace topologically isomorphic to $c_0$ (see [Reference Ding and Zheng10, Theorem 6.12]). The following is a direct consequence of Theorem 5.9.

Let G be a Polish group and let $G_c$ be a closed normal subgroup of G. For $u\in G$ , we define a topological group automorphism $\iota _u:G_c\to G_c$ by $\iota _u(g)=ugu^{-1}$ , and denote

$$ \begin{align*}\mathrm{Inn}_G(G_c)=\{\iota_u:u\in G\}.\end{align*} $$

Recently, we also proved a Rigid theorem on disconnected TSI Polish groups. Its proof can be found in [Reference Zheng30].

It is well known that continuous local injections between Lie groups are also open mappings. Any non-archimedean closed subgroup of a Lie group is countable, since all Lie groups are NSS. Theorem 5.11 can be well applied to the case of Lie groups.

5.3 Embedding $P(\omega )/\text {Fin}$ into $E(G)$

Let $\mathrm {Fin}$ denote the set of all finite subsets of $\omega $ . For $A,B\subseteq \omega $ , we use $A\subseteq ^* B$ to denote $A\setminus B\in \mathrm {Fin}$ . The partially ordered set $(P(\omega ),\subseteq ^*)$ is written $P(\omega )/\text {Fin}$ . The $P(\omega )/\text {Fin}$ is so complicated that every Boolean algebra of size $\le \omega _1$ embeds into it (see [Reference Bella, Dow, Hart, Hrusak, van Mill and Ursino3, Theorem 1.1]). We usually express that some classes of Borel equivalence relations are extremely complicated under the order of Borel reducibility by showing that $P(\omega )/\text {Fin}$ embeds into them. For example, Louveau and Velickovic [Reference Louveau and Velickovic23] showed that $P(\omega )/\text {Fin}$ embeds into both LV-equalities. We proved that the partially ordered set $P(\omega )/\text {Fin}$ embeds into the partially ordered set of all $E(G)$ ’s under the ordering of Borel reducibility.

The following is a direct consequence of Theorem 5.7. We can also see [Reference Ding and Zheng9, Theorem 1.2].

Let $P=(P(0),P(1),\dots )$ be a sequence of integers greater than $1$ . Recall that the P-adic solenoid $S_P$ is defined by

$$ \begin{align*}S_P=\{g\in{\mathbb{T}}^\omega:\forall l\,(g(l)=g(l+1)^{P(l)})\}.\end{align*} $$

In particular, if all $P(i)$ are both prime numbers, then the P-adic solenoid is denoted by $\Sigma _P$ (see [Reference Gumerov15]). The group $\Sigma _P$ is a compact connected abelian Polish group. Let $\mathcal {P}$ denote the set of all primes. The group $S_P$ is topologically isomorphic to $\Sigma _{P'}$ for some $P'\in \mathcal {P}^\omega $ satisfying that $P(l)=P'(i_l)\dots P'(i_{l+1}-1)$ , where $0=i_0<i_1<\cdots <i_l<\cdots $ . For example, we have $S_{(4,6,8,9,\dots ,9,\dots )}\cong \Sigma _{(2,2,2,3,2,2,2,3,3,\dots ,3,3,\dots )}$ .

Let P and Q be in $\mathcal {P}^\omega $ . We write $Q\preceq P$ provided there is a co-finite subset A of $\omega $ and an injection $f:A\to \omega $ such that $Q(n)=P(f(n))$ for each $n\in A$ (for more details, see [Reference Gumerov15, Reference Prajs25]). A proof of the following lemma can be found in [Reference Ding and Zheng9].

Theorem 5.14 and Lemma 5.15 tell us that, $E(\Sigma _P)\leq _B E(\Sigma _Q)$ iff $Q\preceq P$ iff there is a nonzero continuous homomorphism $f:\Sigma _P\rightarrow \Sigma _Q$ . This together with Theorem 3.7 shows $E({\mathbb {R}})<_B E(\Sigma _P)<_B E({\mathbb {T}})$ (see [Reference Ding and Zheng9, Lemma 3.4]).

Theorem 5.16. Let $n\in \mathbb {N}^+$ . Then there are compact connected abelian Polish groups $(G_A)_{ A\subseteq \omega }$ such that $E({\mathbb {R}}^n)<_B E(G_A)<_B E({\mathbb {T}}^n)$ and for $A,B\subseteq \omega $ , we have

$$ \begin{align*}A\subseteq^*B\Longleftrightarrow E(G_A)\leq_B E(G_B).\end{align*} $$

The key idea of the proof is as follows. We construct a map $A\mapsto P_A\in \mathcal {P}^\omega $ such that $A\subseteq ^* B$ iff $P_B\preceq P_A$ . Then let $G_A=(\Sigma _{P_A})^n$ .