1 Introduction
Let G be a Polish group acting on a Polish space in a Borel way. Let
$E^X_G$
be the orbit equivalence relation induced by the action. In other words, for
$x,y \in X$
,
$$ \begin{align*}xE^X_Gy\,\,\Leftrightarrow\,\,\exists g\in G\,\,gx=y. \end{align*} $$
Let E and F be two equivalence relations on Polish spaces X and Y, respectively. A Borel function from X to Y is called a Borel reduction from E to F, if for any
$x_1,x_2\in X$
, we have
$$ \begin{align*}x_1Ex_2\,\,\mbox{if and only if}\,\,f(x_1)Ff(x_2). \end{align*} $$
We say that E is Borel reducible to F if there is a Borel reduction from E to F and denote by
$E\leq _B F$
.
It is natural to ask about the connection between the structure of a Polish group G and the complexity of the equivalence relations it induces. The connection is well-studied if G is countable. For example, if
$G=\mathbb {Z}$
, then all actions induced by G are hyperfinite. In an important breakthrough, Gao and Jackson [Reference Gao and Jackson6] proved that all actions induced by countable Abelian groups are hyperfinite. Weiss conjectured that hyperfiniteness holds for all countable amenable group actions. This conjecture is still open so far.
For every Polish group G, Becker and Kechris [Reference Gao5, Theorem 3.3.4] proved that there is a most complicated orbit equivalence relation induced by G, called a maximal G-action. In other words, there exists a G-action such that all G-actions are Borel reducible to it. Thus, the question regarding connections between a Polish group and its actions can be restated as the following: given a Polish group G, where is its maximal action located in the complexity hierarchy?
A Polish group G is universal if all Polish groups can be continuously embedded as closed subgroup of G. An equivalence relation E is complete in a class
$\mathcal {C}$
of equivalence relations if E is in
$\mathcal {C}$
and all equivalence relations in
$\mathcal {C}$
are Borel reducible to E. Complete orbit equivalence relations exist. Indeed, the maximal action of any universal Polish group is a complete orbit equivalence relation by the Mackey–Hjorth extension theorem [Reference Gao5, Theorem 3.5.2]. There are also natural examples in topology and
$C^*$
-algebras (see, e.g., [Reference Sabok9, Reference Zielinski13]).
It is natural to ask that if there is a connection between completeness of an orbit equivalence relation and the structure of the group which induces this equivalence relation.
In this article, we address the following question posed by Sabok in [Reference Buzzi, Chandgotia, Foreman, Gao, García-Ramos, Gorodetski, Le Maitre, Rodríguez-Hertz and Sabok1].
Question 1.1. [Reference Buzzi, Chandgotia, Foreman, Gao, García-Ramos, Gorodetski, Le Maitre, Rodríguez-Hertz and Sabok1] Does there exist a non-universal Polish group which induces a complete orbit equivalence relation?
We answer this question in the positive.
Theorem 1.2. There exists a non-universal Polish group which induces complete orbit equivalence relation.
In [Reference Ding2], Ding found an example of a surjectively universal Polish group. We note that such group can induce complete orbit equivalence relation, and we show that this group is not universal. As a matter of fact, in [Reference Sabok10], Sabok mentioned without proof that a surjectively universal but not universal Polish group could serve as an example for Question.Footnote 1 However, showing that a surjectively universal group which is not universal exists requires a proof and in the subsequent survey [Reference Buzzi, Chandgotia, Foreman, Gao, García-Ramos, Gorodetski, Le Maitre, Rodríguez-Hertz and Sabok1] Question 1.1 was stated as open.
2 Preliminaries
In this section we discuss the construction and some results about surjectively universal Polish groups from [Reference Ding2, Reference Ding and Gao4].
For a nonempty set X, let
$X^{-1} =\{x^{-1}: x \in X\}$
be a disjoint copy of X, take
$e\notin X\cup X^{-1}$
. Denote
$X\cup X^{-1}\cup \{e\}$
by
$\overline {X}$
. For
$x\in X$
, by
$(x^{-1})^{-1}$
we mean x, and by
$e^{-1}$
we mean e.
We denote the set of nonempty words on
$\overline {X}$
by
$W(X)$
, in other words
$W(X)=\overline {X}^{<\omega }\setminus \{\emptyset \}$
. For
$w\in W(X)$
,
$|w|$
is the length of w. A word
$w\in W(X)$
is irreducible if
$w=e$
or
$w=x_0\ldots x_n$
such that
$n\ge 0$
,
$x_i\ne e$
for every
$0\le i\le n$
and
$x_i\ne x_{i+1}^{-1}$
for every
$0\le i\le n-1$
. We denote the set of irreducible words in
$W(X)$
by
$F(X)$
. For a word
$w\in W(X)$
, the reduced form for w is the irreducible word obtained by successively replacing any occurrence of
$xx^{-1}$
in w by e where
$x\in \overline {X}$
, and eliminating e from any occurrence of the form
$w_1ew_2$
, where at least one of
$w_1$
and
$w_2$
is nonempty. Note that the reduced form of
$w\in W(X)$
is independent from the order of eliminating e or replacing
$xx^{-1}$
to e, denote the reduced form for
$w\in W(X)$
by
$w'$
, then
$w'\in F(X)$
. For
$u,v\in F(X)$
, let
$u\cdot v=(u^\smallfrown v)'$
, then
$(F(X),\cdot )$
is a group, named the free group on X. The neutral element of
$F(X)$
is e.
To define a metric on
$F(X)$
, firstly we assume that X is a metric space with a metric
$d\le 1$
, that is,
$d(x,y)\le 1$
for every
$x,y\in X$
. We can extend d to
$\overline {X}$
by defining
$$ \begin{align*}d(x^{-1}, y^{-1})=d(x,y),\quad d(x^{-1},y)=d(x,e)=d(x^{-1},e)=1\end{align*} $$
for every
$x,y\in X$
. Then to get a metric on
$F(X)$
we need some more concepts.
Definition 2.1. (Ding–Gao [Reference Ding and Gao3], [Reference Gao5, Definition 2.3.2])
For
$m,n\in \mathbb {N}$
and
$m\le n$
, a bijection
$\theta $
on
$\{m,\ldots ,n\}$
is a match if
-
(1)
$\theta ^2=\mathrm {id}$
; -
(2) there is no
$i,j\in \{m,\ldots ,n\}$
such that
$i<j<\theta (i)<\theta (j)$
.
Definition 2.2 (Ding–Gao [Reference Ding and Gao4]).
Let
$\mathbb {R}_+$
be the set of non-negative real numbers. A function
$\Gamma : \overline {X}\times \mathbb {R}_+\rightarrow \mathbb {R}_+$
is a scale on
$\overline {X}$
if the following hold for any
$x\in \overline {X}$
and
$r\in \mathbb {R}_+$
:
-
(i)
$\Gamma (e,r)=r$
,
$\Gamma (x,r)\ge r$
; -
(ii)
$\Gamma (x,r)=0$
iff
$r=0$
; -
(iii)
$\Gamma (x,\cdot )$
is a monotone increasing function with respect to the second variable; -
(iv)
$\mathrm {lim}_{r\rightarrow 0} \Gamma (x,r)=0$
.
If we take
$\Gamma (x,r)=r$
for every
$x\in \overline {X}$
and
$r\in \mathbb {R}_+$
, then
$\Gamma $
is a scale on
$\overline {X}$
, we call it the trivial scale on
$\overline {X}$
.
Let G be a topological group with compatible left-invariant metric
$d_G$
. Define
$\Gamma _G:G\times \mathbb {R}_+ \rightarrow \mathbb {R}_+$
by
$$ \begin{align*}\Gamma_G(g,r) =\mathrm{max}\{ r,\mathrm{sup}\{d_G(1_G,g^{-1}hg):d_G(1_G,h)\le r\}\} .\end{align*} $$
It is easy to see that
$\Gamma _G$
satisfies the conditions (i)–(iii) in the definition of a scale, and (iv) is from the compatibility of
$d_G$
. We will also call
$\Gamma _G$
the scale on G.
Then given a scale
$\Gamma $
on
$\overline {X}$
, we will define a metric on
$F(X)$
related to
$\Gamma $
.
Definition 2.3 (Ding–Gao [Reference Ding and Gao4]).
Let
$\Gamma $
be a scale on
$\overline {X}$
. For
$l\in \mathbb {N}$
,
$w\in W(X)$
with
$|w|=l+1$
and
$\theta $
a match on
$\{0,\ldots ,l\}$
, define
$N^{\theta }_\Gamma (w)$
by induction on l as follows:
-
(0) for
$l =0$
, let
$w =x$
and define
$N^\theta _\Gamma (w)=d(e,x)$
; -
(1) if
$l>0$
and
$\theta (0)=k<l$
, let
$\theta _1 =\theta \upharpoonright \{0,\ldots ,k\}$
,
$\theta _2=\theta \upharpoonright \{k+1,\ldots ,l\},$
and
$w =w_1^\smallfrown w_2$
where
$|w_1|=k +1$
; define
$$ \begin{align*}N^{\theta}_\Gamma(w)=N^{\theta_1}_\Gamma(w_1)+N^{\theta_2}_\Gamma(w_2);\end{align*} $$
-
(2) if
$l>0$
and
$\theta (0)=l$
, let
$\theta _1 =\theta \upharpoonright \{1,\ldots ,l-1\}$
and
$w = x^{-1}w_1y$
where
${x,y \in \overline {X}}$
; then
$|w_1| =l-1$
. Define
$$ \begin{align*}N^{\theta}_\Gamma(w)=d(x,y)+\mathrm{max}\{\Gamma(x,N^{\theta_1}_\Gamma(w_1)),\Gamma(y,N^{\theta_1}_\Gamma(w_1))\}.\end{align*} $$
Then we take
$$ \begin{align*}N_\Gamma(w)=\mathrm{inf}\{N^\theta_\Gamma(w^*):(w^*)'=w,\,\theta\;is\;a\;match\}\end{align*} $$
for
$w\in F(X)$
, and we take
$$ \begin{align*}\delta_\Gamma(u,v)=N_\Gamma(u^{-1}v)\end{align*} $$
for
$u,v\in F(X)$
.
For a metric space
$(X,d)$
and the trivial scale
$\Gamma _0$
on
$\overline {X}$
, we call
$\delta _{\Gamma _0}$
the Graev metric on
$F(X)$
.
For
$u=x_0\ldots x_n$
and
$v=y_0\ldots y_n$
in
$W(X)$
, let
$$ \begin{align*}\rho(u,v)=\Sigma_{0\le i\le n}d(x_i,y_i).\end{align*} $$
Given
$w=x_0\ldots x_n\in W(X)$
and a match
$\theta $
on
$\{0,\ldots ,n\}$
, let
$$ \begin{align*}x^\theta_i= \left \{\begin{array}{@{}lr@{}} x_i,&\theta(i)> i.\\ e,& \theta(i)=i.\\ x^{-1}_{\theta(i)},&\theta(i)< i. \end{array} \right.\end{align*} $$
and
$w^\theta =x^\theta _0\ldots x^\theta _n$
. Then we have
Theorem 2.4. (Ding–Gao [Reference Gao5, Theorem 2.6.5])
Let
$(X,d)$
be a metric space,
$\Gamma _0$
is the trivial scale on
$\overline {X}$
, then
$$ \begin{align*}\delta_{\Gamma_0}(w,e)=\mathrm{ min}\{\rho(w,w^\theta):\;\theta\, is\;a\;match\}\end{align*} $$
for every
$w\in F(X)$
.
Then we have the following theorem in [Reference Ding and Gao4].
Theorem 2.5. (Ding–Gao [Reference Ding and Gao3, Theorem 2.6])
Let
$(X,d)$
be a metric space,
$\Gamma $
is a scale on
$\overline {X}$
, then the
$d_\Gamma $
is a left-invariant metric on
$F(X)$
extending d. Furthermore,
$F(X)$
is a topological group under the topology induced by
$\delta _\Gamma $
.
Denote the topological group
$F(X)$
with the metric
$\delta _\Gamma $
by
$F_\Gamma (X)$
. Let
$\delta ^{-1}_\Gamma (u,v)=\delta _\Gamma (u^{-1},v^{-1})$
for
$u,v\in F(X)$
, and
$d_\Gamma =\delta ^{-1}_\Gamma +\delta _\Gamma $
, then
$d_\Gamma $
is a compatible metric on topological group
$F_\Gamma (X)$
. The completion of
$(F_\Gamma (X),d_\Gamma )$
is a Polish group, denoted by
$\overline {F}_\Gamma (X)$
. Then the main theorem in [Reference Ding2] says that such a group of this form can be a surjectively universal group.
We denote the Baire space
$\omega ^\omega $
by
$\mathcal {N}$
, d is the standard metric on
$\mathcal {N}$
such that
$d(x,y)=\mathrm {max}\{2^{-n}:x(n)\ne y(n)\}$
. For
$n\in \mathbb {N}$
, let
$$ \begin{align*}\mathcal{N}_n=\{x\in\mathcal{N}:\forall m\ge n,x(m)=0\},\end{align*} $$
and
$\mathcal {N}_\omega =\bigcup _{n\in \mathbb {N} }\mathcal {N}_n$
. For
$x\in \mathcal {N}$
, let
$$\begin{align*}\pi_n(x)(m)= \left\{\begin{array}{@{}lr@{}} x(m),&m<n.\\ 0,& m\ge n. \end{array} \right.\end{align*}$$
Then a scale
$\Gamma $
on
$\overline {\mathcal {N}}$
is regular if for every
$x\in \overline {\mathcal {N}}$
,
$r\in \mathbb {R}_+$
, and
$n\in \mathbb {N}$
, we have
$\Gamma (x,r)\ge \Gamma (\pi _n(x),r)$
.
Theorem 2.6. (Ding–Gao [Reference Ding2, Example 4.9])
There is a regular scale
$\Gamma $
on
$\overline {\mathcal {N}}$
such that
$\overline {F}_\Gamma (\mathcal {N})$
is a surjectively universal Polish group.
The next theorem in [Reference Ding and Gao4] tells us how to get a continuous homomorphism from
$\overline {F}_\Gamma (X)$
to another topological group.
Theorem 2.7. (Ding–Gao [Reference Ding and Gao3, Lemma 3.7])
Let G be a topological group and
$d_G$
a compatible left-invariant metric on G. Let
$\Gamma $
be a scale on
$\overline {X}$
. Let
$\varphi :X\rightarrow G$
be a function. Suppose that for any
$x,y\in \overline {X}$
and
$r\in \mathbb {R}_+$
:
-
(1)
$\varphi (e)=1_G$
;
$\varphi (x^{-1})=\varphi (x)^{-1}$
; -
(2)
$d_G(\varphi (x),\varphi (y)) \le d(x,y)$
; -
(3)
$\Gamma _G(\varphi (x),r) \le \Gamma (x,r)$
.
Then
$\varphi $
can be uniquely extended to a continuous group homomorphism
$\Phi :F(X)\rightarrow G$
such that for any
$w \in F(X),$
$$ \begin{align*}d_G(\Phi(w),1_G)\le N_\Gamma(w).\end{align*} $$
3 Proof of the main theorem
By [Reference Ding2], we can fix a regular scale
$\Gamma $
on
$\overline {\mathcal {N}}$
such that
$\overline {F}_\Gamma (\mathcal {N})$
is a surjectively universal Polish group, and
$\Gamma (x,\cdot )$
is continuous for every
$x\in \overline {\mathcal {N}}$
. Recall
$d_\Gamma =\delta _\Gamma +\delta ^{-1}_\Gamma $
on
${F}_\Gamma (\mathcal {N})$
, by abusing the notation we also denote the extension of it on
$\overline {F}_\Gamma (\mathcal {N})$
by
$d_\Gamma $
.
Lemma 3.1. Every surjectively universal Polish group induces a complete orbit equivalence relation.
Proof. Let E be a complete orbit equivalence relation induced by a continuous group action of H on X, where H is a Polish group and X is a Polish space. For every surjectively universal Polish group G, let
$\pi :G\rightarrow H$
be a continuous surjective homomorphism. For
$g\in G$
and
$x\in X$
, let
$g\cdot x=\pi (g)\cdot x$
, this defines a continuous group action of G on X that induces E.
In the rest of this section we will show the group in Theorem 2.6 is not universal. It is mentioned in [Reference Ding2] that the group is abstractly isomorphic to a
$\boldsymbol {\Pi }^0_3$
subgroup of
$S_\infty $
, but we will verify that it is continuously embedded into a totally disconnected topological group.
Lemma 3.2. For every
$n\in \mathbb {N}$
,
$F(\mathcal {N}_n)$
is a discrete closed subgroup of
$\overline {F}_\Gamma (\mathcal {N})$
.
Proof. Let
$\Gamma _0$
be the trivial scale on
$\overline {\mathcal {N}_n}$
. Note that for every
$x\ne y\in \mathcal {N}_n$
, we have
$d(x,y)\ge 2^{-n}$
. Thus, for
$u\ne v\in F(\mathcal {N}_n)$
, we have
$u^{-1}v\ne e$
and consequently
$\rho (u^{-1}v,(u^{-1}v)^\theta )\ge 2^{-n}$
for every match
$\theta $
. Then
$\delta _{\Gamma _0}(u,v)=\delta _{\Gamma _0}(u^{-1}v,e)\ge 2^{-n}$
by Theorem 2.4. By [Reference Ding and Gao4, Lemma 3.6(i)],
$\delta _\Gamma (u,v)\ge \delta _{\Gamma _0}(u,v)$
. So
$d_\Gamma (u,v)\ge 2^{-n}$
for every
$u\ne v\in F(\mathcal {N}_n)$
,
$F(\mathcal {N}_n)$
is a discrete closed subgroup of
$\overline {F}_\Gamma (\mathcal {N})$
.
We denote the topological group
$F(\mathcal {N}_n)$
with the metric
$\delta _\Gamma $
by
$F_\Gamma (\mathcal {N}_n)$
. Then
$F_\Gamma (\mathcal {N}_n)$
is a discrete topological space by Lemma 3.2.
Lemma 3.3. For every
$w\in \overline {\mathcal {N}}$
and every
$r\in \mathbb {R}_+$
, we have
$\Gamma _G(w,r)\le \Gamma (w,r)$
where G is the group
${F}_\Gamma (\mathcal {N})$
together the metric
$\delta _\Gamma $
.
Proof. By definition,
$$ \begin{align*}\Gamma_G(w,r)=\mathrm{max}\{r,\mathrm{sup}\{N_\Gamma(w^{-1}uw):N_\Gamma(u)\le r\}\}.\end{align*} $$
We need to show that for every
$u\in {F}_\Gamma (\mathcal {N})$
with
$N_\Gamma (u)\le r$
, we have
$N_\Gamma (w^{-1}uw)\le \Gamma (w,r)$
. For every
$\epsilon> 0$
, by the definition of
$N_\Gamma (u)$
, there is
$v\in W(\mathcal {N})$
and a match
$\theta $
on
$\{0,\ldots ,|v|-1 \}$
such that
$v'=u$
and
$N^{\theta }_\Gamma (v)\le r+\epsilon $
. Let
$\eta $
be a match on
$\{0,\ldots ,|v|+1 \}$
such that
$\eta (0)=|v|+1$
and
$\eta (i)=\theta (i-1)+1$
for
$1\le i\le |v|$
. Then by definition of
$N^\eta _\Gamma (w^{-1}vw)$
we have
$$ \begin{align*}N_\Gamma(w^{-1}vw)\le N^\eta_\Gamma(w^{-1}vw)=\Gamma(w,N^{\theta}_\Gamma(v))\le \Gamma(w,r+\epsilon).\end{align*} $$
When
$\epsilon $
goes to
$0$
, by continuity of
$\Gamma (w,\cdot )$
, we have
$N_\Gamma (w^{-1}vw)\le \Gamma (w,r)$
. Since we also have
$r\le \Gamma (w,r)$
, we get
$\Gamma _G(w,r)\le \Gamma (w,r)$
.
Lemma 3.4. For every
$n\in \mathbb {N}$
, the map
$\pi _n$
can be uniquely extended to a continuous group homomorphism
$f_n$
from
$\overline {F}_\Gamma (\mathcal {N})$
to
${F}_\Gamma (\mathcal {N}_n)$
such that
$d(g,h)\ge d(f_n(g),f_n(h)) $
for every
$g,h\in \overline {F}_\Gamma (\mathcal {N})$
.
Proof. Firstly we show that
$\pi _n$
can be extended to a group homeomorphism
$f_n$
from
${F}_\Gamma (\mathcal {N})$
to
${F}_\Gamma (\mathcal {N}_n)$
such that
$N_\Gamma (u)\ge N_\Gamma (f_n(u))$
for every
$u\in {F}_\Gamma (\mathcal {N})$
. Let
$\pi _n(e)=e$
and
$\pi _n(x^{-1})=(\pi _n(x))^{-1}$
. We will use Lemma 2.7 to extend
$\pi _n$
to a homomorphism. Items (1) and (2) of Lemma 2.7 are satisfied by definition, and item (3) follows from Lemma 3.3. Then we get such an extension
$f_n$
.
Next we show that
$f_n$
can be extended to
$\overline {F}_\Gamma (\mathcal {N})$
. The group
$\overline {F}_\Gamma (\mathcal {N})$
is the completion of
${F}_\Gamma (\mathcal {N})$
with the metric
$d_\Gamma $
such that for every
$u,v\in {F}_\Gamma (\mathcal {N})$
,
$$ \begin{align*}d_\Gamma (u,v)=\delta_\Gamma(u,v)+\delta_\Gamma(u^{-1},v^{-1}),\end{align*} $$
and
${F}_\Gamma (\mathcal {N}_n)$
is a closed subgroup of
$\overline {F}_\Gamma (\mathcal {N})$
. We have that
$$ \begin{align} d_\Gamma(u,v)=N_\Gamma(u^{-1}v)+N_{\Gamma}(uv^{-1})\ge N_\Gamma(f_n(u^{-1}v))+N_{\Gamma}(f_n(uv^{-1}))\ge d_\Gamma(f_n(u),f_n(v)) \end{align} $$
for every
$u,v\in {F}_\Gamma (\mathcal {N})$
. If
$(x_m)_{m\in \mathbb {N}}$
is a Cauchy sequence in
${F}_\Gamma (\mathcal {N})$
under the metric
$d_\Gamma $
, then
$(f_n(x_m))_{m\in \mathbb {N}}$
is a Cauchy sequence in
${F}_\Gamma (\mathcal {N}_n)$
under the metric
$d_\Gamma $
. So
$f_n$
can be uniquely extended to a continuous group homomorphism from
$\overline {F}_\Gamma (\mathcal {N})$
to
${F}_\Gamma (\mathcal {N}_n)$
.
Finally we argue that for this extension
$f_n$
, we have
$d(g,h)\ge d(f_n(g),f_n(h)) $
for every
$g,h\in \overline {F}_\Gamma (\mathcal {N})$
. By (3.1), we know that
$d(g,h)\ge d(f_n(g),f_n(h)) $
for every
$g,h\in {F}_\Gamma (\mathcal {N})$
. Since
${F}_\Gamma (\mathcal {N})$
is dense in
$\overline {F}_\Gamma (\mathcal {N})$
and
$f_n$
is continuous, we get that
$d(g,h)\ge d(f_n(g),f_n(h)) $
holds for every
$g,h\in \overline {F}_\Gamma (\mathcal {N}).$
Theorem 3.5. There is a continuous embedding from
$\overline {F}_\Gamma (\mathcal {N})$
to
$\prod _{n\in \mathbb {N}}F(\mathcal {N}_n)$
.
Proof. Let f from
$\overline {F}_\Gamma (\mathcal {N})$
to
$\prod _{n\in \mathbb {N}}F(\mathcal {N}_n)$
map g to
$(f_n(g))_{n\in \mathbb {N}}$
for every
$g\in \overline {F}_\Gamma (\mathcal {N})$
. The map f is a continuous homomorphism, we just need to show that it is an injection.
For
$g\ne h\in \overline {F}_\Gamma (\mathcal {N})$
, let
$0<\epsilon < d_\Gamma (g,h)/4$
. Since
$F(\mathcal {N}_\omega )$
is dense in
$\overline {F}_\Gamma (\mathcal {N})$
, we can take
$u,v\in F(\mathcal {N}_\omega )$
such that
$d_\Gamma (u,g)\le \epsilon $
and
$d_\Gamma (v,h)\le \epsilon $
. There is
$n\in \mathbb {N}$
such that
$u,v\in F(\mathcal {N}_n)$
, and since
$f_n$
entends
$\pi _n$
, we have
$f_n(u)=u,f_n(v)=v$
. By Lemma 3.4, we have
$d_\Gamma (f_n(g),u)\le \epsilon $
and
$d_\Gamma (f_n(h),v)\le \epsilon $
, so
$$ \begin{align*}d_\Gamma(f_n(g),f_n(h))\ge d_\Gamma(u,v)-2\epsilon\ge d_\Gamma(g,h)-4\epsilon>0.\end{align*} $$
Then
$f_n(g)\ne f_n(h)$
, so
$f(g)\ne f(h)$
, and hence f is an injection.
Theorem 3.6. The group
$\overline {F}_\Gamma (\mathcal {N})$
is not a universal Polish group.
Proof. We argue that
$(\mathbb {R,+})$
is not isomorphic to a closed subgroup of
$\overline {F}_\Gamma (\mathcal {N})$
. If
$(\mathbb {R,+})$
is isomorphic to a closed subgroup of
$\overline {F}_\Gamma (\mathcal {N})$
, then by Theorem 3.5 there is a continuous embedding from
$(\mathbb {R,+})$
to
$\prod _{n\in \mathbb {N}}F(\mathcal {N}_n)$
. The topological space
$\mathbb {R}$
is connected but
$\prod _{n\in \mathbb {N}}F(\mathcal {N}_n)$
is totally disconnected, a contradiction.
We also have the alternative proof which does use Theorem 3.5. As mentioned in [Reference Ding2, Example 4.9],
$\overline {F}_\Gamma (\mathcal {N})$
is an abstract subgroup of
$S_\infty $
. The isometry group of the Urysohn space
$\mathrm {Iso}(\mathbb {U})$
cannot be continuously embedded to
$\overline {F}_\Gamma (\mathcal {N})$
, because of the automatic continuity property [Reference Sabok11, Reference Sabok12] (see [Reference Malicki7, Reference Rosendal and Suarze8] for new proofs),
$\mathrm {Iso}( \mathbb {U})$
cannot be embedded to
$S_\infty $
as an abstract subgroup since
$\mathrm {yy Iso}(\mathbb {U})$
is connected and
$S_\infty $
is totally disconnected.
Acknowledgments
The authors would like to thank Su Gao and Marcin Sabok for valuable suggestions to an earlier version of this work. Also, the authors are grateful to the suggestions given by the anonymous referee.
Funding
The first author is partially funded by NSFC grant 12271264. The second author is partially funded by NSFC grant 124B2001.
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