Group Metrics for Graph Products of Cyclic Groups

TL;DR

This paper characterizes when graph products of cyclic groups can be embedded into Polish groups, providing conditions related to metrics on the underlying graph and constructing specific ultrametrics.

Contribution

It establishes equivalences for embeddability of graph products into Polish groups and constructs new ultrametrics for certain graph products.

Findings

Equivalence between graph metric properties and embeddability into Polish groups.

Construction of left-invariant separable ultrametrics for specific graph products.

Characterization of when graph products admit a Polish group topology.

Abstract

We complement the characterization of the graph products of cyclic groups $G(\Gamma, \mathfrak{p})$ admitting a Polish group topology of [9] with the following result. Let $G = G(\Gamma, \mathfrak{p})$, then the following are equivalent: (i) there is a metric on $\Gamma$ which induces a separable topology in which $E_{\Gamma}$ is closed; (ii) $G(\Gamma, \mathfrak{p})$ is embeddable into a Polish group; (iii) $G(\Gamma, \mathfrak{p})$ is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for $G = G(\Gamma, \mathfrak{p})$ and $\Gamma$ a closed graph on the Baire space, which is of independent interest.

Equations

$$⟨V\mid a^{\mathfrak{p}(a)}=1,bc=cb:\mathfrak{p}(a)\ne \infty \text{ and }bEc⟩.$$

$$\{\bar{a}\in V^{|\sigma |}:G\models \sigma (\bar{a})=e\}=\{\bar{a}\in G^{|\sigma |}:G\models \sigma (\bar{a})=e\}\cap V^{|\sigma |}$$

$${\eta }_a(n)=\{\begin{array}{l}\mathfrak{p}(a)\text{ if }n=0\\ x(a,n)\text{ if }n>0,\end{array}$$

$$d^{\prime }(a,b)=\frac{1}{lg({\eta }_a\wedge {\eta }_b)+2}.$$

$$d^{\prime }(a_{\nu },b)=\frac{1}{lg({\eta }_{a_{\nu }}\wedge {\eta }_b)+2}⩽\frac{1}{n+2}<\epsilon .$$

$$a^{\prime },b^{\prime }\in \Gamma ,d(a,a^{\prime })<\epsilon ,d(b,b^{\prime })<\epsilon \Rightarrow \{a^{\prime },b^{\prime }\}\notin E_{\Gamma }.$$

$$a^{\prime },b^{\prime }\in \Gamma ,d^{\prime }(a,a^{\prime })<\frac{1}{n+2},d^{\prime }(b,b^{\prime })<\frac{1}{n+2}\Rightarrow \{a^{\prime },b^{\prime }\}\notin E_{\Gamma }.$$

$$d(a,a^{\prime })⩽d(a,{\eta }_a(n))+d(a^{\prime },{\eta }_a(n))<\frac{1}{4^n}+\frac{1}{4^n}<1/n<\epsilon .$$

$$li{m}_{n\to \infty }{\eta }_a(n)=a=li{m}_{n\to \infty }{\eta }_b(n)=b.$$

$$li{m}_{n\to \infty }{\eta }_{c_n}={\eta }_c\Rightarrow li{m}_{n\to \infty }{c}_n=c\text{in }(\Gamma ,d^{\prime }).$$

$$li{m}_{n\to \infty }{\eta }_{a_n}={\eta }_a,li{m}_{n\to \infty }{\eta }_{b_n}={\eta }_b\text{ and }\underset{n<\omega }{\bigwedge }{a}_n{E}_{\Gamma }{b}_n,$$

$${\mathfrak{p}}^{\ast }(\eta )=\{\begin{array}{l}\eta (0)\text{ if }\exists {\eta }_a({\eta }_a(0)=\eta (0))\\ 1\text{ otherwise}.\end{array}$$

$$E_n=\{(\eta ,\nu ):\eta ,\nu \in {\omega }^n\text{ and there are }({\eta }^{\prime },{\nu }^{\prime })\in E\text{ such that }\eta ◃{\eta }^{\prime }\text{ and }\nu ◃{\nu }^{\prime }\},$$

$$G_n\models ({\eta }_1\upharpoonright n{)}^{{\alpha }_1}\cdots ({\eta }_k\upharpoonright n{)}^{{\alpha }_k}\ne e.$$

$$G_m\models ({\eta }_1\upharpoonright m{)}^{{\alpha }_1}\cdots ({\eta }_k\upharpoonright m{)}^{{\alpha }_k}({\theta }_1\upharpoonright m{)}^{{\beta }_1}\cdots ({\theta }_p\upharpoonright m{)}^{{\beta }_p}=ee=e.$$

$$G_n\models ({\eta }_1\upharpoonright n{)}^{{\alpha }_1}\cdots ({\eta }_k\upharpoonright n{)}^{{\alpha }_k}=e\iff G_n\models ({\eta }_k\upharpoonright n{)}^{-{\alpha }_k}\cdots ({\eta }_1\upharpoonright n{)}^{-{\alpha }_1}=e,$$

$$a={\eta }_{a,1}^{\alpha (a,1)}\cdots {\eta }_{a,k_a}^{\alpha (a,k_a)}\text{ and }b={\eta }_{b,1}^{\beta (b,1)}\cdots {\eta }_{b,k_b}^{\beta (b,k_b)}$$

$$D(g,h)=d(g,h)+d(g^{-1},h^{-1}),$$

Full text

1. (i)

there is a metric on $\Gamma$ which induces a separable topology in which $E_{\Gamma}$ is closed; 2. (ii)

$G(\Gamma,\mathfrak{p})$ is embeddable into a Polish group; 3. (iii)

$G(\Gamma,\mathfrak{p})$ is embeddable into a non-Archimedean Polish group.

We also construct left-invariant separable group ultrametrics for $G=G(\Gamma,\mathfrak{p})$ and $\Gamma$ a closed graph on the Baire space, which is of independent interest.

Partially supported by European Research Council grant 338821. No. F1668 on Shelah’s publication list.

1. Introduction

Definition 1**.**

Let $\Gamma=(V,E)$ be a graph and $\mathfrak{p}:V\rightarrow\{p^{n}:p\text{ prime, }n\geqslant 1\}\cup\{\infty\}$ a graph coloring. We define a group $G(\Gamma,\mathfrak{p})$ with the following presentation:

[TABLE]

We call the group $G(\Gamma,\mathfrak{p})$ the $\Gamma$-product111Notice that this is consistent with the general definition of graph products of groups from [6]. In fact every graph product of cyclic groups can be represented as $G(\Gamma,\mathfrak{p})$ for some $\Gamma$ and $\mathfrak{p}$ as above. of the cyclic groups $\{C_{\mathfrak{p}(v)}:v\in\Gamma\}$, or simply the graph product of $(\Gamma,\mathfrak{p})$. These groups have received much attention in combinatorial and geometric group theory. In [9] the authors characterized the graph products of cyclic groups admitting a Polish group topology, showing that $G$ has to have the form $G_{1}\oplus G_{2}$ with $G_{1}$ a countable graph product of cyclic groups and $G_{2}$ a direct sum of finitely many continuum sized vector spaces over a finite field. In the present study we complement the work of [9] with the following results:

Theorem 2**.**

Let $\Gamma=(\omega^{\omega},E)$ be a graph and $\mathfrak{p}:V\rightarrow\{p^{n}:p\text{ prime, }n\geqslant 1\}\cup\{\infty\}$ a graph coloring. Suppose further that $E$ is closed in the Baire space $\omega^{\omega}$, and that $\mathfrak{p}(\eta)$ depends only on $\eta(0)$. Then $G=G(\Gamma,\mathfrak{p})$ admits a left-invariant separable group ultrametric extending the standard metric on the Baire space.

Theorem 3**.**

Let $G=G(\Gamma,\mathfrak{p})$, then the following are equivalent:

1. (a)

there is a metric on $\Gamma$ which induces a separable topology in which $E_{\Gamma}$ is closed; 2. (b)

$G$* is embeddable into a Polish group;* 3. (c)

$G$* is embeddable into a non-Archimedean Polish group;*

Corollary 4**.**

Let $G=G(\Gamma,\mathfrak{p})$, then the following are equivalent:

1. (a)

there is a metric on $\Gamma$ which induces a separable topology in which $E_{\Gamma}$ is closed; 2. (b)

$G$* is embeddable into the automorphism group of the random graph;* 3. (c)

$G$* is embeddable into the automorphism group of Hall’s universal locally finite group.*

The condition(s) occurring in Theorem 3 and Corollary 4 fail e.g. for the $\aleph_{1}$-half graph $\Gamma=\Gamma(\aleph_{1})$, i.e. the graph on vertex set $\{a_{\alpha}:\alpha<\aleph_{1}\}\cup\{b_{\beta}:\beta<\aleph_{1}\}$ with edge relation defined as $a_{\alpha}E_{\Gamma}b_{\beta}$ if and only if $\alpha<\beta$.

Theorem 2 is of independent interest and generalizes results on left-invariant group metrics on free groups on continuum many generators, see [2], [3] and [4].

2. Proofs of the Theorems

Convention 5**.**

In Definition 1 it is usually assumed that for every $a\in\Gamma$ we have $\{a,a\}\not\in E_{\Gamma}$. In order to make our proofs more transparent we will diverge from this convention and assume that our graphs $\Gamma$ are such that $a\in\Gamma$ implies $aE_{\Gamma}a$. This is of course irrelevant from the point of view of the group $G=G(\Gamma,\mathfrak{p})$, since an element $a\in G$ always commutes with itself.

Proposition 6**.**

Let $G$ be a separable topological group which is metrizable (resp. ultrametrizable) by the metric $d$ and $V\subseteq G$. Then the metric (resp. ultrametric) $d\restriction V\times V$ makes $V$ into a separable space such that for every group term $\sigma$ the set $\{\bar{a}\in V^{|\sigma|}:G\models\sigma(\bar{a})=e\}$ is closed in the induced topology.

Proof.

For every group term $\sigma$ the map $\bar{a}\mapsto\sigma(\bar{a})$ is continuous. Thus the set $\{\bar{a}\in G^{|\sigma|}:G\models\sigma(\bar{a})=e\}$ is closed in $(G,d)$, and so the set:

[TABLE]

is closed in $(V,d\restriction V\times V)$. ∎

Notation 7**.**

1. (1)

Given a graph $\Gamma=(V,E)$ and a set $R$, by a map $h:\Gamma\rightarrow R$ we mean a map with domain $V$. Furthermore, given a map $h:\Gamma\rightarrow R$ we let $h(E)=\{\{h(a),h(b)\}:\{a,b\}\in E\}$. 2. (2)

Given $\eta\in X^{\omega}$, $n<\omega$ and $\nu\in X^{n}$, we write $\nu\triangleleft\eta$ to mean that $\eta\restriction n=\nu$. 3. (3)

Given $\eta\neq\eta^{\prime}\in X^{\omega}$, we let $\eta\wedge\eta^{\prime}$ be the unique $\nu\in X^{n}$ such that $\nu\triangleleft\eta$, $\nu\triangleleft\eta^{\prime}$ and $n$ is maximal, and in this case we also let $lg(\eta\wedge\eta^{\prime})=lg(\nu)=n$. 4. (4)

Given a topological space $X$ and $Y\subseteq X$, we denote by $\overline{Y}$ the topological closure of $Y$ in $X$. Also, we denote by $\Delta_{X}$ the set $\{(x,x):x\in X\}$.

Lemma 8**.**

Let $\Gamma$ be a graph and $\mathfrak{p}:\Gamma\rightarrow\omega$ a graph colouring. Suppose that $\Gamma$ admits a separable metric $d$ which makes $E_{\Gamma}$ closed in the induced topology. Then:

1. (1)

$\Gamma$* admits an ultrametric $d^{\prime}$ with the same properties;* 2. (2)

there exists a one-to-one map $h:\Gamma\rightarrow\omega^{\omega}$ and a map $\mathfrak{p}^{*}:\omega^{\omega}\rightarrow\omega$ such that:

1. (a)

$\overline{h(E_{\Gamma})\cup\Delta_{\omega^{\omega}}}\cap h(\Gamma\times\Gamma)=h(E_{\Gamma})$; 2. (b)

$\mathfrak{p}(a)=\mathfrak{p}^{*}(h(a))$, for every $a\in\Gamma$; 3. (c)

$\eta_{1}(0)=\eta_{2}(0)$* if and only if $\mathfrak{p}^{*}(\eta_{1})=\mathfrak{p}^{*}(\eta_{2})$, for every $\eta_{1},\eta_{2}\in\omega^{\omega}$.*

Proof.

Let $(\Gamma,\mathfrak{p})$ and $d$ be as in the statement of the lemma. If $\Gamma$ is countable the lemma is clearly true. Assume then that $\Gamma$ is uncountable. Let $D\subseteq\Gamma$ be a countable dense set of $(\Gamma,d)$, and $\leqslant_{D}$ a well-order of $D$ of order type $\omega$. Renaming the elements of $\Gamma$ we can assume that $D=\omega$ and $\leqslant_{D}$ is the usual order of the natural numbers. For $a\in\Gamma$ we define $\eta_{a}\in\omega^{\omega}$ by letting:

[TABLE]

where:

1. (i)

$x(a,n)$ is at distance $<1/4^{n}$ from $a$; 2. (ii)

$x(a,n)$ is minimal under the condition (i).

We define $d^{\prime}:\Gamma\times\Gamma\rightarrow\mathbb{R}_{\geqslant 0}$ such that:

[TABLE]

Clearly $d^{\prime}$ is an ultrametric. We verify $d^{\prime}$ is as required.

1. $(*)_{1}$

$(\Gamma,d^{\prime})$ is separable.

For each $\nu\in\omega^{<\omega}$ choose $a_{\nu}$ such that $\nu\triangleleft\eta_{a_{\nu}}$, if possible, and arbitrarily otherwise. Let $D^{\prime}=\{a_{\nu}:\nu\in\omega^{<\omega}\}$. We claim that $D^{\prime}$ is dense in $(\Gamma,d^{\prime})$. This suffices, since obviously $D^{\prime}$ is a countable subset of $\Gamma$. Let then $b\in\Gamma$ and $\varepsilon>0$, we shall find $a\in D^{\prime}$ such that $d^{\prime}(a,b)<\varepsilon$. Choose $n>0$ such that $1/(n+2)<\varepsilon$, and let $\nu=\eta_{b}\restriction n$. Now, by the choice of $\nu$, $a_{\nu}\in D^{\prime}$ and $\nu\triangleleft\eta_{a_{\nu}}$. Furthermore, clearly $\nu\trianglelefteqslant\eta_{a_{\nu}}\wedge\eta_{b}$, and so $lg(\eta_{a_{\nu}}\wedge\eta_{b})\geqslant lg(\nu)=n$. Thus we have:

[TABLE]

1. $(*)_{2}$

$E_{\Gamma}$ is closed in $(\Gamma,d^{\prime})$.

Let $a,b\in\Gamma$ and suppose that $\{a,b\}\not\in E_{\Gamma}$. Since $E_{\Gamma}$ is closed in $(\Gamma,d)$, there is $\varepsilon\in(0,1)$ such that:

[TABLE]

Let $n<\omega$ be such that $n>1$ and $1/n<\varepsilon$, we shall prove that:

[TABLE]

Now, for any $a^{\prime}$ as in (2) we have that $lg(\eta_{a}\wedge\eta_{a^{\prime}})>n$, and so $\eta_{a}(n)=\eta_{a^{\prime}}(n)$. Hence:

[TABLE]

Using the same argument we see that for any $b^{\prime}$ as in (2) we have that $d(b,b^{\prime})<\varepsilon$, and so by (1) we conclude that $\{a^{\prime},b^{\prime}\}\not\in E_{\Gamma}$, as wanted.

1. $(*)_{3}$

The map $h:\Gamma\rightarrow\omega^{\omega}$ such that $h(a)=\eta_{a}$ is one-to-one.

If $\eta_{a}=\eta_{b}$, then:

[TABLE]

1. $(*)_{4}$

$\overline{h(E_{\Gamma})}\cap h(\Gamma\times\Gamma)=h(E_{\Gamma})$.

Notice that for $(c_{n})_{n<\omega}\in\Gamma^{\omega}$ and $c\in\Gamma$ we have:

[TABLE]

Thus, if we have:

[TABLE]

then $aE_{\Gamma}b$, since $E_{\Gamma}$ is closed in $(\Gamma,d^{\prime})$.

1. $(*)_{5}$

Let $\mathfrak{p}^{*}:\omega^{\omega}\rightarrow\omega$ be such that:

[TABLE]

Then the map $\mathfrak{p}^{*}$ is clearly as wanted. ∎

We need some basic word combinatorics for $G(\Gamma,\mathfrak{p})$.

Definition 9**.**

Let $(\Gamma,\mathfrak{p})$ be as usual and $G=G(\Gamma,\mathfrak{p})$.

1. (1)

A word $w$ in the alphabet $\Gamma$ is a sequence $(a_{1}^{\alpha_{1}},...,a_{k}^{\alpha_{k}})$, with $a_{i}\neq a_{i+1}\in\Gamma$, for $i=1,...,k-1$, and $\alpha_{1},...,\alpha_{k}\in\mathbb{Z}-\{0\}$. 2. (2)

We denote words simply as $a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$ instead of $(a_{1}^{\alpha_{1}},...,a_{k}^{\alpha_{k}})$. 3. (3)

We call each $a_{i}^{\alpha_{i}}$ a syllable of the word $a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$. 4. (4)

We say that the word $a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$ spells the element $g\in G$ if $G\models g=a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$. 5. (5)

We say that the word $w$ is reduced if there is no word with fewer syllables which spells the same element of $G$. 6. (6)

We say that the consecutive syllables $a_{i}^{\alpha_{i}}$ and $a_{i+1}^{\alpha_{i+1}}$ are adjacent if $a_{i}E_{\Gamma}a_{i+1}$. 7. (7)

We say that the word $w$ is a normal form for $g$ if it spells $g$ and it is reduced.

Fact 10** ([7][Lemmas 2.2 and 2.3]).**

Let $G=G(\Gamma,\mathfrak{p})$.

1. (1)

If the word $a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$ spelling the element $g\in G$ is not reduced, then there exist $1\leqslant p<q\leqslant k$ such that $a_{p}=a_{q}$ and $a_{p}$ is adjacent to each vertex $a_{p+1},a_{p+2},...,a_{q-1}$. 2. (2)

If $w_{1}=a_{1}^{\alpha_{1}}\cdots a_{k}^{\alpha_{k}}$ and $w_{2}=b_{1}^{\beta_{1}}\cdots b_{k}^{\beta_{k}}$ are normal forms for $g\in G$, then $w_{1}$ can be transformed into $w_{2}$ by repetedly swapping the order of adjacent syllables.

Definition/Proposition 11**.**

Let $\Gamma=(\omega^{\omega},E)$, with $E$ closed in the Baire space, and $\mathfrak{p}:V\rightarrow\{p^{n}:p\text{ prime, }n\geqslant 1\}\cup\{\infty\}$ such that $\mathfrak{p}(\eta)$ depends only on $\eta(0)$. For $0<n<\omega$, let:

[TABLE]

and $G_{n}=G((\omega^{n},E_{n}),\mathfrak{p}_{n})$, where $\mathfrak{p}_{n}(\eta)=\mathfrak{p}(\eta^{\prime})(0)$ for any $\eta\triangleleft\eta^{\prime}$. For $g\in G(\Gamma,\mathfrak{p})-\{e\}$ and $\eta_{1}^{\alpha_{1}}\cdots\eta_{k}^{\alpha_{k}}$ a word spelling $g$, we define $n(g)$ as the minimal $0<n<\omega$ such that:

[TABLE]

Finally, for $g\in G(\Gamma,\mathfrak{p})-\{e\}$, we define $d(g)=2^{-n(g)}$, and $d(e)=0$.

Proof.

We have to show that $n(g)$ does not depend on the choice of the word spelling $g$. So let $\eta_{1}^{\alpha_{1}}\cdots\eta_{k}^{\alpha_{k}}$ and $\theta_{1}^{\beta_{1}}\cdots\theta_{m}^{\beta_{m}}$ be words spelling $g\in G$, we want to show that, for every $0<n<\omega$, the words $(\eta_{1}\restriction n)^{\alpha_{1}}\cdots(\eta_{k}\restriction n)^{\alpha_{k}}$ and $(\theta_{1}\restriction n)^{\beta_{1}}\cdots(\theta_{m}\restriction n)^{\beta_{m}}$ spell the same element $g^{\prime}\in G_{n}$. By Fact 10 this is clear, since $\eta_{1}E\eta_{2}$ implies $\eta_{1}\restriction nE_{n}\eta_{2}\restriction n$, and $\mathfrak{p}(\eta)$ depends only on $\eta(0)$. ∎

The following lemma proves Theorem 2.

Lemma 12**.**

Let $\Gamma=(\omega^{\omega},E)$, with $E$ closed in the Baire space, $\mathfrak{p}:V\rightarrow\{p^{n}:p\text{ prime, }n\geqslant 1\}\cup\{\infty\}$ such that $\mathfrak{p}(\eta)$ depends only on $\eta(0)$, and $G=G(\Gamma,\mathfrak{p})$. The function $d:G\times G\rightarrow[0,1)_{\mathbb{R}}$ such that $d(g,h)=d(g^{-1}h)$, for $d:G\rightarrow[0,1)_{\mathbb{R}}$ as in Definition/Proposition 11, is a left-invariant separable group ultrametric extending the usual metric on $\omega^{\omega}$.

Proof.

We show that the function $d:G\rightarrow[0,1)_{\mathbb{R}}$ of Definition/Proposition 11 is an ultranorm, i.e. that it satisfies the following:

1. (i)

$d(g)=0$ iff $g=e$; 2. (ii)

$d(gh)\leqslant max\{d(g),d(h)\}$, for every $g,h\in G$; 3. (iii)

$d(g)=d(g^{-1})$, for every $g\in G$.

We prove (i). Let $g\neq e$ and $\eta_{1}^{\alpha_{1}}\cdots\eta_{k}^{\alpha_{k}}$ a normal form for $g$. Let $0<m<\omega$ be such that for every $1\leqslant i<j\leqslant k$ with $\eta_{i}\neq\eta_{j}$ we have $\eta_{i}E\eta_{j}$ iff $\eta_{i}\restriction mE_{m}\eta_{j}\restriction m$. Then $n(g)\leqslant m$ and so $2^{-m}\leqslant 2^{-n(g)}=d(g)$.

We prove (ii). Without loss of generality $g\neq e$ and $h\neq e$. Let $\eta_{1}^{\alpha_{1}}\cdots\eta_{k}^{\alpha_{k}}$ and $\theta_{1}^{\beta_{1}}\cdots\theta_{p}^{\beta_{p}}$ be normal forms for $g$ and $h$, respectively, and let $t=min\{n(g),n(h)\}$. Then for every $0<m<t<\omega$ we have:

[TABLE]

Hence, $t\leqslant n(gh)$ and so $d(gh)\leqslant max\{d(g),d(h)\}$.

We prove (iii). Let $\eta_{1}^{\alpha_{1}}\cdots\eta_{k}^{\alpha_{k}}$ be a normal form for $g$. It suffices to show that for every $0<n<\omega$ we have:

[TABLE]

but this is trivially true.

The fact that $d$ extends the usual metric on $\omega^{\omega}$ is immediate. Thus we are only left to show the separability of $(G,d)$. For every $n<\omega$, define a relation $R_{n}$ on $G$ by letting $aR_{n}b$ iff there exist normal forms:

[TABLE]

such that $k_{a}=k_{b}$, $\alpha(a,\ell)=\beta(b,\ell)$ and $\eta_{a,\ell}\restriction n=\eta_{b,\ell}\restriction n$. Clearly $R_{n}$ is an equivalence relation on $G$ and it has $\leqslant\aleph_{0}$ equivalence classes. For every $n<\omega$, let $X_{n}$ be a set of representatives of $R_{n}$ equivalence classes. Then $X=\bigcup_{n<\omega}X_{n}$ is countable and dense in $(G,d)$, and so it witnesses the separability of $(G,d)$. ∎

We need two facts before proving Theorem 3.

Fact 13** ([5][Theorem 2.1.3]).**

Let $G$ be a topological group with compatible left-invariant metric (resp. ultrametric) $d$. Let $D$ be defined such that:

[TABLE]

and $\hat{G}$ the completion of the metric space $(G,D)$. Then the multiplication operation of $G$ extends uniquely onto $\hat{G}$ making $\hat{G}$ into a topological group. Furthermore, there is a unique compatible left-invariant metric (resp. ultrametric) $\hat{d}$ on $\hat{G}$ extending $d$.

Definition 14**.**

We say that a Polish group $G$ is non-Archimedean if it has a neighbourhood base of the identity that consists of open subgroups.

Fact 15** ([1][Theorem 1.5.1]).**

Let $G$ be Polish. The following are equivalent:

1. (a)

$G$* is non-Archimedean;* 2. (b)

$G$* is isomorphic to a closed subgroup of $Sym(\omega)$;* 3. (c)

$G$* admits a compatible left-invariant ultrametric;* 4. (d)

$G$* is isomorphic to the automorphism group of a countable first-order structure.*

We finally prove Theorem 3 and Corollary 4.

Proof of Theorem 3.

Suppose that $G(\Gamma,\mathfrak{p})$ is embeddable into a Polish group, then by Proposition 6 there is a separable metric on $\Gamma$ such that $E_{\Gamma}$ is closed in the induced topology. On the other hand, if there is a separable metric $d$ on $\Gamma$ which induces a topology in which $E_{\Gamma}$ is closed, then using Lemma 8 we can embed $(\Gamma,\mathfrak{p})$ in a coloured graph on $\omega^{\omega}$ which satisfies the assumptions of Lemma 12, and so using Facts 13 and 15 we are done. ∎

Proof of Corollary 4.

As well-known, the automorphism group of the random graph embeds $Sym(\omega)$ (this also follows from the main result of [8]). Furthermore, in [10] it is proved that the automorphism group of Hall’s universal locally finite group embeds $Sym(\omega)$. Thus, by Theorem 3 and Fact 15 we are done. ∎