# Conjugacy width in uniform higher rank arithmetic groups of orthogonal type

Nir Avni and Chen Meiri

###### Abstract

Let $n\geq 10$ and let $f_{a}$ be the quadratic form
$f_{a}(x_{1},\ldots,x_{2n})=\sum_{i=1}^{2n}x_{i}^{2}$. Assuming the
Generalized Riemann Hypothesis, we prove that every conjugacy class of
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$
has finite width and describe some consequences. To the best of our knowledge,
this is the first time that a boundedness result was proved for an anisotropic
higher rank arithmetic group. In addition, we show that every higher rank
arithmetic group in which every conjugacy class has a finite width, with some
natural upper bound on this width, has the Congruence Subgroup Property. While
the group $\operatorname{Spin}_{f_{a}}(\mathbb{Z}[\frac{1}{2}])$ was known to
have CSP, we hope that the approach to CSP via conjugacy width will be
fruitful.

In a forthcoming paper, we will extend this result to many other anisotropic
spin groups.

##  1 Introduction

###  1.1 Main result

Let $\Gamma$ be a group. For a subset $X\subseteq\Gamma$, we say that the
width of $X$ is $n$ if every element in the subgroup generated by $X$ is a
product of at most $n$ elements of $X\cup X^{-1}$ and $n$ is the minimal such
natural number. We say that the width of $X$ is infinite if no such $n$ exist.
The width of $X$ in $\Gamma$ is denoted by $\operatorname{wid}_{\Gamma}(X)$.

Group theorists have been studying width for a long time. Of special interest
are determining the widths of verbal sets (see Definition 1.2.1), conjugacy
classes, and generating sets. For example:

  * •

A theorem of Borel [8] implies that the width of any verbal subset in an
algebraic group over an algebraically closed field is at most 2.

  * •

Roman’kov [43] proved that the width of any verbal subset in a finitely
generated nilpotent group is finite.

  * •

In [32] and [33], Nikolov and Segal studied widths of verbal subsets in
general finite groups. For example, they proved that if $w$ is a power word or
the commutator word, then the width of $w$ in a finite group $G$ is bounded by
a function of the number of generators of $G$. This result is the key step in
proving the following conjecture of Serre: In every finitely generated
profinite group, every finite index subgroup is open.

  * •

Larsen and Shalev [22] studied widths of verbal subsets in finite simple
groups. They proved that, for every word $w$,
$\operatorname{wid}_{G}(w(G))\leq 2$ if $G$ is a finite simple group of
sufficiently large size.

  * •

In [19], Jaikin-Zapirain determined which words have finite width in free
pro-$p$ groups.

In this paper, we study width in arithmetic groups. As is well known, there is
a sharp contrast between arithmetic groups of rank one and arithmetic groups
of higher rank. The following results suggest that this contrast is present
also in the context of width:

  * •

Bestvina, Bromberg, and Fujiwara show in [7] that the width of any verbal
subset in an acylindrically hyperbolic group is either 1 or infinite (see [42,
29] for the cases of free groups and hyperbolic groups, respectively). In
particular, this applies to lattices in rank one Lie groups.

  * •

Carter and Keller proved in [13] that the width of the set of elementary
matrices in $\operatorname{SL}_{n}(\mathbb{Z})$, $n\geq 3$, is finite. It
follows that the width of the conjugacy class of the elementary matrix
$e_{1,n}$ is finite. From this, one can deduce that the width of any conjugacy
class and every verbal subset of $\operatorname{SL}_{n}(\mathbb{Z})$ is
finite, see [1]. These results were generalized to Chevalley groups in [50].

For non-Chevalley groups, finite width of the set of elementary matrices was
conjecturally replaced by Bounded Generation, i.e., the property

  * (BG)

There exist finitely many elements $g_{1},\ldots,g_{n}\in G$ such that
$\langle g_{1}\rangle\cdots\langle g_{n}\rangle=G$.

For many years, proving BG for non-Chevalley groups was a central open problem
in the theory of arithmetic groups. The main motivation was that BG implies
the Congruence Subgroup Problem (see [35, 24]). For other consequences of BG,
see, for example, [46].

Bounded Generation for various isotorpic orthogonal groups of higher rank was
proved in [16], but there was no example of a non-isotropic arithmetic group
with BG. It came as a surprise that BG fails for non-isotropic groups. In
fact, Corvaja–Rapinchuk–Ren–Zannier prove in [14] that BG can never hold (even
for Chevalley groups) unless some of the $g_{i}$s are virtually unipotent.

Returning to $\operatorname{SL}_{n}(\mathbb{Z})$ for the moment, finite width
of elementary matrices is equivalent to finite width of the conjugacy class of
$e_{1,n}(1)$. The proof that finite width of elementary matrices implies
finite width of the conjugacy class of $e_{1,n}$ is a simple computation. The
converse, i.e., that finite width of the conjugacy class of $e_{1,n}$ implies
finite width of elementary matrices is also true, as proved by Suslin in [49].
This equivalence holds over arbitrary commutative rings and not just
$\mathbb{Z}$.

The goal of this paper is to show that, in contrast to Bounded Generation,
finite width of all conjugacy classes, can hold also in uniform higher rank
arithmetic groups and that is has many consequences, including the Congruence
Subgroup Property. As there are still cases where the CSP is unknown, we hope
that the approach to CSP via finite conjugacy width will shed some light on
those unknown cases. We stress that all cases where CSP in higher rank groups
is unknown are anisotropic.

For an element $g$ in a group $G$, the generalized conjugacy class of $g$ is
the union of the conjugacy class of $g$, the conjugacy class of $g^{-1}$, and
$\left\\{1\right\\}$. We denote the generalized conjugacy class of $g$ by
$\operatorname{gcl}_{G}(g)$. We say that a group $G$ has finite conjugacy
width if the width of every generalized conjugacy class in $G$ is finite. Our
first result is the following

######  Theorem 1.1.1.

Let $n\geq 10$, let $f_{a}$ be the quadratic form
$x_{1}^{2}+\cdots+x_{2n}^{2}$ on $\mathbb{Z}^{2n}$, and let
$\Gamma=\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$.
Then $\Gamma$ has finite conjugacy width. Moreover, there is a constant $C$
such that, for every $\gamma\in\Gamma$,

$\operatorname{wid}_{\Gamma}\left(\operatorname{gcl}_{\Gamma}(\gamma)\right)\leq
C\cdot\operatorname{wid}_{\operatorname{Spin}_{q}(\mathbb{R})}\left(\operatorname{gcl}_{\operatorname{Spin}_{q}(\mathbb{R})}(\gamma)\right).$

We conjecture that Theorem 1.1.1 holds more generally:

######  Conjecture 1.1.2.

Let $K$ be a global field, let $S$ be a finite set of places of $K$ containing
all archimedean places, and denote the ring of $S$-integers of $K$ by $O_{S}$.
Let $\mathbf{G}$ be a connected and simply connected semisimple $K$-group of
$S$-rank at least two and let $\Gamma$ be a congruence subgroup of
$\mathbf{G}(O_{S})$. There is a constant $C$ such that, for every
$\gamma\in\Gamma$,

$\operatorname{wid}_{\Gamma}\left(\operatorname{gcl}_{\Gamma}(\gamma)\right)\leq
C\cdot\max\left\\{\operatorname{wid}_{\mathbf{G}(K_{v})}\left(\operatorname{gcl}_{\mathbf{G}(K_{v})}(\gamma)\right)\mid
v\in S\right\\}.$ (1)

Our second result relates finite conjugacy width with the Congruence Subgroup
Property:

######  Theorem 1.1.3.

Let $\Gamma$ be as in Conjecture 1.1.2. Then $\Gamma$ has the Congruence
Subgroup Property if and only if its profinite completion $\widehat{\Gamma}$
has finite conjugacy width.

######  Remark 1.1.4.

  1. (1)

It follows from [2] that the right hand side of Inequality (1) is finite.

  2. (2)

Conjecture 1.1.2 implies the Congruence Subgroup Problem. We will prove in
Proposition 6.2.1 that if $\Gamma$ is as in Conjecture 1.1.2 and Inequality
(1) holds, then $\widehat{\Gamma}$ has finite conjugacy width, and, therefore,
$\Gamma$ has the Congruence Subgroup Property.

  3. (3)

Theorem 1.1.1 implies that
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$
is bounded, in the sense of [10]. Conjecture 1.1.2 will imply the same for
every higher rank arithmetic lattice. Note that none of these groups are
meager in the sense of [10].

######  Remark 1.1.5.

If $u$ is a unipotent element, then the map $n\mapsto u^{n}$ is polynomial.
Thus, Bounded Generation with unipotents is a question about solvability of
polynomial Diophantine equations. For a semisimple element $g$, the map
$n\mapsto g^{n}$ is an exponential map in $n$, and Bounded Generation with
semisimple elements is a question about solvability of exponential Diophantine
equations. Exponential Diophantine equations are rarely solvable, and this is
a bird-eye reason for the failure of Bounded Generation without unipotents.

For every element $g$, whether unipotent or semisimple, the map $h\mapsto
hgh^{-1}$ is a polynomial map in the entries of $h$, so finite conjugacy width
is a question about solvability of polynomial Diophantine equations. Thus,
finite conjugacy width is closer to Bounded Generation with unipotents,
lending support to Conjecture 1.1.2.

###  1.2 Applications

####  1.2.1 Verbal width

######  Definition 1.2.1.

  1. (a)

A word $w=u(x_{1},\ldots,x_{k})$ on $k$ letters in an element of the free
group $F(x_{1},\ldots,x_{k})$ with basis $x_{1},\ldots,x_{k}$. For example,
$x_{1}x_{2}^{-2}x_{3}$ is a word.

  2. (b)

Let $\Gamma$ be a group and let $w$ be a non-trivial word on $k$ letters. For
every $\bar{g}=(g_{1},\ldots,g_{k})\in\Gamma^{k}$ we denote by $w(\bar{g})$
the element of $\Gamma$ resulting from substituting each $x_{i}$ appearing in
$w$ by $g_{i}$. For example, if $w=x_{1}x_{2}^{-2}x_{3}$ then
$w(\bar{g})=g_{1}g_{2}^{-2}g_{3}$.

  3. (c)

If $\Gamma$ is a group and $w$ is a word, the verbal subset of $w$ is the set
$w(\Gamma):=\\{w(\bar{g})^{\epsilon}\mid\bar{g}\in\Gamma^{k},\epsilon=\pm
1\\}$. The verbal subgroup of $w$ is the subgroup $\Gamma_{w}:=\langle
w(\Gamma)\rangle$.

  4. (d)

The width $\operatorname{wid}_{w}(\Gamma)$ of $w$ in $\Gamma$ is defined to be
$\operatorname{wid}_{\Gamma}(w(\Gamma))$.

Theorem 1.1.1 gives a new result about verbal width, namely, that the width of
any verbal set in
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$
is finite (assuming GRH). More precisely, Theorem 1.1.1 implies the following:

######  Corollary 1.2.2.

Let $G=\operatorname{Spin}_{f_{a}}$ and
$\Gamma=G\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$. There exists a
constant $C$ such that, for every word $w$,

$\operatorname{wid}_{\Gamma}(w(\Gamma))\leq
C\cdot\operatorname{wid}_{G(\mathbb{R})}\big{(}w(G(\mathbb{R}))\big{)}.$

Corollary 1.2.6 leads to the following conjecture:

######  Conjecture 1.2.3.

Let $\Gamma$ be as in Conjecture 1.1.2. There exists a constant $C$ (possibly
depending on $\Gamma$) such that, for every word $w$,

$\operatorname{wid}_{\Gamma}(w(\Gamma))\leq
C\cdot\operatorname{wid}_{G(\mathbb{A}_{S})\times\overline{\Gamma}}\big{(}w(G(\mathbb{A}_{S})\times\overline{\Gamma})\big{)},$

where $\overline{\Gamma}$ is the congruence completion of $\Gamma$.

######  Remark 1.2.4.

In [11], it was asked if in every higher rank arithmetic group, the commutator
word has a finite width. Corollary 1.2.6 is the first solution of this
question (positive or negative) with respect to a uniform higher rank
arithmetic group.

######  Remark 1.2.5.

  1. (1)

The main theorem of [1] implies that Conjecture 1.2.3 holds for
$\Gamma=\operatorname{SL}_{n}(\mathbb{Z})$, if $n\geq 3$. In this case, the
constant $C$ does not depend on $n$.

  2. (2)

Note the dichotomy between rank 1 groups (where the width is usually infinite
by [7]) and higher rank groups (Conjecture 1.2.3).

####  1.2.2 Model Theory of arithmetic groups

Theorem 1.1.1 also has applications to the model theory of the group
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$.
Namely, it implies that
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$
is bi-interpretable with the ring of integers.

The general notion of bi-interpretability is explained in [2]. It is an
equivalence relation on the class of all structures of all first-order
languages. Roughly, two structures are bi-interpretable if their categories of
definable sets are equivalent.

Bi-interpretation with $\mathbb{Z}$ (as a structure of the language of rings)
has several interesting consequences. For example, a corollary of [20] is that
if a finitely generated group $\Gamma$ is bi-interpretable with $\mathbb{Z}$,
then there exists a first order sentence in the signature of groups such that
(up to isomorphism) $\Gamma$ is the only finitely generated group which
satisfies it. This property was called quasi-finitely axiomatizable in [31].
Another consequence is that if $\Gamma$ is a finitely generated group which is
bi-interpretable with $\mathbb{Z}$, then every finitely generated subgroup of
$\Gamma$ is definable.

In [2], it was proved that if $\Gamma$ is a higher rank arithmetic group of
orthogonal type which satisfies some technical conditions, then
$\Gamma/Z(\Gamma)$ is bi-interpretable with $\mathbb{Z}$. Moreover, $\Gamma$
itself is bi-interpretable with $\mathbb{Z}$ if and only if the power word
$x^{|Z(\Gamma)|}$ has finite width in $\Gamma$. We get that

######  Corollary 1.2.6.

Let
$\Gamma=\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$.
Then $\Gamma$ is bi-interpretable with $\mathbb{Z}$. In particular, $\Gamma$
is quasi-finitely axiomatizable and every finitely generated subgroup of
$\Gamma$ is definable.

######  Remark 1.2.7.

  1. (1)

Using similar arguments to [2, Theorem 1.7] and Theorem 1.1.1, one can show
that every group abstractly commensurable to
$\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$
is bi-interpretable with $\mathbb{Z}$.

  2. (2)

Note again the contrast to rank 1 groups: by [45], if $\Delta$ is a torsion-
free word hyperbolic group, then $\Delta$ and the free product
$\Delta*\mathbb{Z}$ satisfy the same first-order sentences. Moreover, the only
non-trivial, proper, and definable subgroups of $\Delta$ are cyclic.

####  1.2.3 Stability

A famous question of Ulam [51, Chapter VI] asks what happens to algebraic
structures if we relax their defining properties to only hold approximately,
rather than on the nose. Specifying this question to the notion of a
homomorphism, the question is whether every almost-homomorphism is close to an
actual homomorphism. We use the following definition from [6]:

######  Definition 1.2.8.

For every $n$, let $d_{n}$ be the normalized Hamming distance on $S_{n}$. We
say that a group $G$ is flexibly stable if, for any $\epsilon>0$, there is
$\delta>0$ such that, if $f:G\rightarrow S_{n}$ is any function that satisfies

$\left(\forall x,y\in G\right)\quad d_{n}\left(f(xy),f(x)f(y)\right)<\delta,$
(2)

then there exists a natural number $n\leq N\leq n(1+\epsilon)$ and a
homomorphism $F:G\rightarrow S_{N}$ such that

$\left(\forall x\in G\right)\quad d_{N}\left(f(x),F(x)\right)<\epsilon.$ (3)

In the definition, (2) is the condition that $f$ is almost a homomorphism and
(3) is the condition that $f$ is close to a homomorphism. The flexibility
pertains to the ability to replace $n$ by $N$ in the definition.

In [6], Becker and Chapman used Bounded Generation to prove that
$\operatorname{SL}_{n}(\mathbb{Z})$ are flexibly stable. As a corollary of
Theorem 1.1.1, we show:

######  Theorem 1.2.9.

Let $n\geq 20$, let $f_{a}$ be the quadratic form $x_{1}^{2}+\cdots+x_{n}^{2}$
on $\mathbb{Z}^{n}$, and let
$\Gamma=\operatorname{Spin}_{f_{a}}\left(\mathbb{Z}\left[\frac{1}{2}\right]\right)$.
Then $\Gamma$ is flexibly stable.

Conjecture 1.1.2 will imply that higher-rank lattices are flexibly stable.

######  Conjecture 1.2.10.

Let $\Gamma$ be as in Conjecture 1.1.2. Then $\Gamma$ is flexibly stable.

###  1.3 Sketch of the proof: Verbal width and CSP in non-standard models

A key idea of this paper is to relate width in an arithmetic group $\Gamma$ to
variants of the Congruence Subgroup Problem for an ultrapower of $\Gamma$.

Since it plays an important role in this paper, we briefly survey the
Congruence Subgroup Problem. Let $\mathbf{G}$ be a connected, simply
connected, and simple algebraic group over a number field $K$ and let $S$ be a
finite set of places of $K$ containing all archimedean ones. Assume that
$\mathbf{G}(K_{v})$ is not compact, for every finite $v\in S$. Denote the ring
of $S$-integers in $K$ by $O_{S}$ and let $\Gamma=\mathbf{G}(O_{S})$ be an
arithmetic group. For a non-zero ideal $I$ of $O_{S}$, the principal
congruence subgroup $\mathbf{G}(O_{S})[I]$ is the kernel of the homomorphism
$\mathbf{G}(O_{S})\rightarrow\mathbf{G}(O_{S}/I)$. We say that $\Gamma$ has
the Congruence Subgroup Property if there is a constant $C$ with the following
property: if $\Delta\subseteq\Gamma$ is a finite-index subgroup, then there is
an ideal $I$ such that
$\left[\mathbf{G}(O_{S})[I]:\Delta\cap\mathbf{G}(O_{S})[I]\right]\leq C$. The
Congruence Subgroup Problem asks whether $\Gamma$ has the Congruence Subgroup
Property if the $S$-rank of $\mathbf{G}$ is at least two.

It is known that most higher rank arithmetic groups have the Congruence
Subgroup Property. The main case which is still open is when $\mathbf{G}$ is
the norm one elements of a division algebra over $K$. In the modern proof of
this fact, which stems from [4, 5], the Congruence Subgroup Problem is
reformulated as follows. Let $\widehat{\mathbf{G}(O_{S})}$ be the profinite
completion of $\mathbf{G}(O_{S})$. By the assumptions on $\mathbf{G}$, the
congruence completion of $\mathbf{G}(O_{S})$ is
$\mathbf{G}\left(\widehat{O_{S}}\right)$. Since $\widehat{\mathbf{G}(O_{S})}$
is compact, the map
$\widehat{\mathbf{G}(O_{S})}\rightarrow\mathbf{G}\left(\widehat{O_{S}}\right)$
is onto. Consider the short exact sequence

$1\rightarrow
C\rightarrow\widehat{\mathbf{G}(O_{S})}\rightarrow\mathbf{G}\left(\widehat{O_{S}}\right)\rightarrow
1.$ (4)

The Congruence Subgroup Problem is equivalent to the claim that $C$ is finite.

There are two parts of the proof of the Congruence Subgroup Property for
$\mathbf{G}(O_{S})$. The first part is to show that the short exact sequence
(4) is a central extension. The second part is to show that (4) has a
continuous splitting. By Property (T), the abelization of $\mathbf{G}(O_{S})$
is finite, so the finiteness of $C$ follows from the continuous splitting of
(4).

The proof of Theorem 1.1.1 also starts by reformulating finite conjugacy width
in terms of a short exact sequence, now related to an ultrapower of $\Gamma$.
As a part of the proof, we show that finite conjugacy width in
$\operatorname{Spin}_{q}(O_{S})$ follows from a uniform bound on the width of
the commutator word in all congruence subgroups of
$\operatorname{Spin}_{q}(O_{S})$. We describe now how to bound these widths.
For simplicity, we only consider the width in
$\operatorname{Spin}_{f_{a}}(O)$.

Denote $\Gamma=\operatorname{Spin}_{f_{a}}(O)$. Let $\Gamma^{*}$ be the
ultrapower of $\Gamma$ taken along a non-principal ultrafilter and let $O^{*}$
be the ultrapower of $O$ along the same ultrafilter. It is easy to see that
$\Gamma^{*}=\operatorname{Spin}_{f_{a}}(O^{*})$. A standard model theoretic
argument implies that the commutator word has a finite width in
$\operatorname{Spin}_{f_{a}}(O)$ if and only if the abelianization of
$\Gamma^{*}$ is finite.

The congruence subgroups
$\Gamma^{*}[I]:=\operatorname{Spin}_{f_{a}}(O^{*})[I]$ of
$\Gamma^{*}=\operatorname{Spin}_{f_{a}}(O^{*})$, where $I$ ranges over all
non-zero ideals of $O^{*}$, form a basis of identity neighborhoods for a
topology on $\Gamma^{*}$. We refer to this topology as the congruence
topology. Denote the completion of $\Gamma^{*}$ with the congruence topology
by $\overline{\Gamma^{*}}$. We define a stronger topology on $\Gamma^{*}$, in
which sets of the form $\Gamma^{*}[I]\cap(\Gamma^{*})^{\prime}$, where
$(\Gamma^{*})^{\prime}$ is the derived subgroup, form a basis of identity
neighborhoods. We call this topology the commgrunece topology. Denote the
completion of $\Gamma^{*}$ with respect to the commgruence topology by
$\widehat{\Gamma^{*}}$.

We prove in Lemma 3.2.8 that the natural map
$\varphi:\widehat{\Gamma^{*}}\rightarrow\overline{\Gamma^{*}}$ is surjective
and that the short exact sequence

$1\rightarrow
C\rightarrow\widehat{\Gamma^{*}}\rightarrow\overline{\Gamma^{*}}\rightarrow 1$
(5)

is a split topological central extension with a discrete center. In contrast
to the standard case, neither $\widehat{\Gamma^{*}}$ nor
$\overline{\Gamma^{*}}$ is compact, so the surjectivity in (5) is not obvious.
Also, $\Gamma^{*}$ does not have Property (T), so the splitting of (5) does
not imply the finiteness of $C$. However, once the splitting of (5) is
established, it is easy to see that $C$ is isomorphic to the abelization of
$\Gamma^{*}$, so the centrality of (5) follows.

In general, we need to show that $C$ is finite, but, in the case we discuss,
we will actually show that $C$ is trivial. Let $K^{*}$ be the ultrapower of
$K$ over the same ultrafilter as before. It is easy to see that the congruence
topology extends to a topology on $\operatorname{Spin}_{f_{a}}(K^{*})$ under
which $\operatorname{Spin}_{f_{a}}(K^{*})$ is a topological group. In §3, we
show the same for the commgruence topology. The argument here is more involved
and, in a sense, replaces the centrality argument in the classical case.

At this point, we get a short exact sequence

$1\rightarrow
C\rightarrow\widehat{\operatorname{Spin}_{f_{a}}(K^{*})}\rightarrow\overline{\operatorname{Spin}_{f_{a}}(K^{*})}\rightarrow
1$ (6)

with the same kernel $C$ of (5). The advantage of (6) is that
$\operatorname{Spin}_{f_{a}}(K^{*})$ is perfect, as we prove in Corollary
A.4.9. Thus, in order to show that $C$ is trivial, it is enough to prove that
(6) admits a continuous splitting.

The short exact sequence (6) has a continuous splitting over the open subgroup
$\overline{\Gamma^{*}}$ and has an abstract (not necessarily continuous)
splitting over the subgroup $\operatorname{Spin}_{f_{a}}(K^{*})$ with a dense
image. We call a topological central extension

$1\rightarrow D\rightarrow
E\rightarrow\overline{\operatorname{Spin}_{f_{a}}(K^{*})}\rightarrow 1$ (7)

discrete metaplectic if $D$ is discrete, there is an abstract section of (7)
over $\operatorname{Spin}_{f_{a}}(K^{*})$ whose image is dense, and there is a
continuous splitting of (7) over an open subgroup of
$\overline{\operatorname{Spin}_{f_{a}}(K^{*})}$. In the proof of Theorem
1.1.1, we show that any discrete metaplectic extension has a continuous
splitting. For simplicity, we will assume here that there is a continuous
splitting of (7) over all of $\overline{\operatorname{Spin}_{f_{a}}(O^{*})}$.

Let $\overline{K^{*}}$ be the congruence completion of $K^{*}$ and let $f_{s}$
be the sum of $n$ hyperbolic planes. Since
$\overline{\operatorname{Spin}_{f_{a}}(K^{*})}=\operatorname{Spin}_{f_{a}}\left(\overline{K^{*}}\right)$
is isomorphic to the split form
$\operatorname{Spin}_{f_{s}}\left(\overline{K^{*}}\right)$, we can rewrite (7)
as

$1\rightarrow D\rightarrow
E\rightarrow\overline{\operatorname{Spin}_{f_{s}}(K^{*})}\rightarrow 1$ (8)

and note that (8) has a continuous splitting over
$\overline{\operatorname{Spin}_{f_{s}}(O^{*})}=\overline{\operatorname{Spin}_{f_{a}}(O^{*})}$
and an abstract splitting over
$\overline{\operatorname{Spin}_{f_{a}}(K^{*})}$.

In order to show that (8) splits, it is enough to show that it has an abstract
splitting over $\operatorname{Spin}_{f_{s}}(K^{*})$. Indeed, let $\alpha$ be
such a splitting. Since every root element is a commutator, Bounded Generation
for $\operatorname{Spin}_{f_{s}}(O)$ implies that
$\operatorname{Spin}_{f_{s}}(O^{*})$ is perfect. Thus, any splitting of (8)
over $\operatorname{Spin}_{f_{s}}(O^{*})$ coincides with $\alpha$. Since there
is a continuous splitting over
$\overline{\operatorname{Spin}_{f_{s}}(O^{*})}$, it agrees with $\alpha$ over
$\operatorname{Spin}_{f_{s}}(O^{*})=\operatorname{Spin}_{f_{s}}(K^{*})\cap\overline{\operatorname{Spin}_{f_{s}}(O^{*})}$.
Thus, $\alpha$ is continuous and extends to a splitting of (8).

Let $h_{1},h_{2}$ be two coroots of $\operatorname{Spin}_{f_{s}}$
corresponding to two roots of angle $120^{\circ}$. For every
$a_{1},a_{2}\in(K^{*})^{\times}$, the elements $h_{1}(a_{1}),h_{2}(a_{2})$
commute. It follows from [25, Theorem 8.2] and the perfectness of
$\operatorname{Spin}_{f_{s}}(K^{*})$ that the central extension (8) has a
splitting over $\operatorname{Spin}_{f_{s}}(K^{*})$ if some lifts of
$h_{1}(a_{1}),h_{2}(a_{2})$ to $E$ commute.

One way to ensure that some lifts of $h_{1}(a_{1}),h_{2}(a_{2})$ commute is to
find a pair $(g_{1},g_{2})$ of commuting elements in
$\operatorname{Spin}_{f_{a}}(K^{*})$ that is sufficiently close to
$(h_{1}(a_{1}),h_{2}(a_{2}))$. Indeed, let
$\mu:\operatorname{Spin}_{f_{a}}(K^{*})\rightarrow E$ be a section. If
$g_{1},g_{2}\in\operatorname{Spin}_{f_{a}}(K^{*})$ commute, then the lifts
$\mu(g_{1}),\mu(g_{2})$ also commute. If $(h_{1}(a_{1}),h_{2}(a_{2}))$ is
close to $(g_{1},g_{2})$, then one can choose the lifts of
$h_{1}(a_{1}),h_{2}(a_{2})$ to be close to $\mu(g_{1}),\mu(g_{2})$, so their
commutator is close to the identity. Since $D$ is discrete, if the lifts of
$h_{1}(a_{1}),h_{2}(a_{2})$ are chosen close enough to
$\mu(g_{1}),\mu(g_{2})$, then $h_{1}(a_{1})$ and $h_{2}(a_{2})$ commute.

In general, we will construct, for every $a_{1},a_{2}$, a sequence
$(g_{1,1},g_{2,1}),\ldots,(g_{1,m},g_{2,m})$ of commuting pairs in
$\operatorname{Spin}_{f_{a}}(K^{*})$ and lifts
$\widetilde{h_{1}(a_{1})},\widetilde{h_{2}(a_{2})}$ such that

$\left[\widetilde{h_{1}(a_{1})},\widetilde{h_{2}(a_{2})}\right]=\left[\mu(g_{1,1}),\mu(g_{2,1})\right]\cdots\left[\mu(g_{1,m}),\mu(g_{2,m})\right]=1.$
(9)

A main difficulty in finding the sequence of commuting pairs, not present in
the classical case, is that, a-priori, $D$ may have a divisible factor. It is
in this point of the proof that we use the Generalized Riemann Hypothesis.
Specifically, we use a variant of Artin’s Primitive Root Conjecture from [23]
in the argument. We believe that results similar to [18] together with minor
modifications of the arguments in this paper will give an unconditional
result.

We end this section with a weaker version of Conjecture 1.1.2 which resembles
Margulis’ Normal Subgroup Theorem.

######  Conjecture 1.3.1.

Let $\Gamma$ be as in Conjecture 1.1.2. Fix a non-principal ultrafilter
$\mathcal{N}$ on the natural numbers. Let $O^{*},K^{*}$ denote the
$\mathcal{N}$-ultrapowers of $O$ and $K$, respectively.

For every $g\in\mathbf{G}(O^{*})$ such that the normal subgroup of
$\mathbf{G}(K^{*})$ generated by $g$ is $\mathbf{G}(K^{*})$, the normal
subgroup of $\mathbf{G}(O^{*})$ generated by $g$ contains a finite index
subgroup of a principal congruence subgroup.

###  1.4 Acknowledgements

We thank Danny Neftin, Eugene Plotkin, Pavel Gvozdevsky, Peter Sarnak, and
V.N. Venkataramana for their help. N.A. was supported by NSF grant
DMS-1902041, C.M. was supported by ISF grant 1226/19 and BSF grant 2020119.

##  2 The congruence topology on ultrapowers

###  2.1 Setting for Section 2

Sections 2, 3, 4, and Appendix A concern algebraic groups over ultraproducts
and the results there are stated in varying order of generality. We will start
each of those sections by spelling out our assumptions. The relation between
the settings is summarized in the following diagram:

$\framebox{\text{\parbox{71.13188pt}{Generality of Section
\ref{sec:ultra}}}}\supseteq\framebox{\text{\parbox{71.13188pt}{Generality of
Section
\ref{sec:word_comp}}}}\supseteq\framebox{\text{\parbox{71.13188pt}{Generality
of Appendix
\ref{sec:spin}}}}\supseteq\framebox{\text{\parbox{71.13188pt}{Generality of
Section \ref{sec:fcw}}}}$

In this section, we fix

  * •

A number field $K$ and a finite set $S$ of places of $K$, containing all
archimedean places.

  * •

An affine group scheme $G$ over the ring of $S$-integers of $K$ and an affine
embedding of $G$ into $\operatorname{Mat}_{D}$, for some $D\in\mathbb{N}$.

  * •

A non-principal ultrafilter $\mathcal{N}$ on the set of natural numbers
$\mathbb{N}$.

We make the following assumption:

  1. (1)

$G_{K}$ is semisimple, connected, simply connected, and satisfies
$\operatorname{rank}_{S}G:=\sum_{v\in S}\operatorname{rank}(G_{K_{v}})\geq 1$.

Throughout the paper, we will use the following notations:

  * •

$O$ is the ring of $S$-integers of $K$.

  * •

$\mathcal{I}_{K}$ is the set of non-zero ideals of $O$ and $\mathcal{P}_{K}$
is the set of non-zero prime ideals of $O$. For every $I\in\mathcal{I}_{K}$
and $\mathcal{Q}\subseteq\mathcal{P}_{K}$, denote

$\mathcal{P}_{K}(I):=\\{P\in\mathcal{P}_{K}\mid I\subseteq P\\}\text{ and
}\mathcal{I}_{K}(\mathcal{Q}):=\\{I\in\mathcal{I}_{K}\mid(\forall
P\in\mathcal{P}_{K}\smallsetminus\mathcal{Q})\ I+P=O\\}.$

  * •

${\bf A}:=\underset{v\notin S}{\prod^{\prime}}K_{v}$ is the ring of finite
$S$-adeles. We view $K$ as a subring of ${\bf A}$ via the diagonal embedding.

  * •

${\bf O}$ is the closure of $O$ in ${\bf A}$. We have ${\bf O}=\\{x\in{\bf
A}\mid(\forall P\in\mathcal{P}_{K})\ \operatorname{val}_{P}(a)\geq 0\\}$,
where $\operatorname{val}_{P}:{\bf A}\rightarrow\mathbb{Z}\cup\\{\infty\\}$ is
the $P$-adic valuation.

  * •

$N^{*}$, $Z^{*}$, $O^{*}$, $K^{*}$, ${\bf A}^{*}$ and ${\bf O}^{*}$ are the
$\mathcal{N}$-ultrapowers of $\mathbb{N}$, $\mathbb{Z}$, $O$, $K$, ${\bf A}$
and ${\bf O}$, respectively. We identify the $\mathcal{N}$-ultrapower of
$\mathbb{Z}\cup\infty$ with $Z^{*}\cup\\{\infty\\}$.

  * •

There are natural embeddings $N^{*}\hookrightarrow Z^{*}\hookrightarrow
O^{*}\hookrightarrow K^{*}$ and $O^{*}\hookrightarrow{\bf
O}^{*}\hookrightarrow{\bf A}^{*}\hookleftarrow K^{*}$. Abusing notations, we
regard these as inclusions. Note that $K^{*}$ is the quotient field of the
integral domain $O^{*}$.

  * •

For a sequence $a_{1},a_{2},\ldots$ of elements of $K$, $[a_{n}]_{n}$ is the
equivalence class of $(a_{n})_{n}$ in $K^{*}$. Unless explicitly stated
otherwise, when we write $a=[a_{n}]_{n}\in K^{*}$, we assume that each $a_{n}$
belongs to $K$. We use a similar convention for elements of $N^{*}$, $Z^{*}$,
$O^{*}$, ${\bf A}^{*}$ and ${\bf O}^{*}$.

  * •

Given a sequence $A_{1},A_{2},\ldots$ of subsets of $K$, denote

$[A_{n}]_{n}:=\\{[a_{n}]_{n}\in K^{*}\mid(\exists X\in\mathcal{N}\ \forall
n\in X)\ a_{n}\in A_{n}\\}.$

Note that $[A_{n}]_{n}=[B_{n}]_{n}$ if and only if $\\{n\in\mathbb{N}\mid
A_{n}=B_{n}\\}\in\mathcal{N}$.

  * •

$\leq$ is the linear order on $Z^{*}$ defined by $a\leq b$ if and only if
$b-a\in N^{*}$. We extend $\leq$ to $Z^{*}\cup\\{\infty\\}$ by defining
$m\leq\infty$, for every $m\in Z^{*}$.

  * •

$\mathcal{I}_{K^{*}}:=\\{aO^{*}+bO^{*}\mid a,b\in O^{*}\setminus\\{0\\}\\}$
and $\mathcal{P}_{K^{*}}:=\\{P\in\mathcal{I}_{K^{*}}\mid P\text{ is a prime
ideal}\\}$. Since every ideal of $O$ is generated by two elements,
$\mathcal{I}_{K^{*}}$ and $\mathcal{P}_{K^{*}}$ are the sets of definable and
definable prime ideals of $O^{*}$, where definable means definable with
parameters. We will not use this fact in this paper.

  * •

For every $I\in\mathcal{I}_{K^{*}}$ and
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$, denote

$\mathcal{P}_{K^{*}}(I):=\\{P\in\mathcal{P}_{K^{*}}\mid I\subseteq P\\}\text{
and }\mathcal{I}_{K^{*}}(\mathcal{Q}):=\\{I\in\mathcal{I}_{K^{*}}\mid(\forall
P\in\mathcal{P}_{K^{*}}\smallsetminus\mathcal{Q})\ I+P=O^{*}\\}.$

Note that, if $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ and
$I\in\mathcal{I}_{K^{*}}$, then $I\in\mathcal{P}_{K}^{*}(\mathcal{Q})$ if and
only if $\\{P\in\mathcal{P}_{K^{*}}\mid I\subseteq P\\}\subseteq\mathcal{Q}$.

  * •

For every $I\in\mathcal{I}_{K^{*}}$, there exists a sequence
$(I_{n})_{n}\in(\mathcal{I}_{K})^{\mathbb{N}}$ such that $I=[I_{n}]_{n}$.
Similarly, for every $P\in\mathcal{P}_{K^{*}}$, there exists a sequence
$(P_{n})_{n}\in(\mathcal{P}_{K})^{\mathbb{N}}$ such that $P=[P_{n}]_{n}$.
Unless explicitly stated otherwise, when we write
$I=[I_{n}]_{n}\in\mathcal{I}_{K^{*}}$ (respectively,
$P=[P_{n}]_{n}\in\mathcal{P}_{K^{*}}$) we assume that each $I_{n}$ belongs to
$\mathcal{I}_{K}$ (respectively, $P_{n}$ belongs to $\mathcal{P}_{K}$).

  * •

We identity $G(K^{*})$ with the $\mathcal{N}$-ultrapower on $G(K)$ via the map
$[g_{n}]_{n}\mapsto\big{(}[(g_{n})_{i,j}]_{n}\big{)}_{1\leq i,j\leq D}$.

Finally, note that $\mathcal{I}_{K}^{*}$ is closed under finite intersections
since $[I_{n}]_{n}\cap[J_{n}]_{n}=[I_{n}\cap J_{n}]_{n}$.

###  2.2 Internal subsets

######  Definition 2.2.1.

  * •

Given a sequence $(\mathcal{Q}_{n})_{n}$ of subsets of $\mathcal{P}_{K}$,
denote

$[\mathcal{Q}_{n}]_{n}:=\\{[P_{n}]_{n}\in\mathcal{P}_{K^{*}}\mid(\exists
X\in\mathcal{N}\ \forall n\in X)\ P_{n}\in\mathcal{Q}_{n}\\}.$

Note that $[\mathcal{Q}_{n}]_{n}=[\mathcal{R}_{n}]_{n}$ if and only if
$\\{n\in\mathbb{N}\mid\mathcal{Q}_{n}=\mathcal{R}_{n}\\}\in\mathcal{N}$.

  * •

A subset $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ is called internal if there
exists a sequence $(\mathcal{Q}_{n})_{n}$ of subsets of $\mathcal{P}_{K}$ such
that $\mathcal{Q}=[\mathcal{Q}_{n}]_{n}$. If, in addition, each
$\mathcal{Q}_{n}$ can be chosen to be finite, then $\mathcal{Q}$ is called
small internal.

  * •

Unless explicitly stated otherwise, when we write
$\mathcal{Q}=[\mathcal{Q}_{n}]_{n}$, we assume that each $\mathcal{Q}_{n}$ is
a subset of $\mathcal{P}_{K}$. If, in addition, $\mathcal{Q}$ is small
internal, we also assume that each $\mathcal{Q}_{n}$ is finite.

######  Lemma 2.2.2.

A subset $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ is small internal if and
only if there exists $I\in\mathcal{I}_{K^{*}}$ such that
$\mathcal{Q}=\mathcal{P}_{K^{*}}(I)$.

###### Proof.

We first prove the “if” direction. Let $I=[I_{n}]\in\mathcal{I}_{K^{*}}$. For
every $n\in\mathbb{N}$, let $\mathcal{Q}_{n}:=\\{P\in\mathcal{P}_{K}\mid
I_{n}\subseteq P\\}$. Then $\mathcal{P}_{K^{*}}(I)=[\mathcal{Q}_{n}]_{n}$, and
$\mathcal{P}_{K^{*}}(I)$ is small internal.

For the “only if” direction, assume that $\mathcal{Q}=[\mathcal{Q}_{n}]_{n}$,
where, for each $n\in\mathbb{N}$, $\mathcal{Q}_{n}\subseteq\mathcal{P}_{K}$ is
a finite set. For every $n\in\mathbb{N}$, denote $I_{n}:=O$ if
$\mathcal{Q}_{n}=\emptyset$ and $I_{n}:=\prod_{Q\in\mathcal{Q}_{n}}Q$
otherwise. Then $\mathcal{Q}=\mathcal{P}_{K^{*}}\left([{I}_{n}]_{n}\right)$. ∎

######  Lemma 2.2.3.

The collection of internal subsets of $\mathcal{P}_{K^{*}}$ is closed under
finite intersections, finite unions, and complements. The collection of small
internal subsets of $\mathcal{P}_{K^{*}}$ is closed under finite intersections
and finite unions.

###### Proof.

Assume that $\mathcal{Q}=[\mathcal{Q}_{n}]_{n}$ and
$\mathcal{R}=[\mathcal{R}_{n}]_{n}$. Then,

$\mathcal{Q}\cap\mathcal{R}=[\mathcal{Q}_{n}\cap\mathcal{R}_{n}]_{n},\
\mathcal{Q}\cup\mathcal{R}=[\mathcal{Q}_{n}\cup\mathcal{R}_{n}]_{n}\text{ and
}\mathcal{Q}^{c}=[\mathcal{P}_{K}\smallsetminus\mathcal{Q}_{n}]_{n}.$

∎

###  2.3 Valuations

######  Lemma 2.3.1 (Properties of the valuation map).

Let $P\in\mathcal{P}_{K^{*}}$.

  1. (1)

There exists a function $\operatorname{val}_{P}:{\bf A}^{*}\rightarrow
Z^{*}\cup\\{\infty\\}$ such that, if $P=[P_{n}]_{n}$ and $a=[a_{n}]_{n}\in
K^{*}$, then
$\operatorname{val}_{P}(a)=[\operatorname{val}_{P_{n}}(a_{n})]_{n}\in
Z^{*}\cup\\{\infty\\}$.

  2. (2)

For every $a,b\in{\bf A}^{*}$,
$\operatorname{val}_{P}(a+b)\geq\min(\operatorname{val}_{P}(a),\operatorname{val}_{P}(b))$
and
$\operatorname{val}_{P}(ab)=\operatorname{val}_{P}(a)+\operatorname{val}_{P}(b)$.
If $\operatorname{val}_{P}(a)<\operatorname{val}_{P}(b)$, then
$\operatorname{val}_{P}(a+b)=\operatorname{val}_{P}(a)$.

  3. (3)

For every $m\in N^{*}$, $a\in K^{*}\smallsetminus 0$ and $b\in{\bf
A}^{*}\smallsetminus 0$,
$\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(a)>m\\}$ and
$\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(b)<-m\\}$ are small
internal and $\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(b)>m\\}$ is
internal.

###### Proof.

1 Suppose that $a=[a_{n}]_{n}=[a_{n}^{\prime}]_{n}\in{\bf A}^{*}$ and
$P=[P_{n}]_{n}=[P_{n}^{\prime}]_{n}$. Then, $X_{1}:=\\{n\mid
a_{n}=a_{n}^{\prime}\\}\in\mathcal{N}$, $X_{2}:=\\{n\mid
P_{n}=P_{n}^{\prime}\\}\in\mathcal{N}$ and $X_{1}\cap
X_{2}\subseteq\\{n\mid\operatorname{val}_{P_{n}}(a_{n})=\operatorname{val}_{P_{n}^{\prime}}(a_{n})\\}$
so

$\\{n\mid\operatorname{val}_{P_{n}}(a_{n})=\operatorname{val}_{P_{n}^{\prime}}(a_{n})\\}\in\mathcal{N}\text{
and
}[\operatorname{val}_{P_{n}}(a_{n})_{n}]=[\operatorname{val}_{P_{n}^{\prime}}(a_{n}^{\prime})]_{n}.$

2 Write $P=[P_{n}]_{n}$, $a=[a_{n}]_{n}$, and $b=[b_{n}]_{n}$. Then,

$\operatorname{val}_{P}(a+b)=[\operatorname{val}_{P_{n}}(a_{n}+b_{n})]_{n}\geq[\operatorname{val}_{P_{n}}(a_{n})+\operatorname{val}_{P_{n}}(b_{n})]=[\operatorname{val}_{P_{n}}(a_{n})]_{n}+[\operatorname{val}_{P_{n}}(b_{n})]_{n}=\operatorname{val}_{P}(a)+\operatorname{val}_{P}(b)$

and

$\operatorname{val}_{P}(ab)=[\operatorname{val}_{P_{n}}(a_{n}b_{n})]_{n}=[\operatorname{val}_{P_{n}}(a_{n})\operatorname{val}_{P_{n}}(b_{n})]=[\operatorname{val}_{P_{n}}(a_{n})]_{n}[\operatorname{val}_{P_{n}}(b_{n})]_{n}=\operatorname{val}_{P}(a)\operatorname{val}_{P}(b).$

If $\operatorname{val}_{P}(a)<\operatorname{val}_{P}(b)$, then

$X:=\\{n\mid\operatorname{val}_{P_{n}}(a_{n})<\operatorname{val}_{P_{n}}(b_{n})\\}\in\mathcal{N}$

and

$X\subseteq\\{n\mid\operatorname{val}_{P_{n}}(a_{n}+b_{n})=\operatorname{val}_{P_{n}}(a_{n})\\}$

so

$\\{n\mid\operatorname{val}_{P_{n}}(a_{n}+b_{n})=\operatorname{val}_{P_{n}}(a_{n})\\}\in\mathcal{N}$

and

$\operatorname{val}_{P}(a)=[\operatorname{val}_{P_{n}}(a_{n})]_{n}=[\operatorname{val}_{P_{n}}(a_{n}+b_{n})]_{n}=\operatorname{val}_{P}(a+b).$

3 Write $a=[a_{n}]$, $b=[b_{n}]_{n}$, and $m=[m_{n}]$. We can assume that, for
every $n$, $a_{n},b_{n}\neq 0$. For every $n$, denote

$P_{n}:=\\{P\in\mathcal{P}_{K}\mid\operatorname{val}_{P}(a_{n})>m_{n}\\},$
$P_{n}^{\prime}:=\\{P\in\mathcal{P}_{K}\mid\operatorname{val}_{P}(b_{n})<-m_{n}\\}$

and

$P_{n}^{\prime\prime}:=\\{P\in\mathcal{P}_{K}\mid\operatorname{val}_{P}(b_{n})>m_{n}\\}.$

For each $n$, $P_{n}$ and $P_{n}^{\prime}$ are finite subsets and

$[P_{n}]_{n}=\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(a)>m\\},$
$[P_{n}^{\prime}]_{n}=\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(b)<-m\\}$

and

$[P_{n}^{\prime\prime}]_{n}=\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(b)>m\\}.$

∎

######  Remark 2.3.2.

The type of argument in the proof of Lemma 2.3.1 is called a coordinate-wise
argument. In the rest of this paper, we will omit coordinate-wise arguments
that only use well known properties of $K$ and ${\bf A}$.

######  Definition 2.3.3.

We refer to the map $\operatorname{val}_{P}:{\bf A}^{*}\rightarrow
Z^{*}\cup\\{\infty\\}$ defined in Lemma 2.3.1 as the $P$-valuation map.

######  Lemma 2.3.4.

For every $m=[m_{n}]\in N^{*}$, there is a well defined $m$-th power map
${}^{m}:\mathcal{I}_{K^{*}}\rightarrow\mathcal{I}_{K^{*}}$ defined by
$[I_{n}]_{n}^{m}=[I_{n}^{m_{n}}]_{n}$. If $m,m^{\prime}\in N^{*}$, then

  1. (1)

$I^{m}I^{m^{\prime}}=I^{m+m^{\prime}}$.

  2. (2)

$(I^{m})^{m^{\prime}}=I^{mm^{\prime}}$.

  3. (3)

For every $a\in{\bf O}^{*}$ and $P\in\mathcal{P}_{K^{*}}$,
$\operatorname{val}_{P}(a)=\max\\{m\in N^{*}\mid a\in P^{m}\\}$.

###### Proof.

The proof is a simple coordinate-wise argument and is omitted. ∎

######  Proposition 2.3.5 (Strong Approximation Theorem for $K^{*}$).

Let $k\in\mathbb{N}$. For every $1\leq i\leq k$, let $m_{i}\in N^{*}$,
$I_{i}\in\mathcal{I}_{K^{*}}$, and $a_{i}\in K^{*}$. Assume that
$\mathcal{P}_{K^{*}}(I_{i})$ and $\mathcal{P}_{K^{*}}(I_{j})$ are disjoint,
for every $1\leq i\neq j\leq k$. There exists an element $a\in K^{*}$ such
that:

  1. (1)

For every $1\leq i\leq k$ and every $P\in\mathcal{P}(I_{i})$,
$\operatorname{val}_{P}(a-a_{i})\geq m_{i}$.

  2. (2)

For every $P\in\cap_{i=1}^{k}\mathcal{P}(I_{i})^{c}$,
$\operatorname{val}_{P}(a)\geq 0$.

###### Proof.

The claim follows from the Strong Approximation Theorem for $K$ via a simple
coordinate-wise argument. ∎

###  2.4 The congruence topology on ${\bf A}^{*}$

######  Definition 2.4.1.

If $P=[P_{n}]_{n}\in\mathcal{P}_{K^{*}}$ and
$I=[I_{n}]_{n}\in\mathcal{I}_{K^{*}}$, a simple coordinate-wise argument shows
that $[\operatorname{val}_{P_{n}}(I_{n})]_{n}$ is independent of the choice of
the sequences $P_{n},I_{n}$. We denote it by $\operatorname{val}_{P}(I)$.

Note that $\operatorname{val}_{P}(I)=0$ if and only if $I$ is not contained in
$P$ and that, for every $m\in N^{*}$ and $I\in\mathcal{I}_{K^{*}}$,
$\operatorname{val}_{P}(I^{m})=m\operatorname{val}_{P}(I)$.

######  Lemma 2.4.2.

Let $a\in{\bf A}^{*}$, $I,J\in\mathcal{I}_{K^{*}}$ and
$P\in\mathcal{P}_{K^{*}}$. Then:

  1. (1)

$\operatorname{val}_{P}(aI)=\operatorname{val}_{P}(a)+\operatorname{val}_{P}(I)$
and
$\operatorname{val}_{P}(IJ)=\operatorname{val}_{P}(I)+\operatorname{val}_{P}(J)$.

  2. (2)

There exists $m\in N^{*}$ such that $-m<\operatorname{val}_{Q}(a)<m$, for
every $Q\in\mathcal{P}_{K^{*}}$.

######  Notation 2.4.3.

For $I\in\mathcal{I}_{K^{*}}$, denote ${\bf I}:=\\{x\in{\bf A}^{*}\mid(\forall
P\in\mathcal{P}_{K^{*}})\
\operatorname{val}_{P}(x)\geq\operatorname{val}_{P}(I)\\}$.

The ring $O^{*}$ is not Noetherian and ideals in $\mathcal{I}_{K^{*}}$ are not
necessarily products of $N^{*}$-powers of ideals in $\mathcal{P}_{K^{*}}$.
Nevertheless, the following lemma shows that ideals are determined by their
values under valuations.

######  Lemma 2.4.4.

Let $I\in\mathcal{I}_{K^{*}}$ then:

  1. (1)

$I=\\{x\in O^{*}\mid(\forall P\in\mathcal{P}_{K^{*}})\
\operatorname{val}_{P}(x)\geq\operatorname{val}_{P}(I)\\}$. In particular:

     * •

$I={\bf I}\cap K^{*}$.

     * •

For every $I_{1},I_{2}\in\mathcal{I}_{K^{*}}$, $I_{1}\subseteq I_{2}$ if and
only if, for every $P\in\mathcal{P}_{K^{*}}$,
$\operatorname{val}_{P}(I_{1})\leq\operatorname{val}_{P}(I_{2})$.

  2. (2)

${\bf I}$ is an ideal of ${\bf O}^{*}$.

###### Proof.

All claims follow from coordinate-wise arguments. ∎

######  Lemma 2.4.5.

The set $\mathcal{C}:=\\{{\bf I}\mid I\in\mathcal{I}_{K^{*}}\\}$ is a base of
open neighborhoods of $0$ for a Hausdorff topology on ${\bf A}^{*}$ under
which ${\bf A}^{*}$ is a topological ring. Under this topology, for every
$I\in\mathcal{I}_{K^{*}}$, ${\bf I}$ is closed.

###### Proof.

The collection $\mathcal{C}$ is closed under intersections and every ${\bf
I}\in\mathcal{C}$ is an additive subgroup of ${\bf A}^{*}$. Thus, in order to
show the first claim, it is enough to show that, if $a\in{\bf A}^{*}$ and
$I\in\mathcal{I}_{K^{*}}$, then there exists $J\in\mathcal{I}_{K^{*}}$ such
that $a{\bf J}\subseteq{\bf I}$. By Lemmas 2.2.2 and 2.3.1, there exists an
ideal $J^{\prime}\in\mathcal{I}_{K^{*}}$ such that
$\mathcal{P}_{K^{*}}(J^{\prime})=\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(a)<0\\}$.
By Lemma 2.4.2, there exists $m\in N^{*}$ such that
$\operatorname{val}_{P}(a)>-m$, for every $P\in\mathcal{P}_{K^{*}}$. Denote
$J=(J^{\prime})^{m}I$. For any $x\in{\bf J}$,
$\operatorname{val}_{P}(ax)\geq\operatorname{val}_{P}(a)+m\operatorname{val}_{P}(J^{\prime})+\operatorname{val}_{P}(I)\geq\operatorname{val}_{P}(I)$,
so $a{\bf J}\subseteq{\bf I}$.

Since ${\bf I}$ is an ideal of ${\bf A}^{*}$, ${\bf I}={\bf
A}^{*}\smallsetminus\bigcup_{a\in{\bf A}^{*}\setminus{\bf I}}(a+{\bf I})$ is
closed. ∎

######  Definition 2.4.6.

We call the topology of ${\bf A}^{*}$ defined in Lemma 2.4.5 and the induced
subset topology on ${\bf O}^{*}$ the congruence topologies. We also call the
induced subset topologies on $O^{*}$ and $K^{*}$ the congruence topologies.
Since every non-zero ideal of $O^{*}$ contains an ideal in
$\mathcal{I}_{K^{*}}$, this definition of the congruence topology on $O^{*}$
on $K^{*}$ is equivalent to the standard one.

######  Lemma 2.4.7.

Let $I\in\mathcal{I}_{K^{*}}$. Then:

  1. (1)

$O^{*}$ is dense in ${\bf O}^{*}$.

  2. (2)

$K^{*}$ is dense in ${\bf A}^{*}$.

  3. (3)

$I$ is dense in ${\bf I}$

###### Proof.

The claim follows from the Strong Approximation Theorem for $K^{*}$
(Proposition 2.3.5). ∎

###  2.5 Completions of $K^{*}$ and ${\bf A}^{*}$

We refer the reader to Chapter 3 of [9] for the definitions and properties of
uniform spaces and their completions. In particular, recall that a Hausdorff
topological ring has a natural uniform structure induced by the additive
structure: if $\mathcal{U}$ is a base of open symmetric neighborhoods of 0,
then a base of entourages is given by $\mathcal{V}:=\\{V_{U}\mid
U\in\mathcal{U}\\}$, where we denote $V_{U}:=\\{(x,y)\mid x-y\in U\\}$. The
symmetry of $U$ ensures that $(x,y)\in V_{U}\iff(y,x)\in V_{U}$. Clearly,
every topological homomorphism between Hausdorff topological rings is a
uniformly continuous map.

Every Hausdorff abelian topological group has a completion. In a topological
ring, multiplication can be continuously extended to the Hausdorff completion
of the additive group (see §5 of Chapter 3 [9]), so every Hausdorff
topological ring has a completion.

######  Definition 2.5.1.

Denote the completion of ${\bf A}^{*}$ with respect to the congruence topology
by $\mathbb{K}$ and denote the closure of ${\bf O}^{*}$ in $\mathbb{K}$ by
$\mathbb{O}$. For $I\in\mathcal{I}_{K^{*}}$, denote the closure of $I$ in
$\mathbb{K}$ by $\mathbb{I}$.

######  Remark 2.5.2.

  1. (1)

Since $K^{*}$ is dense in ${\bf A}^{*}$, $\mathbb{K}$ is also the completion
of $K^{*}$ with respect to the congruence topology.

  2. (2)

Since $O^{*}$ is dense in ${\bf O}^{*}$ , $\mathbb{O}$ is also the closure of
$O^{*}$. Since a closed subspace of a complete space is complete, $\mathbb{O}$
is the completion of ${\bf O}^{*}$ and of $O^{*}$.

  3. (3)

$\\{\mathbb{I}\mid I\in\mathcal{I}_{K^{*}}\\}$ is a base of open neighborhoods
of $0$ for $\mathbb{K}$.

######  Remark 2.5.3.

The set $Z^{*}\cup\left\\{\infty\right\\}$ has a complete uniform structure
given by the base $\left\\{V_{n}\right\\}_{n\in Z^{*}}$, where
$V_{n}=\left\\{(x,y)\mid\text{either $x,y>n$ or $x=y$}\right\\}$. For every
$P\in\mathcal{P}_{K^{*}}$, the map $\operatorname{val}_{P}:{\bf
A}^{*}\rightarrow Z^{*}\cup\\{\infty\\}$ extends continuously to a map
$\operatorname{val}_{P}:\mathbb{K}\rightarrow Z^{*}\cup\\{\infty\\}$ with
similar properties. In particular, for every $I\in\mathcal{I}_{K^{*}}$,
$\mathbb{I}=\\{x\in\mathbb{K}\mid(\forall P\in\mathcal{P}_{K^{*}})\
\operatorname{val}_{P}(x)\geq\operatorname{val}_{P}(I)\\}$.

######  Lemma 2.5.4.

The natural embedding of
$i:O^{*}\rightarrow\underset{\longleftarrow}{\lim}(O^{*}/I)$ has a unique
extension to an isomorphism of the topological rings
$\mathbb{O}\cong\underset{\longleftarrow}{\lim}(O^{*}/I)$, where the inverse
limit is taken over $\mathcal{I}_{K^{*}}$ and every $O^{*}/I$ is given the
discrete topology.

###### Proof.

As an inverse limit of complete topological rings,
$\underset{\longleftarrow}{\lim}(O^{*}/I)$ is complete. Since the collection
$\\{I\mid I\in\mathcal{I}_{K^{*}}\\}$ is a base for identity neighborhoods of
$O^{*}$, $i$ is a homeomorphism onto its image $i(O^{*})$ and $i(O^{*})$ is
dense in $\underset{\longleftarrow}{\lim}(O^{*}/I)$. The claim now follows
from the universal property of completions. ∎

###  2.6 Completions of groups - general properties

Recall ([9, §3]) that a Hausdorff topological group has natural left and right
uniform structures. A Base of entourages for the left (respectively, right)
uniformity is $\\{V^{l}_{U}\mid U\in\mathcal{U}\\}$ (respectively,
$\\{V^{r}_{U}\mid U\in\mathcal{U}\\}$), where $\mathcal{U}$ is a base of
symmetric identity neighborhoods and $V^{l}_{U}:=\\{(x,y)\mid x^{-1}y\in
U\\}$, $V^{r}_{U}:=\\{(x,y)\mid xy^{-1}\in U\\}$. In general, the right
uniformity and the left uniformity do not coincide so a left Cauchy filter
(i.e. a Cauchy filter with respect to the left uniformity) is not necessarily
a right Cauchy filter. However, $\mathcal{F}$ is a left Cauchy filter if and
only if $\mathcal{F}^{-1}:=\\{F^{-1}\mid F\in\mathcal{F}\\}$ is a right Cauchy
filter, where $F^{-1}:=\\{x^{-1}\mid x\in F\\}$. Every continuous homomorphism
between topological groups is a uniformly continuous map with respect to the
left uniformities of both groups and also with respect to the right
uniformities. If $\Gamma$ is a Hausdorff topological group, the following are
equivalent:

  1. (1)

$\Gamma$ has a completion with respect to its left uniformity.

  2. (2)

$\Gamma$ has a completion with respect to its right uniformity.

  3. (3)

Every left Cauchy filter is also a right Cauchy filter.

  4. (4)

Every right Cauchy filter is also a left Cauchy filter.

Moreover, if a completion with respect to the left (respectively, right)
uniformity exists then it is also a completion with respect to the right
(respectively, left) uniformity.

######  Remark 2.6.1.

Let $\Gamma$ be a Hausdorff topological group and let $\mathcal{U}$ be the
minimal uniformity on $\Gamma$ which contains the left and right uniformities.
The topology generated by $\mathcal{U}$ is the original topology of $\Gamma$
and $\Gamma$ always has a completion with respect to $\mathcal{U}$. If the
completion with respect to the left uniformity exists, then it is equal to the
completion with respect to $\mathcal{U}$. The only reason we mention the left
and right uniformities is to be able to quote [9]. In all the cases we are
interested in, the left are right uniformities are equal.

######  Lemma 2.6.2.

Let $\Gamma$ be a topological group and let $\Lambda$ be a subgroup of
$\Gamma$. If $\Gamma$ has a base $\mathcal{U}$ of identity neighborhoods
consisting of normal subgroups of $\Lambda$, then every left Cauchy filter is
a right Cauchy filter. Thus, $\Gamma$ has a completion.

###### Proof.

Let $\mathcal{F}$ be a left Cauchy filter. There exists $F\in\mathcal{F}$ such
that $\\{x^{-1}y\mid x,y\in F\\}\subseteq\Lambda$. Fix such $F$ and fix some
$f\in F$. To show that $\mathcal{F}$ is also a right Cauchy filter, we have to
find, for every $U\in\mathcal{U}$, an element $F^{\prime}\in\mathcal{F}$ such
that $xy^{-1}\in U$, for every $x,y\in F^{\prime}$.

Let $U\in\mathcal{U}$. Since conjugation is continuous, there exists
$U^{\prime}\in\mathcal{U}$ such that $fU^{\prime}f^{-1}\subseteq U$. Since
$\mathcal{F}$ is a left Cauchy filter, there exists $F^{\prime}\in F$ such
that $\\{x^{-1}y\mid x,y\in F^{\prime}\\}\subseteq U^{\prime}$. By replacing
$F^{\prime}$ with $F\cap F^{\prime}$, we can further assume that
$F^{\prime}\subseteq F$. Let $x,y\in F^{\prime}$ and denote
$g:=f^{-1}x\in\Lambda$. Since $y^{-1}x\in U^{\prime}$ and since $U^{\prime}$
is normal in $\Lambda$,

$xy^{-1}\in
xU^{\prime}x^{-1}=fgU^{\prime}g^{-1}f^{-1}=fU^{\prime}f^{-1}\subseteq U.$

∎

The following two lemmas and corollary are well known, so their proofs are
omitted.

######  Lemma 2.6.3.

Let $\Gamma$ be a group and let $\mathcal{U}$ be a collection of subgroups of
$\Gamma$ with the following properties:

  1. (1)

$\cap_{U\in\mathcal{U}}U=1$.

  2. (2)

For every $U_{1},U_{2}\in\mathcal{U}$ there exists $U_{3}\in\mathcal{U}$ such
that $U_{3}\subseteq U_{1}\cap U_{2}$.

  3. (3)

For every $g\in\Gamma$ and every $U_{1}\in\mathcal{U}$ there exists
$U_{2}\in\mathcal{U}$ such that $gU_{2}g^{-1}\subseteq U_{1}$.

There exists a unique topology on $G$, called the topology generated by
$\mathcal{U}$, under which $G$ is a Hausdorff topological group and
$\mathcal{U}$ is a base of open neighborhoods of the identity.

######  Lemma 2.6.4.

Let $\Gamma$ and $\Lambda$ be Hausdorff topological groups. Let
$A\subseteq\Gamma$ be a dense subgroup and let $U\subseteq\Gamma$ be an open
subgroup. Let $\alpha:A\rightarrow\Lambda$ be an abstract homomorphisms and
let $\beta:U\rightarrow\Lambda$ be a continuous homomorphism. Assume that
$\alpha\restriction_{A\cap U}=\beta\restriction_{A\cap U}$. Then there is a
unique continuous homomorphism $\gamma:\Gamma\rightarrow\Lambda$ extending
both $\alpha$ and $\beta$.

######  Corollary 2.6.5.

Let $\Gamma$ and $\Lambda$ be Hausdorff topological groups. Let $A\leq\Gamma$
and $B\leq\Lambda$ be dense subgroups and let $\alpha:A\rightarrow B$ be an
isomorphism of abstract groups. Let $U\leq\Gamma$ and $V\leq\Lambda$ be open
subgroups and let $\beta:U\rightarrow V$ be an isomorphism of topological
groups. Assume that $\alpha\restriction_{A\cap U}=\beta\restriction_{A\cap
U}$. Then there exists a unique continuous isomorphism of topological groups
$\gamma:\Gamma\rightarrow\Lambda$ such that $\gamma\restriction_{A}=\alpha$
and $\gamma\restriction_{U}=\beta$.

###  2.7 The congruence completion of $G(K^{*})$

If $R$ is a Hausdorff topological ring, the topology on $R$ induces a topology
on $G(R)\subseteq Mat_{D}(R)$ in which $G(R)$ is a Hausdorff topological
group. Unless stated otherwise, we equip $G(R)$ with this topology. This
applies, in particular, to $G(K^{*})$, $G({\bf A}^{*})$, and $G(\mathbb{K})$.
We refer to these topologies as the congruence topologies.

######  Definition 2.7.1.

Let $I\in\mathcal{I}_{K^{*}}$. Denote

  1. (1)

$\pi_{I}:G(O^{*})\rightarrow G(O^{*}/I)$, $\pi_{\bf I}:G({\bf
O}^{*})\rightarrow G({\bf O}^{*}/{\bf I})$ and
$\pi_{\mathbb{I}}:G(\mathbb{O})\rightarrow G(\mathbb{O}/{\mathbb{I}})$ are the
modulo-$I$, modulo-${\bf I}$, and modulo-$\mathbb{I}$ reduction maps,
respectively.

  2. (2)

$G(O^{*})[I]:=\ker\pi_{I}$, $G({\bf O}^{*})[I]:=\ker\pi_{\bf I}$, and
$G(\mathbb{O})[I]:=\ker\pi_{\mathbb{I}}$.

A principal congruence subgroup is a subgroup of $G(K^{*})$ of the form
$G(O^{*})[I]$. A congruence subgroup is a subgroup of $G(K^{*})$ that contains
a principal congruence subgroup.

The following lemma is clear.

######  Lemma 2.7.2.

The sets $\mathcal{C}_{O^{*}}:=\\{G(O^{*})[I]\mid I\in\mathcal{I}_{K^{*}}\\}$,
$\mathcal{C}_{{\bf O}^{*}}:=\\{G({\bf O}^{*})[I]\mid
I\in\mathcal{I}_{K^{*}}\\}$, and
$\mathcal{C}_{\mathbb{O}}:=\\{G(\mathbb{O})[I]\mid I\in\mathcal{I}_{K^{*}}\\}$
are bases for open neighborhoods of the identity for the congruence topology
of $G(K^{*})$, $G({\bf A}^{*})$, and $G(\mathbb{K})$, respectively.

The standard Strong Approximation Theorem states that $G(K)$ is dense in
$G({\bf A})$ or, more precisely, that the embedding of $G(K)$ in $G({\bf A})$
uniquely extends to a topological isomorphism from the congruence completion
of $G(K)$ to $G({\bf A})$. Our next goal is to prove the analogous statement:
the embedding of $G(K^{*})$ in $G(\mathbb{K})$ uniquely extends to a
topological isomorphism from the congruence completion of $G(K^{*})$ to
$G(\mathbb{K})$.

######  Proposition 2.7.3 (Strong Approximation in $X(K^{*})$).

Let $X$ be an affine scheme over $O$. If $X(K)$ is dense in $X({\bf A})$ then
$X(K^{*})$ is dense in $X(\mathbb{K})$.

###### Proof.

Let $X$ be the zero locus of the map $x\mapsto(p_{1}(x),\ldots,p_{k}(x))$,
where $p_{1},\ldots,p_{k}\in O[x_{1},\ldots,x_{m}]$. For $x\in K^{m}$ and
$I\in\mathcal{I}_{K}$, let $\psi_{K}(x,I)$ be the first order formula saying
that $\left(x+I^{m}\right)\cap X(K)\neq\emptyset$ and let $\phi_{K}(x,I)$ be
the first order formula

$\phi_{K}(x,I):=\text{``$\left(\forall J\in\mathcal{I}_{K}\right)$
$\left(\exists y\in K^{m}\right)$ $x-y\in I^{m}$ and $p(y)\in J^{k}$"}.$

Since ${\bf O}$ is compact, the following are equivalent

  1. (1)

$\phi_{K}(x,I)$ holds in $K$.

  2. (2)

$x+(I{\bf A})^{m}\cap X({\bf A})\neq\emptyset$ (i.e., there is a point $y\in
X({\bf A})$ such that $x\equiv y\text{ (mod $I{\bf A}$)}$).

The assumption that $X(K)$ is dense in $X({\bf A})$ implies that the statement
$(\forall x,I)\left(\psi_{K}(x,I)\rightarrow\phi_{K}(x,I)\right)$ holds. By
Los’s theorem, the statement $(\forall
x,I)\left(\psi_{K^{*}}(x,I)\rightarrow\phi_{K^{*}}(x,I)\right)$ (where
$\psi_{K^{*}},\phi_{K^{*}}$ are obtained from $\psi_{K},\phi_{K}$ by replacing
$K$ by $K^{*}$) also holds.

To prove the proposition, we need to show that, for every $a\in X(\mathbb{K})$
and $I\in\mathcal{I}_{K^{*}}$, the intersection $a+\mathbb{I}^{m}\cap
X(K^{*})$ is non-empty. For every $J\in\mathcal{I}_{K^{*}}$ satisfying
$J\subseteq I$, choose a point
$b_{J}\in\left(a+\mathbb{J}^{m}\right)\cap(K^{*})^{m}$. Since $b_{J}\equiv
a\text{ (mod $\mathbb{I}$)}$, we get that $b_{J}-b_{I}\in I^{m}$. Since
$p(b_{J})\equiv p(a)\text{ (mod $\mathbb{J}$)}$, we get that $p(b_{J})\in
J^{k}$. It follows that $\phi_{K^{*}}(b_{I},I)$ holds. Therefore
$\psi_{K^{*}}(b_{I},I)$ holds, so there is a point
$c\in\left(b_{I}+I^{m}\right)\cap X(K^{*})\subseteq a+\mathbb{I}^{m}\cap
X(K^{*})$. ∎

######  Corollary 2.7.4 (Strong Approximation Theorem for $G(K^{*})$).

The natural embedding of $G(K^{*})$ in $G(\mathbb{K})$ has a unique extension
to a topological isomorphism from the congruence completion of $G(K^{*})$ to
$G(\mathbb{K})$. Similarly, The natural embedding of $G(O^{*})$ in
$G(\mathbb{O})$ has a unique extension to a topological isomorphism from the
congruence completion of $G(O^{*})$ to $G(\mathbb{O})$.

###### Proof.

Lemma 2.6.2 implies that $G(K^{*})$ has a completion with respect to the
congruence topology. The embedding of $G(K^{*})$ in $G(\mathbb{K})$ is an
embedding of topological groups with respect to the congruence topology of
$G(K^{*})$. Proposition 2.7.3 implies that $G(K^{*})$ is dense in
$G(\mathbb{K})$. Since $G(\mathbb{K})$ is complete it follows from the
universal property of completions that the embedding of $G(K^{*})$ in
$G(\mathbb{K})$ has a unique extension to a topological isomorphism from the
congruence completion of $G(K^{*})$ to $G(\mathbb{K})$. Since $G(O^{*})$ is
dense in $G(\mathbb{O})$ and $G(\mathbb{O})$ is closed and open, the second
statement follows from the first. ∎

######  Remark 2.7.5.

From now on we view $G(\mathbb{O})$ and $G(\mathbb{K})$ as the congruence
completions of $G(O^{*})$ and $G(K^{*})$, respectively.

######  Lemma 2.7.6.

The natural embedding of $G(O^{*})$ in $\underset{\longleftarrow}{\lim}\
G(O^{*}/I)$ uniquely extends to an isomorphism of the topological groups
$G(\mathbb{O})$ and $\underset{\longleftarrow}{\lim}\ G(O^{*}/I)$ where the
inverse limit is taken over $\mathcal{I}$ and every $G(O^{*}/I)$ is given the
discrete topology.

###### Proof.

The proof is similar to the proof of Lemma 2.5.4 and is omitted. ∎

###  2.8 The $\mathcal{Q}$-projection maps

We identify $G({\bf A})$ with $\prod^{\prime}_{P\in\mathcal{P}_{K}}G(K_{P})$
in the natural way. More precisely, elements of $G({\bf A})$ are identified
with sequences
$(g_{P})_{P\in\mathcal{P}_{K}}\in\prod_{P\in\mathcal{P}_{K}}G(K_{P})$ for
which $g_{p}\in G(O_{P})$, for all but finitely many $P$.

######  Definition 2.8.1.

  1. (1)

Let $\mathcal{Q}\subseteq\mathcal{P}_{K}$. For every $g=(g_{P})_{P}\in G({\bf
A})$, define $g_{\mathcal{Q}}=(g_{\mathcal{Q},P})_{P}\in G({\bf A})$ by:

$\begin{cases}g_{\mathcal{Q},P}:=g_{P}&\text{ if }P\in\mathcal{Q}\\\
g_{\mathcal{Q},P}:=1&\text{ if }P\notin\mathcal{Q}\end{cases}.$

  2. (2)

For every internal subset
$\mathcal{Q}=[\mathcal{Q}_{n}]_{n}\subseteq\mathcal{P}_{K^{*}}$ and every
$g=[g_{n}]_{n}\in G({\bf A}^{*})$, denote
$g_{\mathcal{Q}}=[(g_{n})_{\mathcal{Q}_{n}}]_{n}$. This definition does not
depend on the choice of representatives.

  3. (3)

If $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ is internal, the
$\mathcal{Q}$-projection maps is the map $\pi_{\mathcal{Q}}:G({\bf
A}^{*})\rightarrow G({\bf A}^{*})$ defined by
$\pi_{\mathcal{Q}}(g)=g_{\mathcal{Q}}$. We denote the image of
$\pi_{\mathcal{Q}}$ by $G({\bf A}^{*})_{\mathcal{Q}}$.

######  Lemma 2.8.2.

Let $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ be an internal subset.

  1. (1)

$\pi_{\mathcal{Q}}$ is a continuous homomorphism.

  2. (2)

$G({\bf A}^{*})_{\mathcal{Q}}=\ker(\pi_{\mathcal{Q}^{c}})$, where
$\mathcal{Q}^{c}:=\mathcal{P}_{K^{*}}\smallsetminus\mathcal{Q}$.

  3. (3)

The map $(\pi_{\mathcal{Q}}\times\pi_{\mathcal{Q}^{c}}):G({\bf
A}^{*})\rightarrow G({\bf A}^{*})_{\mathcal{Q}}\times G({\bf
A}^{*})_{\mathcal{Q}^{c}}$ is an isomorphism of topological groups.

###### Proof.

The proof follows from a coordinate-wise argument and is omitted. ∎

##  3 Word completions of $G(K^{*})$ and discrete $G(\mathbb{K})$-metaplectic
extensions

###  3.1 Setting for Section 3

In this section, we keep the assumptions and notations of 2.1 and make the
following additional assumptions on $G$:

  1. (2)

For every non-trivial word $w$ there exist $N_{w}\in\mathbb{N}$ such that
every $J^{\prime}\in\mathcal{I}_{K}$ contains $J\in\mathcal{I}_{K}$ for which:

    1. (2a)

$G(O)[J]\subseteq(G(O)[J^{\prime}])_{w}$ where $G(O)[J]$ is the $J$-th
congruence subgroup of $G(O)$.

    2. (2b)

For every non-zero ideals $I\subseteq J^{\prime}$ of $O$,
$\operatorname{wid}_{w}(G(O)[J^{\prime}]/G(O)[I])\leq N_{w}$.

    3. (2c)

For every $I\in\mathcal{I}_{K}$, there exists $I^{\prime}\in\mathcal{I}_{K}$
which is contained in $I$ for which

$\\{[h,g]\mid h\in G(O)[J]\text{ and }g\in G(O)[I^{\prime}]\\}\subseteq
w(G(O)[I])^{N_{w}}.$

  2. (3)

There exists $N_{c}\in\mathbb{N}$ such that:

    1. (3a)

Every element of $G(K)$ is a product of at most $N_{c}$ commutators.

    2. (3b)

For every ideal $I\in\mathcal{I}_{K}$,
$(\operatorname{gcl}_{G(K)}(G(O)[I]))^{N_{c}}=G(K)$.

###  3.2 Word completions of $G(K^{*})$ and discrete
$G(\mathbb{K})$-metaplectic extensions

The following definition is used only in Lemmas 3.2.2 and 3.2.3 below.

######  Definition 3.2.1.

Let $\Gamma$ be a group and $X$ a symmetric subset of $\Gamma$ which contains
the identity.

  1. (1)

A $(\Gamma,X)$-separating set is a subset $T$ of $\Gamma$ which contains the
identity element and such that for every distinct $t_{1},t_{2}\in T$,
$t_{1}X\cap t_{2}X=\emptyset$.

  2. (2)

A $(\Gamma,X)$-separating set is called maximal if it is not properly
contained in any other $(\Gamma,X)$-separating set.

  3. (3)

A $(\Gamma,X)$-covering set is a subset $T$ of $\Gamma$ which contains the
identity element and such that $XT=\Gamma$. Note that is $T$ is a maximal
$(\Gamma,X)$-separating set then $T$ is a $(\Gamma,X^{2})$-covering set

######  Lemma 3.2.2.

Let $t\in\mathbb{N}$, $\Delta\lhd\Lambda$ be groups and $X\subseteq\Lambda$ be
a symmetric subset which contains the identity such that $\langle
X\rangle=X\Delta$. If $T$ is a maximal $(\Delta,\Delta\cap X)$-separating set
of size $t$ then $\Delta\subseteq X^{2\cdot 6^{t}}$.

###### Proof.

We can assume that $T\neq\\{\operatorname{id}\\}$. Denote $Y:=X^{2}$. Then
$Y\supseteq X$, $\langle Y\rangle=Y\Delta$ and $T$ is a
$(\Delta,Y\cap\Delta)$-covering set. By induction on the size of a covering
set, it is enough to prove that there exists a proper subset $T^{\prime}$ of
$T$ which is a $(\Delta,Y^{6}\cap\Delta)$-covering set. We divide the proof
into cases.

  1. (1)

If $Y^{2}\supseteq\Delta$ then $T^{\prime}=\\{1\\}$ is an
$(\Delta,Y^{2}\cap\Delta)$-covering set.

  2. (2)

Assume that $(Y^{4}\cap\Delta)\setminus Y\neq\emptyset$ and choose
$g\in(Y^{4}\cap\Delta)\setminus Y$. Since $T$ is an a
$(\Delta,Y\cap\Delta)$-covering set and $g\notin Y$, there exists $1\neq t\in
T$ such that $g\in tY$. Thus, $t\in Y^{5}$, $tY\subseteq Y^{6}$ and
$T^{\prime}=T\setminus\\{t\\}$ is an $(\Delta,Y^{6}\cap\Delta)$-covering set.

  3. (3)

Assume that $Y\cap\Delta\neq\Delta$. If $Y^{2}\supseteq\Delta$ we are back in
case (1) so we can assume that $Y^{2}\not\supseteq\Delta$. Since $\langle
Y\rangle=Y\Delta$, $Y^{3}\supsetneq Y^{2}$ and there are $g_{1}\in Y$ and
$g_{2}\in\Delta$ such that $g_{1}g_{2}\in Y^{3}\setminus Y^{2}$. Thus,
$g_{2}\in(Y^{4}\cap\Delta)\setminus Y$ and we are back at step (2).

∎

######  Lemma 3.2.3.

Let $r\in\mathbb{N}$, $\Gamma$ a group, $w$ a word and $\mathcal{N}$ a non-
principal ultrafilter on $\mathbb{N}$. Assume that for every $n$,
$\Delta_{n}\lhd\Lambda_{n}$ are subgroups of $\Gamma$ such that
$\Delta_{n}\subseteq(\Lambda_{n})_{w}$ and
$\operatorname{wid}_{w}(\Lambda_{n}/\Delta_{n})\leq r$. Let $\Gamma^{*}$ be
the ultrapower of $\Gamma$ over $\mathcal{N}$ and $\Lambda^{*}$ and
$\Delta^{*}$ be the ultraproducts of the $\Lambda_{n}$s and $\Delta_{n}$s over
$\mathcal{N}$. Then $[\Delta^{*}:(\Lambda^{*})_{w}\cap\Delta^{*}]<\infty$ if
and only there exists $s\in\mathbb{N}$ such that

$\\{n\mid\operatorname{wid}_{w}(\Lambda_{n})\leq s\\}\in\mathcal{N}.$ (10)

###### Proof.

By assumption, for every $n$,
$\operatorname{wid}_{w}(\Lambda_{n}/\Delta_{n})\leq r$. Thus, there exists $s$
for which equation (10) holds if and only if there exists $s$ for which the
following Equation holds

$\\{n\mid\Delta_{n}\subseteq w(\Lambda_{n})^{s}\\}\in\mathcal{N}.$ (11)

Thus it is enough to prove that
$[\Delta^{*}:(\Lambda^{*})_{w}\cap\Delta^{*}]<\infty$ if and only there exists
$s\in\mathbb{N}$ for which Equation (11) holds.

The “if” is clear, if Equation (11) holds, then $\Delta^{*}\subseteq
w(\Lambda^{*})^{s}$. For the “only if” part assume that there is no $s$ for
which Equation (11) holds. For every $n$,

  1. (a)

Let $m_{n}\in\mathbb{N}\cup\\{\infty\\}$ be

$\sup\\{m\in\mathbb{N}\mid\text{There exists a $(\Delta_{n},\Delta_{n}\cap
w(\Lambda_{n})^{r+m})$-separating set of size at least $m$}\\}.$

  2. (b)

For every $n\in\mathbb{N}$ for which $m_{n}<\infty$, let $X_{n}$ be a
$(\Delta_{n},\Delta_{n}\cap w(\Lambda_{n})^{r+m_{n}})$-separating set of size
$m_{n}+r$.

  3. (c)

For every $n\in\mathbb{N}$ for which $m_{n}=\infty$, let $X_{n}$ be a
$(\Delta_{n},\Delta_{n}\cap w(\Lambda_{n})^{r+n})$-separating set of size
$r+n$.

If $m_{n}$ is finite, say $m_{n}\leq s$ for some $s\in\mathbb{N}$, then every
maximal $(\Delta_{n},\Delta_{n}\cap w(\Gamma_{n})^{r+m_{n}+1})$-separating set
has at most $r+m_{n}$ elements so Lemma 3.2.2 implies that

$w(\Gamma_{n})^{2(r+s+1)6^{r+s}}\supseteq(w(\Gamma_{n})^{r+m_{n}+1})^{2\cdot
6^{r+m_{n}}}\supseteq\Delta.$

Hence, the assumption implies that for every $s\in\mathbb{N}$,

$\\{n\mid m_{n}\leq s\\}\notin\mathcal{N}.$ (12)

By considering the cases $m_{n}<\infty$ and $m_{n}=\infty$ separately, we get
that for every $s\in\mathbb{N}$,

$\\{n\geq s\mid|X_{n}|\leq s+r\\}\subseteq\\{n\mid m_{n}\leq
s\\}\notin\mathcal{N}$

so

$\\{n\mid|X_{n}|\leq s+r\\}\notin\mathcal{N}.$

It follows that the ultraproduct $X^{*}$ of the $X_{n}$ over $\mathcal{N}$ is
infinite (in fact, $|X^{*}|\geq 2^{\aleph_{0}}$).

Equation (12) and the definition of the $X_{n}$s imply that $X^{*}$ is a
$(\Delta^{*},\Delta^{*}\cap(\Lambda^{*})_{w})$-separating set. Hence,

$[\Delta^{*}:\Delta^{*}\cap(\Lambda^{*})_{w}]\geq|X^{*}|=\infty.$

∎

######  Notation 3.2.4.

  1. (1)

To shorten notations, denote $\Gamma^{*}:=G(O^{*})$,
$\overline{\Gamma^{*}}:=G(\mathbb{O})$ and, for every
$I\in\mathcal{I}_{K^{*}}$, $\Gamma^{*}[I]:=G(O^{*})[I]$ and
$\overline{\Gamma^{*}}[I]:=G(\mathbb{O})[I]$.

  2. (2)

Let $w$ be a word and $J^{\prime}=[J_{n}^{\prime}]_{n}\in\mathcal{I}_{K^{*}}$.
Let $N_{w}\in\mathbb{N}$ and for every $n$, $J_{n}\in\mathcal{I}_{K}$ be as in
Item 2 of Setting 3.1, where each $J_{n}$ corresponds to $J^{\prime}_{n}$.
Denote $J:=[J_{n}]_{n}\in\mathcal{I}_{K^{*}}$.

######  Lemma 3.2.5.

Under Notation 3.2.4, the following hold:

  1. (1)

For every $I\in\mathcal{I}_{K^{*}}$, $\Gamma^{*}[J]\subseteq
w(\Gamma^{*}[J^{\prime}])^{N_{w}}\Gamma^{*}[I]$. In particular, for every
$I_{1},I_{2}\in\mathcal{I}$ for which $I_{1}\subseteq I_{2}\subseteq J$, the
diagram

${\frac{\Gamma^{*}[J]}{(\Gamma^{*}[J^{\prime}])_{w}\cap\Gamma^{*}[I_{1}]}}$${\frac{\Gamma^{*}[J]}{(\Gamma^{*}[J^{\prime}])_{w}\cap\Gamma^{*}[J]}\times\frac{\Gamma^{*}[J]}{\Gamma^{*}[I_{1}]}}$${\frac{\Gamma^{*}[J]}{(\Gamma^{*}[J^{\prime}])_{w}\cap\Gamma^{*}[I_{2}]}}$${\frac{\Gamma^{*}[J]}{(\Gamma^{*}[J^{\prime}])_{w}\cap\Gamma^{*}[J]}\times\frac{\Gamma^{*}[J]}{\Gamma^{*}[I_{2}]}}$

is commutative and the horizontal arrows are isomorphisms.

  2. (2)

For every $I\in\mathcal{I}_{K^{*}}$, there exists
$I^{\prime}\in\mathcal{I}_{K^{*}}$ such that $I^{\prime}\subseteq I$ and

$\\{[h,g]\mid h\in\Gamma^{*}[J]\text{ and
}g\in\Gamma^{*}[I^{\prime}]\\}\subseteq(\Gamma^{*}[I])^{N_{w}}.$

  3. (3)

The following conditions are equivalent

     * •

There exists $s\in\mathbb{N}$ such that
$\\{n\mid\operatorname{wid}_{w}(G(O)[J_{n}^{\prime}])\leq s\\}\in\mathcal{N}$.

     * •

$[\Gamma^{*}[J]:\Gamma^{*}[J]\cap(\Gamma^{*}[J^{\prime}])_{w}]<\infty$.

###### Proof.

Item 2 follows from item 2c of Setting 3.1 and a coordinate-wise argument .
Items 2a and 2b of of Setting 3.1 and a coordinate-wise argument imply that
for every $I\subseteq J$, $w(\Gamma^{*}[J^{\prime}])^{N_{w}}\cap\Gamma^{*}[J]$
projects onto $\Gamma^{*}[J]/\Gamma^{*}[I]$ so Item 1 holds. Item 3 follows
from Item 2 of Setting 3.1 and Lemma 3.2.3 applied to $\Gamma:=G(O)$,
$\Lambda_{n}:=G(O)[J_{n}^{\prime}]$ and $\Delta_{n}:=G(O)[J_{n}]$. Indeed,
there are natural identifications of the ultrapower of $G(O)$ over
$\mathcal{N}$ with $\Gamma^{*}=G(O^{*})$ and of the ultraproducts
$\Lambda^{*}$ and $\Delta^{*}$ with
$\Gamma^{*}[J^{\prime}]=G(O^{*})[J^{\prime}]$ and $\Gamma^{*}[J]=G(O^{*})[J]$,
respectively. ∎

######  Definition 3.2.6.

Under Notation 3.2.4, for every $I\in\mathcal{I}_{K^{*}}$, denote
$\Gamma^{*}_{w,J^{\prime}}[I]:=(\Gamma^{*}[J^{\prime}])_{w}\cap\Gamma^{*}[I]$.
Every $\Gamma^{*}_{w,J^{\prime}}[I]$ is normal in $\Gamma^{*}$ and the
collection $\mathcal{W}(w,J^{\prime}):=\\{\Gamma^{*}_{w,J^{\prime}}[I]\mid
I\in\mathcal{I}_{K^{*}}\\}$ satisfies the assumptions of Lemma 2.6.3. Thus,
$\mathcal{W}(w,J^{\prime})$ is a base for identity neighborhoods for a
Hausdorff topology in $G(O^{*})$. We call this topology the
$(w,J^{\prime})$-topology.

######  Definition 3.2.7.

Under Notation 3.2.4,

  1. (1)

Lemma 2.6.2 implies that $\Gamma^{*}$ has a completion
${\widehat{\Gamma^{*}}}^{(w,J^{\prime})}$ with respect to the
$(w,J^{\prime})$-topology. When no confusion occurs, we simply write
$\widehat{\Gamma^{*}}$ instead of ${\widehat{\Gamma^{*}}}^{(w,J^{\prime})}$.

  2. (2)

Since the $(w,J^{\prime})$-topology is finer than the congruence topology, the
identity map of $\Gamma^{*}$ uniquely extend to a continuous homomorphism
$\rho_{w,J^{\prime}}:\widehat{\Gamma^{*}}\rightarrow\overline{\Gamma^{*}}$.
When no confusion occurs, we simply write $\rho$ instead of
$\rho_{w,J^{\prime}}$.

  3. (3)

For every $I\in\mathcal{I}$, let $\widehat{\Gamma^{*}}[I]$ be the closure of
$\Gamma^{*}[I]$ in $\widehat{\Gamma^{*}}$.

######  Lemma 3.2.8.

Under Notation 3.2.4, denote
$\widehat{\Gamma^{*}}:={\widehat{\Gamma^{*}}}^{(w,J^{\prime})}$ and
$\rho:=\rho_{w,J^{\prime}}$. Then:

  1. (1)

$\ker\rho\restriction_{\widehat{\Gamma^{*}}[J]}=\ker\rho\simeq\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]$
where $\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]$ is given the discrete
topology.

  2. (2)

The sequence

$1\longrightarrow\ker(\rho\restriction_{\widehat{\Gamma^{*}}[J]})\longrightarrow\widehat{\Gamma^{*}}[J]\overset{\rho}{\longrightarrow}\overline{\Gamma^{*}}[J]\longrightarrow
1$ (13)

is a topological central extension that has a continuous splitting.

###### Proof.

Let $I\in\mathcal{I}_{K^{*}}$ and let $T\subseteq\Gamma^{*}$ be a set of coset
representatives for $\Gamma^{*}/\Gamma^{*}[J]$. Since $\Gamma^{*}[I]$ is an
open subgroup of $\Gamma^{*}$ in both topologies,
$\overline{\Gamma^{*}}=\underset{{g\in T}}{\sqcup}g\overline{\Gamma^{*}}[J]$
and $\widehat{\Gamma^{*}}=\underset{g\in T}{\sqcup}g\widehat{\Gamma^{*}}[J]$
where $\sqcup$ is disjoint union. Hence, for every $I\in\mathcal{I}_{K^{*}}$,
$\ker\rho\subseteq\widehat{\Gamma^{*}}[I]$. Thus,
$\ker\rho\restriction_{\widehat{\Gamma^{*}}[J]}=\ker\rho$.

The map $g\mapsto(g\Gamma^{*}[I])_{I\in\mathcal{I}}$ uniquely extends to a
topological isomorphism

$\psi_{c}:\overline{\Gamma^{*}}[J]\rightarrow\underset{\longleftarrow}{\lim}\
\Gamma^{*}[J]/\Gamma^{*}[I],$

where the inverse limit is over $\\{I\in\mathcal{I}_{K^{*}}\mid I\subseteq
J\\}$ and every $\Gamma^{*}[J]/\Gamma^{*}[I]$ is given the discrete topology.
Similarly, the map $g\mapsto(g\Gamma^{*}_{w,J^{\prime}}[I])_{I\in\mathcal{I}}$
uniquely extends to a topological isomorphism

$\psi_{w}:\widehat{\Gamma^{*}}[J]\rightarrow\underset{\longleftarrow}{\lim}\
\Gamma^{*}[J]/\Gamma_{w,J^{\prime}}^{*}[I],$

where the inverse limit is over $\\{I\in\mathcal{I}_{K^{*}}\mid I\subseteq
J\\}$ and every $\Gamma^{*}[J]/\Gamma_{w,J^{\prime}}^{*}[I]$ is given the
discrete topology. Lemma 3.2.5 implies that exists an isomorphism of
topological groups

$\psi_{1}:\underset{\longleftarrow}{\lim}\
\Gamma^{*}[J]/\Gamma_{w,J^{\prime}}^{*}[I]\rightarrow\underset{\longleftarrow}{\lim}\
\left((\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J])\times(\Gamma^{*}[J]/\Gamma^{*}[I])\right).$

There is a natural isomorphism of topological groups

$\psi_{2}:\underset{\longleftarrow}{\lim}\
\left((\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J])\times(\Gamma^{*}[J]/\Gamma^{*}[I])\right)\rightarrow(\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J])\times\underset{\longleftarrow}{\lim}\
(\Gamma^{*}[J]/\Gamma^{*}[I]).$

Since $\psi_{w}$, $\psi_{1}$, $\psi_{2}$ and $(id,\psi_{c})$ are isomorphisms,

$(\operatorname{id},(\psi_{c})^{-1})\circ\psi_{2}\circ\psi_{1}\circ\psi_{w}:\widehat{\Gamma^{*}}[J]\rightarrow(\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J])\times\overline{\Gamma^{*}}[J]$
(14)

is an isomorphism of topological groups, where
$\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]$ is given the discrete topology.
Moreover, $\rho\restriction_{\widehat{\Gamma^{*}}[J]}$ is equal to the
composition of
$(\operatorname{id},(\psi_{c})^{-1})\circ\psi_{2}\circ\psi_{1}\circ\psi_{w}$
and projection to the right coordinate. In particular,
$\ker(\rho\restriction_{\widehat{\Gamma^{*}}[J]})\simeq\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]$
and the sequence (13) has a continuous splitting.

In order to prove that $\ker\rho$ is central in $\widehat{\Gamma^{*}}[J]$, it
is enough to prove that $\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]$ is
abelian. Item 2 of Lemma 3.2.5 applied to $I=J$ implies that there exists
$J^{\prime\prime}\in\mathcal{I}_{K^{*}}$ such that
$[\Gamma^{*}[J],\Gamma^{*}[J^{\prime\prime}]]\subseteq(\Gamma^{*}[J^{\prime}])_{w}$
so
$(\Gamma^{*}[J^{\prime\prime}]\Gamma^{*}_{w,J^{\prime}}[J])/\Gamma^{*}_{w,J^{\prime}}[J]$
is abelian. Applying Item 1 of Lemma 3.2.5 with respect to
$I_{1}=J^{\prime\prime}$ we get that the map
$\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J^{\prime\prime}]\rightarrow(\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J])\times(\Gamma^{*}[J]/\Gamma^{*}[J^{\prime\prime}])$
is an isomorphism so

$\Gamma^{*}[J]/\Gamma^{*}_{w,J^{\prime}}[J]=(\Gamma^{*}[J^{\prime\prime}]\Gamma^{*}_{w,J^{\prime}}[J])/\Gamma^{*}_{w,J^{\prime}}[J]$

is abelian. ∎

We already know that the congruence topology of $\Gamma^{*}=G(O^{*})$ extends
to a topology of $G(K^{*})$. The next lemma show that the
$(w,J^{\prime})$-topology also extends to $G(K^{*})$.

######  Proposition 3.2.9.

Under Notation 3.2.4, the collection
$\mathcal{W}=\mathcal{W}(w,J^{\prime}):=\\{\Gamma_{w,J^{\prime}}^{*}[I]\mid
I\in\mathcal{I}_{K^{*}}\\}$ satisfies the assumptions of Lemma 2.6.3.

###### Proof.

It is clear that items (1) and (2) of Lemma 2.6.3 hold. We claim that, for
every $I\in\mathcal{I}_{K^{*}}$, every $g\in G(K^{*})$, and every
$h\in\Gamma^{*}[J]$, there exists $I_{3}\in\mathcal{I}_{K^{*}}$ such that

$ghg^{-1}(\Gamma^{*}_{w,J^{\prime}}[I_{3}])gh^{-1}g^{-1}\subseteq\Gamma^{*}_{w,J^{\prime}}[I].$

Let $I\in\mathcal{I}_{K^{*}}$, $g\in G(K^{*})$ and $h\in\Gamma^{*}[J]$. We can
assume that $I\subseteq J^{\prime}$, so
$(\Gamma^{*}[I])_{w}\subseteq\Gamma^{*}_{w,J^{\prime}}[I]$. Choose
$I_{1},I_{2},I_{3}\in\mathcal{I}_{K^{*}}$ such that

  * •

$g\Gamma^{*}[I_{1}]g^{-1}\subseteq\Gamma^{*}[I]$.

  * •

$(\Gamma^{*}[I_{1}])_{w}\supseteq[\Gamma^{*}[J],\Gamma^{*}[I_{2}]]$ ($I_{2}$
exists by Lemma 3.2.52).

  * •

$g^{-1}\Gamma^{*}[I_{3}]g\subseteq\Gamma^{*}[I_{2}]$ and $I_{3}\subseteq I$.

Let $f\in\Gamma_{w,J^{\prime}}^{*}[I_{3}]$. Then:

$g(hg^{-1}fgh^{-1})g^{-1}=g[h,g^{-1}fg]g^{-1}f\in
g((\Gamma^{*}[I_{1}])_{w})g^{-1}f\subseteq(\Gamma^{*}[I])_{w}f\subseteq\Gamma_{w,J^{\prime}}^{*}[I].$

The proof of the claim is complete.

We now prove Item (3) of Lemma 2.6.3. Let $g\in G(K^{*})$ and let
$I\in\mathcal{I}_{K^{*}}$ satisfy $I\subseteq J^{\prime}$. We have to show
that there exists $I^{\prime}\in\mathcal{I}_{K^{*}}$ such that
$g\Gamma^{*}_{w,J^{\prime}}[I^{\prime}]g^{-1}\subseteq\Gamma^{*}_{w,J^{\prime}}[I]$.
Item 3 of Lemma 3.2.5 implies that there are $g_{1},\ldots,g_{N}\in G(K^{*})$
and $h_{1},\ldots,h_{N}\in\Gamma^{*}[J]$ such that
$g=(g_{1}h_{1}g_{1}^{-1})\ldots(g_{N}h_{N}g_{N}^{-1})$. The claim proved above
shows that there exists a sequence $I_{N}\subseteq\cdots\subseteq
I_{1}\subseteq I_{0}:=I$ such that, for every $1\leq m\leq N$,
$I_{m}\in\mathcal{I}_{K^{*}}$ and
$(g_{m}h_{m}g_{m}^{-1})\Gamma^{*}_{w}[I_{m}](g_{m}h_{m}^{-1}g_{m}^{-1})\subseteq\Gamma^{*}_{w}[I_{m-1}]$.
Thus, $g\Gamma^{*}_{w}[I_{N}]g^{-1}\subseteq\Gamma^{*}_{w,J^{\prime}}[I]$. ∎

######  Definition 3.2.10.

Under Notation 3.2.4,

  1. (1)

By Proposition 3.2.9, $\mathcal{W}(w,J^{\prime})$ is a base of identity
neighborhoods for a Hausdorff topology on $G(K^{*})$ which we call the
$(w,J^{\prime})$-word topology.

  2. (2)

Lemma 2.6.2 implies that $G(K^{*})$ has a completion
$\widehat{G(K^{*})}^{\mathcal{W}(w,J^{\prime})}$ with respect to the
$(w,J^{\prime})$-word topology. When no confusion can occur, we will write
$\widehat{G(K^{*})}$ instead of
$\widehat{G(K^{*})}^{\mathcal{W}(w,J^{\prime})}$.

  3. (3)

Recall that we identify $\overline{\Gamma^{*}}=G(\mathbb{O})$ with the
congruence completion of $\Gamma^{*}$. Since the $(w,J^{\prime})$-topology is
finer then the congruence topology, the map
$\rho_{W,J^{\prime}}:\widehat{\Gamma^{*}}\rightarrow\overline{\Gamma^{*}}$
uniquely extend to a continuous homomorphism
$\rho_{w,J^{\prime}}:\widehat{G(K^{*})}\rightarrow G(\mathbb{K})$.

######  Lemma 3.2.11.

Under Notation 3.2.4,

  1. (1)

Every element of $G(K^{*})$ is a product of $N_{c}$ commutators. In
particular, $G(K^{*})$ is perfect.

  2. (2)

For every $I\in\mathcal{I}_{K^{*}}$,
$(\operatorname{gcl}_{G(K^{*})}(\Gamma^{*}[I]))^{N_{c}}=G(K^{*})$.

###### Proof.

The proof follows from Item 3 of Setting 3.1 and a coordinate-wise argument
and is omitted. ∎

######  Proposition 3.2.12.

Under Notation 3.2.4, denote $\mathcal{W}:=\mathcal{W}(w,J^{\prime})$,
$\widehat{G(K^{*})}:=\widehat{G(K^{*})}^{\mathcal{W}(w,J^{\prime})}$, and
$\rho:=\rho_{w,J^{\prime}}$. Then:

  1. (1)

The sequence

$1\longrightarrow\ker\rho\longrightarrow\widehat{G(K^{*})}\overset{\rho}{\longrightarrow}G(\mathbb{K})\longrightarrow
1$ (15)

is the a topological central extension and the following hold:

    1. (1a)

$\ker\rho=\ker(\rho\restriction_{\widehat{\Gamma^{*}}})$. In particular,
$\ker\rho$ is discrete.

    2. (1b)

$\rho$ has a section over $G(K^{*})$.

    3. (1c)

$\rho$ has a continuous section over $G(\mathbb{O})[J]$.

  2. (2)

If the sequence (15) has a continuous splitting then:

    1. (2a)

$\ker\rho=1$.

    2. (2b)

There exists $s\in\mathbb{N}$ such that
$\\{n\mid\operatorname{wid}_{w}(\Gamma[J_{n}])\leq s\\}\in\mathcal{N}.$

###### Proof.

(1) Since $\Gamma^{*}=G(O^{*})$ is open in $G(K^{*})$ in the congruence
topology and in the $\mathcal{W}$-topolgy, Item (1a) holds. The restriction of
$\rho$ to $G(K^{*})$ is the identity map so Item (1b) holds. Since
$G(\mathbb{O})[J]=\overline{\Gamma^{*}}[J]$, Item (1c) follows from Lemma
3.2.8.

We now prove that $\ker\rho$ is central in $\widehat{G(K^{*})}$. By the choice
of $J$, $\ker\rho$ is central in
$\widehat{\Gamma^{*}}[J]=\widehat{G(O^{*})}[J]$. Since $\ker\rho$ is a normal
subgroup, it is also central in the normal subgroup generated by all the
conjugates of $\widehat{G(O^{*})}[J]$. Item (2) of Lemma 3.2.11 implies that
the group generated by the all the conjugates of $G(O^{*})[J]$ contains
$G(K^{*})$ so $G(K^{*})$ centralizes $\ker\rho$. Since the centralizer of a
subset is closed and $G(K^{*})$ is dense in $\widehat{G(K^{*})}$, $\ker\rho$
is central in $\widehat{G(K^{*})}$.

In order to finish the proof of Item (1), it is enough to show that the
induced map $\bar{\rho}:\widehat{G(K^{*})}/\ker\rho\rightarrow G(\mathbb{K})$
is an isomorphism of topological groups. Clearly, the restriction of
$\bar{\rho}$ to $(G(K^{*})\ker\rho)/\ker\rho$ is an isomorphism of abstract
groups onto $G(K^{*})$ and by the choice of $J$, the restriction of
$\bar{\rho}$ to $(\widehat{G(O^{*})}[J]\ker\rho)/\ker\rho$ is an isomorphism
of topological groups onto $G(\mathbb{O})[J]$. Thus, Coroallary 2.6.5 implies
that $\bar{\rho}:\widehat{G(K^{*})}/\ker\rho\rightarrow G(\mathbb{K})$ is an
isomorphism of topological groups.

We now prove item (2a). Assume that
$\mu:G(\mathbb{K})\rightarrow\widehat{G(K^{*})}$ is a continuous section of
$\rho$. Then $\mu\circ\rho:G(\mathbb{K})\rightarrow\ker\rho$ is a continuous
epimorphism. Lemma 3.2.11 implies that $G(K^{*})$ is perfect. Since $\ker\rho$
is abelian, $G(K^{*})\subseteq\ker(\mu\circ\rho)$. Since $\ker(\mu\circ\rho)$
is a closed subgroup of $\widehat{G(K^{*})}$ and $G(K^{*})$ is dense in
$\widehat{G(K^{*})}$, $\ker\rho=\widehat{G(K^{*})}$. Since $\mu\circ\rho$ is
an epimorphism, $\ker\rho=1$.

Item (2b) follows from Item 3 of Lemma 3.2.5, Item 1 of Lemma 3.2.8, Item (1a)
and Item (2a).

∎

######  Definition 3.2.13.

A discrete $G(K^{*})$-metaplectic extension is a topological central extension

$1\longrightarrow D\longrightarrow
E\overset{\rho}{\longrightarrow}G(\mathbb{K})\longrightarrow 1$

satisfying the following

  1. (1)

$\rho$ has a section $\mu$ over $G(K^{*})$ such that $\mu(G(K^{*}))$ is dense
in $E$.

  2. (2)

There exists $J\in\mathcal{I}$ such that $\rho$ has a continuous section over
$G(\mathbb{O})[J]$.

  3. (3)

$D$ discrete.

######  Example 3.2.14.

By Proposition 3.2.12, the short exact sequence (15) is a discrete
$G(K^{*})$-metaplectic extension.

######  Corollary 3.2.15.

Under Notation 3.2.4, if every discrete $G(K^{*})$-metaplectic extension has a
continuous splitting, then there exists $s\in\mathbb{N}$ such that, for every
non-zero ideal $J^{\prime}\lhd O$,
$\operatorname{wid}_{w}(G(O)[J^{\prime}])\leq s$.

###### Proof.

Assume that the conclusion does not hold. For every $n$ choose a non-zero
ideal $J_{n}^{\prime}\lhd O$ such that
$\operatorname{wid}_{w}(G(O)[J_{n}^{\prime}])\geq n$ and let
$J^{\prime}=[J^{\prime}_{n}]$. Proposition 3.2.12 implies that the sequence
(15) is a discrete $G(K^{*})$-metaplectic sequence. By assumption, (15) has a
continuous splitting. By 3.2.12 again,
$\\{n\mid\operatorname{wid}_{w}(G(O)[J_{n}])\leq s\\}\in\mathcal{N}$, for some
$s$. But the last set is finite and $\mathcal{N}$ is non-principal, a
contradiction. ∎

##  4 Finite conjugacy width

In this section, we prove Theorem 1.1.1, postponing some results to the next
section.

###  4.1 Setting for Section 4

In this section, we continue to follow the notations in 2.1, but we restrict
to the case $K=\mathbb{Q}$ and $S=\left\\{2,\infty\right\\}$ (so,
$O=\mathbb{Z}[1/2]$).

We fix $n\geq 10$ and denote $f_{a}:=\sum_{i=1}^{2n}x_{i}^{2}$ and
$f_{s}:=\sum_{i=1}^{n}(x_{2i}^{2}-x^{2}_{2i-1})$. By Appendix A, the groups
$G=\operatorname{Spin}_{f_{a}},\operatorname{Spin}_{f_{s}}$ satisfy the
assumptions in §3.1, so all the results of §3 hold. Note also that $f_{s}$ is
properly equivalent over $O$ to the form $\sum_{i=1}^{n}x_{2i}x_{2i-1}$ and
that $f_{a}$ and $f_{s}$ are properly equivalent over ${\bf O}$.

Let $f$ be either $f_{a}$ or $f_{s}$. We recall the definition of the Clifford
algebra of $f$ over $O$; for more details, see [34, Chapter V]. The Clifford
algebra $\operatorname{Cliff}_{f}(O)$ is the quotient of the tensor algebra
$\oplus_{k\geq 0}(O^{2n})^{\otimes k}$ by the ideal generated by the set
$\\{a\otimes a-f(a)\mid a\in O^{2n}\\}$. $\operatorname{Cliff}_{f}(O)$ is a
unital associative $O$-algebra and satisfies the following properties:

  * •

The map $O^{2n}\rightarrow\operatorname{Cliff}_{f}(O)$ is injective.

  * •

$\operatorname{Cliff}_{f}(O)$ is generated as a unital $O$-algebra by
$O^{2n}$.

  * •

For every $a,b\in O^{2n}$, $ab+ba=f(a+b)-f(a)-f(b)$.

  * •

Let $e_{1},\ldots,e_{2n}$ be the standard basis of $O^{2n}$ and, for every
$T\subseteq\\{1,\ldots,2n\\}$, denote $e_{T}:=e_{i_{1}}\cdots
e_{i_{k}}\in\operatorname{Cliff}_{f}(O)$ where $T=\\{i_{1},\ldots,i_{k}\\}$
and $i_{1}<i_{2}<\cdots<i_{k}$ (in particular, $e_{\emptyset}:=1$). Then
$\operatorname{Cliff}_{f}(O)$ is a free $O$-module of dimension $4^{n}$ and
$\\{e_{T}\mid T\subseteq\\{1,\ldots,2n\\}\\}$ is a free basis for
$\operatorname{Cliff}_{f}(O)$.

  * •

There exists a unique $O$-linear map
${}^{\prime}:\operatorname{Cliff}_{f}(O)\rightarrow\operatorname{Cliff}_{f}(O)$
such that, for every $T=\\{i_{1},\ldots,i_{k}\\}\subseteq\\{1,\ldots,2n\\}$
with $i_{1}<\ldots<i_{k}$, $e^{\prime}_{T}=e_{i_{k}}\cdots e_{i_{i}}$. The map
′ is called the canonical involution of $\operatorname{Cliff}_{f}(O)$.

  * •

For every $O$-algebra $R$, let
$\operatorname{Cliff}_{f}(R):=\operatorname{Cliff}_{f}(O)\otimes_{O}R$. In
this way, we regard $\operatorname{Cliff}_{f}$ as an affine $O$-ring scheme.
It is clear that
${}^{\prime}:\operatorname{Cliff}_{f}(O)\rightarrow\operatorname{Cliff}_{f}(O)$
extends to an automorphism of the $O$-scheme $\operatorname{Cliff}_{f}$. For
every $O$-algebra $R$, let $\operatorname{Cliff}_{f,0}(R)$ be the subalgebra
of $\operatorname{Cliff}_{f}(R)$ spanned by $\\{e_{T}\mid|T|\text{ is
even}\\}$. The spin group of $f$, denoted by $\operatorname{Spin}_{f}$, is the
$O$-group scheme defined by

$\operatorname{Spin}_{f}(R):=\left\\{x\in\operatorname{Cliff}_{f,0}(R)\mid
xx^{\prime}=1,x\operatorname{Span}_{R}\\{e_{1},\ldots,e_{2n}\\}x^{\prime}=\operatorname{Span}_{R}\\{e_{1},\ldots,e_{2n}\\}\right\\},$

for every $O$-algebra $R$.

  * •

For every $g\in\operatorname{Spin}_{f}(R)$, let $\tau_{g}:R^{n}\rightarrow
R^{n}$ be the map defined by $\tau_{g}(a):=gag^{\prime}$, for every $a\in
R^{n}$. Then $\tau_{g}\in\operatorname{SO}_{f}(R)$ and the map
$g\mapsto\tau_{g}$ is a homomorphism from $\operatorname{Spin}_{f}(R)$ to
$\operatorname{SO}_{f}(R)$.

Since the quadratic forms $f_{a}$ and $f_{s}$ are equivalent over ${\bf O}$,
there is an ${\bf O}$-linear isometry $\psi$ from the quadratic space $({\bf
O}^{2n},f_{a})$ to $({\bf O}^{2n},f_{s})$. The ultrapower of $\psi$ is an
isometry from $(({\bf O}^{*})^{2n},f_{a})$ to $(({\bf O}^{*})^{2n},f_{s})$.
Abusing notations, we denote this ultrapower by $\psi$. Since ${\bf O}^{*}$ is
a subring of $\mathbb{K}$, $\psi$ extends to an isometry from
$(\mathbb{K}^{2n},f_{a})$ to $(\mathbb{K}^{2n},f_{s})$. Abusing notations
again, we denote this extension by $\psi$. Note that
$\psi:\mathbb{K}^{2n}\rightarrow\mathbb{K}^{2n}$ is a homeomorphism if
$\mathbb{K}^{2n}$ is given the congruence topology and that, for every
$I\in\mathcal{I}_{K^{*}}$, $\psi(\mathbb{I}^{2n})=\mathbb{I}^{2n}$. Finally,
the isometry $\psi$ uniquely extends to an algebra isomorphism from
$\operatorname{Cliff}_{f_{a}}(\mathbb{K})$ to
$\operatorname{Cliff}_{f_{s}}(\mathbb{K})$. Again, we denote this isomorphism
by $\psi$. The restriction of $\psi$ is an isomorphism of topological groups
from $\operatorname{Spin}_{f_{a}}(\mathbb{O})$ onto
$\operatorname{Spin}_{f_{s}}(\mathbb{O})$. For every
$I\in\mathcal{I}_{K^{*}}$,
$\psi(\operatorname{Spin}_{f_{a}}(\mathbb{O})[I])=\operatorname{Spin}_{f_{s}}(\mathbb{O})[I]$.

###  4.2 The coroots $h_{1}$ and $h_{2}$

In this subsection we restrict to the case where $f=f_{s}$. The Lie algebra of
$\operatorname{Spin}_{f_{s}}$, denoted by $\mathfrak{spin}_{f_{s}}$, is the
span of the elements $e_{i}e_{j}$, where $i<j$. The subspace $\mathfrak{t}$
spanned by $e_{1}e_{2},e_{3}e_{4},\ldots,e_{2n-1}e_{2n}$ is a maximal torus in
$\mathfrak{spin}_{f_{s}}$. The elements

$X=e_{1}e_{3}+e_{2}e_{3}-e_{1}e_{4}-e_{2}e_{4}\text{ and
}Y=e_{3}e_{5}+e_{4}e_{5}-e_{3}e_{6}-e_{4}e_{6}$

are root vectors:

$\left[e_{1}e_{2},X\right]=-2X\quad\quad\left[e_{3}e_{4},X\right]=2X\quad\quad\left[e_{2k-1,2k},X\right]=0,$

for all $k\neq 1,2$, and

$\left[e_{3}e_{4},Y\right]=-2Y\quad\quad\left[e_{5}e_{6},Y\right]=2Y\quad\quad\left[e_{2l-1,2l},Y\right]=0,$

for all $l\neq 2,3$. We denote the root corresponding to $X$ by $\alpha_{1}$
and the root corresponding to $Y$ by $\alpha_{2}$.

Let $T=\exp\mathfrak{t}$ be the corresponding maximal torus. The map

$(t_{1},\ldots,t_{n})\mapsto\prod_{i}\left(\frac{t_{i}+t_{i}^{-1}}{2}+\frac{t_{i}-t_{i}^{-1}}{2}e_{2i-1}e_{2i}\right)$

is an isogeny between $\mathbb{G}_{m}^{n}$ and $T$ whose kernel is the
subgroup $\left\\{v\in\left\\{\pm 1\right\\}^{n}\mid v_{1}\cdots
v_{n}=1\right\\}$. We have

$Ad\left(e^{te_{1}e_{2}}\right)X=\exp\left(t\cdot
ad(e_{1}e_{2})\right)X=e^{-2t}X.$

and

$Ad\left(e^{te_{3}e_{4}}\right)X=\exp\left(t\cdot
ad(e_{3}e_{4})\right)X=e^{2t}X.$

Since
$e^{te_{1}e_{2}}=\left(\frac{e^{t}+e^{-t}}{2}\right)+\left(\frac{e^{t}-e^{-t}}{2}\right)e_{1}e_{2}$
and
$e^{te_{3}e_{4}}=\left(\frac{e^{t}+e^{-t}}{2}\right)+\left(\frac{e^{t}-e^{-t}}{2}\right)e_{3}e_{4}$,
we get that

$\alpha_{1}\left(\prod_{i}\left(\frac{t_{i}+t_{i}^{-1}}{2}+\frac{t_{i}-t_{i}^{-1}}{2}e_{2i-1}e_{2i}\right)\right)=\left(\frac{t_{2}}{t_{1}}\right)^{2}.$

The coroot $h_{1}$ corresponding to $\alpha_{1}$ is the map
$h_{1}:\mathbb{G}_{m}\rightarrow T$ such that, for any $O$-algebra $R$ and any
$t\in R^{\times}$,

$h_{1}(t)=\frac{t+2+t^{-1}}{4}+\frac{t-t^{-1}}{4}e_{1}e_{2}-\frac{t-t^{-1}}{4}e_{3}e_{4}-\frac{t-2+t^{-1}}{4}e_{1}e_{2}e_{3}e_{4}=$
$=\left(\frac{s+s^{-1}}{2}+\frac{s-s^{-1}}{2}e_{1}e_{2}\right)\cdot\left(\frac{s+s^{-1}}{2}-\frac{s-s^{-1}}{2}e_{3}e_{4}\right),$

where $s$ is a square root of $t$ (possibly in a quadratic extension).
Similarly,

$h_{2}(t)=\frac{t+2+t^{-1}}{4}+\frac{t-t^{-1}}{4}e_{3}e_{4}-\frac{t-t^{-1}}{4}e_{5}e_{6}-\frac{t-2+t^{-1}}{4}e_{3}e_{4}e_{5}e_{6}=$
$=\left(\frac{s+s^{-1}}{2}+\frac{s-s^{-1}}{2}e_{3}e_{4}\right)\cdot\left(\frac{s+s^{-1}}{2}-\frac{s-s^{-1}}{2}e_{5}e_{6}\right).$

###  4.3 Matsumoto’s theorem

In this subsection, we recall

######  Definition 4.3.1 (Steinberg and generalized Steinberg symbols).

Let $R$ be a ring in which 2 is invertible and let

$1\rightarrow D\rightarrow
E\overset{\eta}{\rightarrow}\operatorname{Spin}{f_{s}}(R)\rightarrow 1$

be a central extension.

  1. (1)

For every $a_{1},a_{2}\in R^{\times}$, choose
$\widetilde{h_{1}(a_{1})}\in\eta^{-1}(h_{1}(a_{1}))$ and
$\widetilde{h_{2}(a_{2})}\in\eta^{-1}(h_{2}(a_{2}))$ and denote

$\\{a_{1}:a_{2}\\}:=[\widetilde{h_{1}(a_{1})},\widetilde{h_{2}(a_{2})}].$

Since $D$ is central in $E$, $\\{a_{1}:a_{2}\\}$ depends only on $a_{1}$ and
$a_{2}$ and not on the choices of $\widetilde{h_{1}(a_{1})}$ and
$\widetilde{h_{2}(a_{2})}$. The map

$\\{-:-\\}:R^{\times}\times R^{\times}\rightarrow D$

is called a Steinberg symbol.

  2. (2)

More generally, for every commuting
$g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}(R)$ choose
$\widetilde{g_{1}}\in\eta^{-1}(g_{1})$ and
$\widetilde{g_{2}}\in\eta^{-1}(g_{2})$ and denote

$[g_{1}:g_{2}]:=[\widetilde{g_{1}},\widetilde{g_{2}}].$

The map

$[-:-]:\\{(g_{1},g_{2})\in\operatorname{Spin}_{f_{s}}(R)\mid[g_{1},g_{2}]=1\\}\rightarrow
D$

is called a generalized Steinberg symbol. Since $D$ is central in $E$,
$[g_{1}:g_{2}]$ depends only on $g_{1}$ and $g_{2}$ and not on the choices of
$\widetilde{g_{1}}$ and $\widetilde{g_{2}}$. Note that for every
$a_{1},a_{2}\in R^{\times}$,

$\\{a_{1}:a_{2}\\}=[h_{1}(a_{1}):h_{2}(a_{2})].$

  3. (3)

In the special case where $R={\bf A}^{*}$, for every commuting
$g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})$ and every internal
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ denote,

$[g_{1}:g_{2}]_{\mathcal{Q}}:=[(g_{1})_{\mathcal{Q}}:(g_{2})_{\mathcal{Q}}].$

The following is a special case of [25, Lemma 5.4 and Theorem 8.2] for the
Chevalley group $\operatorname{Spin}_{f_{s}}$ and the coroots $h_{1}$ and
$h_{2}$.

######  Theorem 4.3.2.

Let $F$ be a field of characteristic $\neq$ 2 and let

$1\rightarrow D\rightarrow
E\overset{\eta}{\rightarrow}\operatorname{Spin}_{f_{s}}(F)\rightarrow 1$ (16)

be a central extension. The extension (16) splits if and only if, for every
$a_{1},a_{2}\in F^{\times}$, $[h_{1}(a):h_{2}(b)]=1$.

###  4.4 Basic properties of Steinberg symbols

The following lemma states properties of generalized Steinberg symbols that
hold over every $O$-algebra.

######  Lemma 4.4.1.

Let $R$ be a ring in which 2 is invertible and let

$1\rightarrow D\rightarrow
E\overset{\eta}{\rightarrow}\operatorname{Spin}{f_{s}}(R)\rightarrow 1$

be a central extension.

  1. (1)

For every $g\in\operatorname{Spin}_{f_{s}}(R)$, $[g:1]=[1:g]=1$.

  2. (2)

For every commuting $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}(R)$,
$[g_{1}:g_{2}]=[g_{2}:g_{1}]^{-1}$.

  3. (3)

For every $g_{1},g_{2},g_{3}\in\operatorname{Spin}_{f_{s}}(R)$, if $g_{1}$
commutes with $g_{2}$ and $g_{3}$ then
$[g_{1}:g_{2}g_{3}]=[g_{1}:g_{2}][g_{1}:g_{3}]$.

  4. (4)

For every $g_{1},g_{2},g_{3}\in\operatorname{Spin}_{f_{s}}(R)$, if $g_{1}$ and
$g_{2}$ commute with $g_{3}$ then
$[g_{1}g_{2}:g_{3}]=[g_{1}:g_{3}][g_{2}:g_{3}]$.

  5. (5)

For every commuting $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}(R)$,
$[g_{1}:g_{2}]^{-1}=[g_{1}^{-1}:g_{2}]=[g_{1}:g_{2}^{-1}]$.

  6. (6)

For every commuting $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}(R)$ and every
$l\in\operatorname{Spin}_{f_{s}}(R)$,

$[g_{1}:g_{2}]=[lg_{1}l^{-1}:lg_{2}l^{-1}]=l[g_{1}:g_{2}]l^{-1}.$

  7. (7)

If $a,1-a\in R^{\times}$ then $[h_{1}(a),h_{2}(1-a)]=1$.

  8. (8)

If $R$ is a topological ring and the extension is a topological central
extension then the generalized Steinberg symbol $[\cdot:\cdot]$ is continuous.

###### Proof.

  1. (1)

Let $\widetilde{g}$ be a lift of $g$ to $E$. Then $[g:1]=[\widetilde{g},1]=1$
and similarly, $[1:g]=1$.

  2. (2)

For every group elements $x$ and $y$, $[x,y]=[y,x]^{-1}$.

  3. (3)

For every group elements $x$, $y$ and $z$, $[x,yz]=[x,y]y[x,z]y^{-1}$. The
claim follows from the centrality of $D$.

  4. (4)

For every group elements $x$, $y$ and $z$, $[xy,z]=x[y,z]x^{-1}[x,z]$. The
claim follows from the centrality of $D$.

  5. (5)

Follows from Items 1, 3 and 4.

  6. (6)

The claim follows from the centrality of $D$.

  7. (7)

This a standard property of Steinberg symbols (the proof in [25, Lemma 5.5] is
for fields, but it works in general).

  8. (8)

Let $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}$ be commutating elements. Fix
lifts $\tilde{g}_{1},\tilde{g_{2}}$ of $g_{1},g_{2}$, respectively, and let
$U\subseteq D$ be an open neighborhood of
$[\tilde{g}_{1},\tilde{g_{2}}]=[g_{1}:g_{2}]$. By the continuity of
multiplication, there exist open subsets $U_{1}\ni\tilde{g}_{1}$ and
$U_{2}\ni\tilde{g}_{2}$ of $E$ such that $\\{[l_{1},l_{2}]\mid l_{1}\in
U_{1}\land\ l_{2}\in U_{2}\\}\subseteq U$. Since the extension is topological,
$\eta$ is open and the induced isomorphism
$\bar{\eta}:E/D\rightarrow\operatorname{Spin}_{f_{s}}(R)$ is a homeomorphism.
Thus, every $g_{3}\in\eta(U_{1})$ and every $g_{4}\in\eta(U_{2})$ have lifts
$\tilde{g}_{3}$ and $\tilde{g}_{4}$ such that $\tilde{g}_{3}\in U_{1}$ and
$\tilde{g}_{4}\in U_{2}$ so $[g_{3}:g_{4}]=[\tilde{g}_{3},\tilde{g}_{4}]\in
U$.

∎

We now restrict the attention to discrete
$\psi(\operatorname{Spin}_{f_{a}}(K^{*}))$-metaplectic extensions.

######  Setting 4.4.2.

Under Setting 4.1, fix a discrete
$\operatorname{Spin}_{f_{a}}(K^{*})$-metaplectic extension

$1\longrightarrow D\longrightarrow
E\overset{\rho}{\longrightarrow}\operatorname{Spin}_{f_{a}}(\mathbb{K})\longrightarrow
1.$ (17)

We get a discrete $\psi(\operatorname{Spin}_{f_{a}}(K^{*}))$-metaplectic
extension,

$1\longrightarrow D\longrightarrow
E\overset{\psi\circ\rho}{\longrightarrow}\operatorname{Spin}_{f_{s}}(\mathbb{K})\longrightarrow
1.$ (18)

We fix an ideal $L\in\mathcal{I}_{K^{*}}$ such that the extension (17) has a
continuous splitting over $\operatorname{Spin}_{f_{s}}(\mathbb{O})[L]$.

We need two preliminaries Lemmas.

######  Lemma 4.4.3.

There exists a constant $N$ such that for every non-empty internal subset
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ and every
$g\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})$ such that for every
$P\in\mathcal{Q}$, $g_{\\{P\\}}\notin Z(\operatorname{Spin}_{f_{s}}({\bf
A}^{*}))\cdot\operatorname{Spin}_{f_{s}}({\bf O}^{*})[P]$,

$\left(\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{Q}}}(g_{\mathcal{Q}})\right)^{N}=\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{Q}}.$

###### Proof.

Applying [2, Lemma 3.18] and [2, Lemma 3.22] for
$\underline{H}=\operatorname{Spin}_{f_{s}}$, there is a constant $N$ such that
the following hold

  1. (1)

For every local field $F$ and every non-central element
$x\in\operatorname{Spin}_{f_{s}}(F)$,
$\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}(F)}(x)^{N}=\langle\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}(F)}(x)\rangle=\operatorname{Spin}_{f_{s}}(F)$.

  2. (2)

There is a finite set $\mathcal{R}_{0}\subseteq\mathcal{P}_{K}$ such that, for
every $P\in\mathcal{P}_{K}\smallsetminus\mathcal{R}_{0}$, if
$x\in\operatorname{Spin}_{f_{s}}(O_{P})\smallsetminus
Z(\operatorname{Spin}_{f_{s}}(K_{P}))\cdot\operatorname{Spin}_{f_{s}}(O_{P})[P]$,
then
$\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}(O_{P})}(x)^{N}=\operatorname{Spin}_{f_{s}}(O_{P})$.

We first show that $\left(\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}({\bf
A})_{\mathcal{Q}}}(g_{\mathcal{Q}})\right)^{N}=\operatorname{Spin}_{f_{s}}({\bf
A})_{\mathcal{Q}}$. Suppose that $\mathcal{Q}\subseteq\mathcal{P}_{K}$, that
$g\in\operatorname{Spin}_{f_{s}}({\bf A})$ is such that, for every
$P\in\mathcal{Q}$, $g_{\\{P\\}}\notin Z(\operatorname{Spin}_{f_{s}}({\bf
A}^{*}))\cdot\operatorname{Spin}_{f_{s}}({\bf O}^{*})[P]$, and that
$h\in\operatorname{Spin}_{f_{s}}({\bf A})$. Let $\mathcal{R}_{1}$ be the set
of primes $P\in\mathcal{Q}$ for which
$h_{P}\notin\operatorname{Spin}_{f_{s}}({\bf O})$. Applying (1) for the
finitely many primes in $\mathcal{R}_{0}\cup\mathcal{R}_{1}$ and (2) for all
other primes in $\mathcal{Q}$, we get that
$h\in\operatorname{gcl}_{\operatorname{Spin}_{f_{s}}({\bf
A})_{\mathcal{Q}}}(g)^{N}$.

The claim of the lemma follows by taking ultrapowers. ∎

######  Lemma 4.4.4.

Under Setting 4.1, there exists a constant $N$ such that for every
$I\in\mathcal{I}_{K}$, every element of $\operatorname{Spin}_{f_{s}}({\bf
O})[I^{2}]$ is a product of at most $N$ elements from $\\{[g_{1},g_{2}]\mid
g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf O})[I]\\}$.

###### Proof.

By Lemma A.3.4, the commutator widths of all congruence subgroups
$\operatorname{Spin}_{f_{s}}({\bf O})[I]$ are uniformly bounded. It remains to
prove that the commutator subgroup of $\operatorname{Spin}_{f_{s}}({\bf
O})[I]$ is $\operatorname{Spin}_{f_{s}}({\bf O})[I^{2}]$, for every $I$. By
the Chinese Remainder theorem, it is enough to prove that, for every prime $P$
and every natural number $k$, the commutator subgroup of
$\operatorname{Spin}_{f_{s}}(O_{P})[P^{k}]$ is equal to
$\operatorname{Spin}_{f_{s}}(O_{P})[P^{2k}]$.

To simplify notation, let $G=\operatorname{Spin}_{f_{s}}$ and let
$\mathfrak{g}=\mathfrak{spin}_{f_{s}}$ be its Lie algebra. Since $f_{s}$ is
regular over ${\bf O}$, there is an isomorphism $G(O_{P})/G(O_{P})[P]\cong
G(O/P)$ and a compatible isomorphism of representations
$G(O_{P})[P^{k}]/G(O_{P})[P^{k+1}]\rightarrow\mathfrak{g}(O/P)$. If $k>0$,
then, under this isomorphism, the group theoretic commutator map
$G(O_{P})[P^{k}]/G(O_{P})[P^{k+1}]\times
G(O_{P})[P^{k}]/G(O_{P})[P^{k+1}]\rightarrow
G(O_{P})[P^{2k}]/G(O_{P})[P^{2k+1}]$ is identified with the Lie bracket
$\mathfrak{g}(O/P)\times\mathfrak{g}(O/P)\rightarrow\mathfrak{g}(O/P)$.

  * •

If $k=0$, the quasi-simplicity of $G(O/P)$ implies that there are elements
$h_{1},h_{2}\in G(O/P)$ such that $[h_{1},h_{2}]\notin Z(G(O/P))$, and,
therefore, $[h_{1},h_{2}]$ normally generates. Taking $g_{1},g_{2}\in
G(O_{P})$ to be lifts of $h_{1},h_{2}$, the normal subgroup generated by
$[g_{1},g_{2}]$ projects to $G(O/P)$ and, therefore, is equal to $G(O_{P})$.

  * •

If $k>0$, the simplicity of $\mathfrak{g}(O/P)$ implies that there are
elements $X_{1},X_{2}\in\mathfrak{g}(O/P)$ such that $[X_{1},X_{2}]\neq 0$.
Since $\mathfrak{g}(O/P)$ is an irreducible $G(O/P)$-module, the
$G(O/P)$-module generated by $[X_{1},X_{2}]$ is $\mathfrak{g}(O/P)$. If
$g_{1},g_{2}\in G(O_{P})[P^{k}]$ satisfy that $g_{1},g_{2}$ are identified
with $X_{1},X_{2}$, then the normal subgroup generated by $[g_{1},g_{2}]$
projects to $G(O_{P})[P^{2k}]/G(O_{P})[P^{2k+1}]$ and, therefore, is equal to
$G(O_{P})[P^{2k}]$.

∎

######  Lemma 4.4.5.

Under Setting 4.4.2,

  1. (1)

For every commuting $g_{1},g_{2}\in\psi(\operatorname{Spin}_{f_{a}}(K^{*}))$,
$[g_{1}:g_{2}]=1$.

  2. (2)

For every commuting $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L]$, $[g_{1}:g_{2}]=1$.

In particular, for every internal subset $\mathcal{Q}$ of
$\mathcal{P}_{K^{*}}$ which is disjoint from $\mathcal{P}_{K^{*}}(L)$ and
every commuting $g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$,
$[g_{1}:g_{2}]_{\mathcal{Q}}=1$.

  3. (3)

Let $a_{1},a_{2}\in({\bf A}^{*})^{\times}$ and let
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ be a non-empty internal subset.
There exists $I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$ such that for every
$b_{1},b_{2}\in({\bf A}^{*})^{\times}$ for which
$h_{1}(a_{1}^{-1}b_{1})_{\mathcal{Q}},h_{2}(a_{2}b_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}.$

  4. (4)

Let $\mathcal{Q},\mathcal{R}\subseteq\mathcal{P}_{K^{*}}$ be disjoint non-
empty internal subsets. For every $g_{1}\in\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{Q}}$ and every $g_{2}\in\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{R}}$, $g_{1}$ and $g_{2}$ commute and $[g_{1}:g_{2}]=1$.

  5. (5)

Let $\mathcal{Q},\mathcal{R}\subseteq\mathcal{P}_{K^{*}}$ be disjoint non-
empty internal subsets. For every commuting
$g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})$,

$[g_{1},g_{2}]_{\mathcal{Q}\cup\mathcal{R}}=[g_{1},g_{2}]_{\mathcal{Q}}[g_{1},g_{2}]_{\mathcal{R}}.$

  6. (6)

Let $a_{1},a_{2}\in({\bf A}^{*})^{\times}$ and denote

$\mathcal{Q}:=\mathcal{P}_{K^{*}}(L)\cup\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(a_{1})\neq
0\lor\operatorname{val}_{P}(a_{2})\neq 0\\}.$

There exists $I\in\mathcal{P}_{K^{*}}(\mathcal{Q})$ such that for every
$b_{1},b_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(a_{1}b_{1}^{-1}),h_{1}(a_{2}b_{2}^{-1})\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,

$[h_{1}(a_{1}):h_{2}(a_{2})]=[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}$

  7. (7)

Let $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ be internal subset and let
$g_{1},g_{2}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})_{\mathcal{Q}}$ be
commuting elements such that $g_{1}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ or $g_{2}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L^{2}]$.
Then $[g_{1}:g_{2}]=1.$

  8. (8)

Let $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ be internal subset and
$a_{1},a_{1}^{\prime},a_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(a_{1}^{\prime}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ and
$h_{2}(a_{2})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$. Then,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2})]_{\mathcal{Q}}.$

  9. (9)

Let $a_{1}\in({\bf A}^{*})^{\times}$, $a_{2}\in({\bf O}^{*})^{\times}$ and
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$. There exists
$I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$ such that for every
$b_{1},b_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ and
$h_{2}(b_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}.$
(19)

In particular, if $\mathcal{Q}$ is disjoint for $\mathcal{P}_{K^{*}}(L)$,
$h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ and
$h_{2}(b_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$, Equation (19) holds.

###### Proof.

Items 1 and 2 follow from the fact that $\psi\circ\rho$ has sections over
$\psi(\operatorname{Spin}_{fa}(K^{*}))$ and over
$\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L]$. By Item 8 of Lemma 4.4.1,
generalized Steinberg symbols are continuous. Thus, Item 3 holds.

4 Let $g_{1}\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})_{\mathcal{Q}}$ and
$g_{2}\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})_{\mathcal{R}}$. It is clear
the $g_{1}$ commutes with every element of $\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{R}}$. Choose
$g_{3}^{\prime},g_{3}^{\prime\prime}\in\operatorname{Spin}_{f_{s}}({\bf
A}^{*})_{\mathcal{R}}$ such that for every $P\in\mathcal{R}$,
$[g_{3}^{\prime},g_{3}^{\prime\prime}]_{\\{\mathcal{P}\\}}\notin
Z(\operatorname{Spin}_{f_{s}}({\bf A}^{*})_{\\{\mathcal{P}\\}})$. Denote
$g_{3}:=[g_{3}^{\prime}:g_{3}^{\prime\prime}]$. Since $D$ is abelian,

$[g_{1}:g_{3}]=[g_{1}:[g_{3}^{\prime}:g_{3}^{\prime\prime}]]=[g_{1}:g^{\prime}_{3}][g_{1}:g^{\prime\prime}_{3}][g_{1}:(g^{\prime}_{3})]^{-1}[g_{1}:(g^{\prime\prime}_{3})]^{-1}=1.$

By Lemma 4.4.3, there exist $N$, $\epsilon_{1},\ldots,\epsilon_{N}\in\\{\pm
1\\}$ and $l_{1},\ldots,l_{N}\in\operatorname{Spin}_{f_{s}}({\bf
A})_{\mathcal{R}}$ such that
$g_{2}=(l_{1}g_{3}^{\epsilon_{1}}l_{1}^{-1})\cdots(l_{N}g_{3}^{\epsilon_{N}}l_{N}^{-1})$.
For every $1\leq i\leq N$,

$[g_{1}:l_{i}g_{3}^{\epsilon_{i}}l_{i}^{-1}]=l_{i}[g_{1}:g_{3}^{\epsilon_{i}}]l_{i}^{-1}=[g_{1}:g_{3}^{\epsilon_{i}}]=1$

so

$[g_{1}:g_{2}]=[g_{1}:l_{1}g_{3}^{\epsilon_{1}}l_{1}^{-1}]\cdots[g_{1}:l_{N}g_{3}^{\epsilon_{N}}l_{N}^{-1}]=1$

5 For every $g\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$,
$g_{\mathcal{Q}\cup\mathcal{R}}=g_{\mathcal{Q}}g_{\mathcal{R}}$. By Lemma
4.4.3 and Item 7 of Lemma 4.4.5 ,

$[g_{1}:g_{2}]_{\mathcal{Q}\cup\mathcal{R}}=[(g_{1})_{\mathcal{Q}}:(g_{2})_{\mathcal{Q}}][(g_{1})_{\mathcal{R}}:(g_{2})_{\mathcal{Q}}][(g_{1})_{\mathcal{Q}}:(g_{2})_{\mathcal{R}}][(g_{1})_{\mathcal{R}}:(g_{2})_{\mathcal{R}}]=[g_{1}:g_{2}]_{\mathcal{Q}}[g_{1}:g_{2}]_{\mathcal{R}}.$

6 By symmetry, we can assume that $g_{1}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$. By Item 5, for every $a_{1},a_{2}\in({\bf A}^{*})^{\times}$
and every small internal $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$,

$[h_{1}(a_{1}):h_{2}(a_{2})]=[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}^{c}}.$

Thus, the claim follows from Items 2 and 3.

7 Let $\widetilde{g_{2}}\in(\psi\circ\rho)^{-1}(g_{2})$ and denote by
$\chi_{g_{2}}$ and $\chi_{\widetilde{g_{2}}}$ the conjugation maps $g\mapsto
g_{2}gg_{2}^{-1}$ and $g\mapsto\widetilde{g_{2}}g\widetilde{g_{2}}^{-1}$,
respectively. There exists a continuous section
$\eta:\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L]\rightarrow E$ of
$\psi\circ\rho$ over $\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L]$. Since
$\operatorname{Spin}_{f_{s}}(\mathbb{O})[L]$ is normal in
$\operatorname{Spin}_{f_{s}}(\mathbb{O})$, the homomorphism
$\eta^{\prime}:=\chi_{{\widetilde{g_{2}}}}\circ\eta_{1}\circ\chi_{g_{2}}^{-1}$
is another continuous section of $\psi\circ\rho$ over
$\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L]$. The continuous sections $\eta$
and $\eta^{\prime}$ must agree on the commutator subgroup of
$\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L]$. By Lemma 4.4.4, the commutator
subgroup contains $\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L^{2}]$. Since
$g_{2}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$ and thus normalizes
$\operatorname{Spin}_{f_{s}}({\bf O}^{*})[L^{2}]$,

$[g_{1}:g_{2}]=[\eta(g_{1}):\widetilde{g_{2}}]=\eta(g_{1})\chi_{\widetilde{g_{2}}}(\eta(g_{1})^{-1})=\eta(g_{1})\eta^{\prime}(\chi_{g_{2}}(g_{1}^{-1}))=$
$\eta(g_{1})\eta^{\prime}(g_{2}g_{1}^{-1}g_{2}^{-1})=\eta(g_{1})\eta(g_{2}g_{1}^{-1}g_{2}^{-1})=\eta([g_{1},g_{2}])=1.$

8 By Item 7,

$[h_{1}(a_{1}^{\prime}a_{1}^{-1}):h_{2}(a_{2})]_{\mathcal{Q}}=1\text{ so
}[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2})]_{\mathcal{Q}}.$

9 Since $[h_{1}(a_{1}):h_{2}(1)]=1$, by Item 3, there exists
$I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$ such that for every $c\in({\bf
A}^{*})^{\times}$ for which
$h_{2}(c)_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})[I]$,
$[h_{1}(a_{1}):h_{2}(c)]_{\mathcal{Q}}=1$.

Let $b_{1},b_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ and
$h_{2}(b_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$. The following two properties hold:

  * •

Since $h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ and
$h_{2}(b_{2})_{\mathcal{Q}},h_{2}(b_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{S}}({\bf
O}^{*})$, by Item 7,

$[h_{1}(b_{1}a_{1}^{-1}):h_{2}(b_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}a_{1}^{-1}):h_{2}(b_{2}a_{2}^{-1})]_{\mathcal{Q}}=1.$

  * •

Since $h_{2}(b_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$, by the choice of $I$,

$[h_{1}(a_{1}):h_{2}(b_{2}a_{2}^{-1})]_{\mathcal{Q}}=1.$

Thus,

$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}=[h_{1}((b_{1}a_{1}^{-1})a_{1}):h_{1}((b_{2}a_{2}^{-1})a_{2})]_{\mathcal{Q}}=$
$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}[h_{1}(a_{1}):h_{2}(b_{2}a_{2}^{-1})]_{\mathcal{Q}}[h_{1}(b_{1}a_{1}^{-1}):h_{2}(b_{2})]_{\mathcal{Q}}[h_{1}(b_{1}a_{1}^{-1}):h_{2}(b_{2}a_{2}^{-1})]_{\mathcal{Q}}=$
$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}.$

The “in particular” part is true since if $\mathcal{Q}$ is disjoint for
$\mathcal{P}_{K^{*}}(L)$ and
$h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ then
$h_{1}(b_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$. ∎

######  Lemma 4.4.6.

Under Setting 4.4.2, let
$\mathcal{R}\subseteq\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(L)$
be an internal subset and let
$a_{1},a_{1}^{\prime},a_{2},a_{2}^{\prime}\in({\bf A})^{\times}$ be elements
such that, for every $P\in\mathcal{R}$,

$\operatorname{val}_{P}(a_{1})=\operatorname{val}_{P}(a_{1}^{\prime})=1,\
\operatorname{val}_{P}(a_{2})=0\text{ and
}\operatorname{val}_{P}(a_{2}-a_{2}^{\prime})\geq 1.$

Then,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{R}}.$

###### Proof.

We first prove the claim under the stronger assumption that, for every
$P\in\mathcal{R}$, $\operatorname{val}_{P}(a_{2}-a_{2}^{\prime})=1$. By a
coordinate-wise argument, there exists $a_{1}^{\prime\prime}\in({\bf
A}^{*})^{\times}$ such that $a_{2}^{\prime}=a_{2}(1-a_{1}^{\prime\prime})$
and, for every $P\in\mathcal{R}$,
$\operatorname{val}_{P}(a_{1}^{\prime\prime})=1$. By Item 8 of Lemma 4.4.5,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime\prime}):h_{2}(a_{2})]_{\mathcal{R}}\text{
and
}[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{R}}.$

This and Item 7 of Lemma 4.4.1 imply that

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime\prime}):h_{2}(a_{2})]_{\mathcal{R}}=$
$[h_{1}(a_{1}^{\prime\prime}),h_{2}(a_{2}(1-a_{1}^{\prime\prime}))]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime\prime}),h_{2}(a_{2}^{\prime})]_{\mathcal{R}}=[h_{1}(a_{1}^{\prime}),h_{2}(a_{2}^{\prime})]_{\mathcal{R}}.$

We now prove the general case. Denote
$\mathcal{Q}:=\\{P\in\mathcal{R}\mid\operatorname{val}_{P}(a_{2}-a_{2}^{\prime})\geq
2\\}$. Note that $\mathcal{Q}$ is an internal subset. By the special case
proved above,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{R}\smallsetminus\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{R}\smallsetminus\mathcal{Q}}.$

Because of Lemma 4.4.55, it is enough to prove that

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{Q}}.$

Remembering that $O=\mathbb{Z}[\frac{1}{2}]$, a coordinate-wise argument shows
that there exists $a_{2}^{\prime\prime}\in({\bf A})^{\times}$ such that
$\operatorname{val}_{P}(a_{2}-a_{2}^{\prime\prime})=\operatorname{val}_{P}(a_{2}^{\prime}-a_{2}^{\prime\prime})=1$,
for every $P\in\mathcal{Q}$. Thus,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime\prime})]_{\mathcal{Q}}=[h_{1}(a_{1}^{\prime}):h_{2}(a_{2}^{\prime})]_{\mathcal{Q}}.$

∎

###  4.5 Steinberg symbols are trivial

######  Definition 4.5.1.

Let $a\in({\bf A}^{*})^{\times}$ and let
$\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ be an internal subset. The support
of $a$ in $\mathcal{Q}$ is
$\operatorname{supp}_{\mathcal{Q}}(a):=\\{P\in\mathcal{Q}\mid\operatorname{val}_{P}(a)\neq
0\\}$.

######  Definition 4.5.2.

Under Setting 4.4.2, let $a_{1},a_{2},b_{1},b_{2}\in({\bf A}^{*})^{\times}$
and let $I_{1},I_{2},J\in\mathcal{I}_{K^{*}}$. Assume that the sets
$\mathcal{Q}_{1}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(I_{1}I_{2}J)}(a_{1})$
and
$\mathcal{Q}_{2}:=\operatorname{supp}_{=\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(I_{1}I_{2}J)}(a_{2})$
are disjoint.

We say that the pair $(b_{1},b_{2})$ is an $(I_{1},I_{2},J)$-approximation of
$(a_{1},a_{2})$ if there exist commuting
$g_{1},g_{2}\in\psi(\operatorname{Spin}_{f_{a}}(K^{*}))$ and
$l\in\operatorname{Spin}_{f_{s}}({\bf A}^{*})$ such that

  1. (1)

$h_{i}(a_{i}b_{i}^{-1})_{\mathcal{P}_{K^{*}}(I_{1}I_{2})}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I_{i}]$, for $i=1,2$.

  2. (2)

$(lg_{i}l^{-1})_{\mathcal{P}_{K^{*}}(I_{1}I_{2})\cup\mathcal{Q}_{1}\cup\mathcal{Q}_{2}}=h_{i}(b_{i})_{\mathcal{P}_{K^{*}}(I_{1}I_{2})\cup\mathcal{Q}_{1}\cup\mathcal{Q}_{2}}$,
for $i=1,2$.

  3. (3)

$(lg_{1}l^{-1})_{(\mathcal{Q}_{1}\cup\mathcal{P}_{K^{*}}(I_{1}I_{2}))^{c}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ and
$(lg_{2}l^{-1})_{(\mathcal{Q}_{2}\cup\mathcal{P}_{K^{*}}(I_{1}I_{2}))^{c}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[J]$.

######  Lemma 4.5.3.

Under Setting 4.4.2, let $a_{1},a_{2},b_{1},b_{2}\in({\bf A}^{*})^{\times}$,
and let $I_{1},I_{2},J\in\mathcal{I}_{K^{*}}$. Denote
$\mathcal{Q}:=\mathcal{P}_{K^{*}}(I_{1}I_{2})$ and assume that the following
conditions hold:

  1. (1)

For every $c_{1},c_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(a_{1}c_{1}^{-1})\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})[I_{1}]$ and
$h_{2}(a_{2}c_{2}^{-1})\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})[I_{2}]$,

$[h_{1}(a_{1}),h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(c_{1}),h_{2}(c_{2})]_{\mathcal{Q}}.$

  2. (2)

$(b_{1},b_{2})$ is a $(I_{1},I_{2},J)$-approximation of $(a_{1},a_{2})$.

  3. (3)

Either $\mathcal{P}_{K^{*}}(L)\subseteq\mathcal{P}_{K^{*}}(I_{1}I_{2})$ or
$L^{2}\subseteq J$.

Then,

$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}):h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{2}}$

where
$\mathcal{Q}_{1}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(I_{1}I_{2}J)}(a_{1})$
and
$\mathcal{Q}_{2}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(I_{1}I_{2}J)}(a_{2})$.

###### Proof.

By the definition of $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ and the
definition of $(I_{1},I_{2},J)$-approximation, $\mathcal{Q}$,
$\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ are pairwise disjoint. Let $g_{1}$,
$g_{2}$, and $l$ be as in as in the definition of
$(I_{1},I_{2},J)$-approximation and denote
$\mathcal{R}:=\big{(}\mathcal{Q}\cup\mathcal{Q}_{1}\cup\mathcal{Q}_{2}\big{)}^{c}$.
Since $g_{1},g_{2}\in\psi(\operatorname{Spin}_{f_{a}}({\bf A}^{*}))$ commute,

$1=[g_{1}:g_{2}]=[lg_{1}l^{-1}:lg_{2}l^{-1}].$

Thus,

$[lg_{1}l^{-1}:lg_{2}l^{-1}]_{\mathcal{Q}}=[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{\mathcal{Q}_{1}}[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{\mathcal{Q}_{2}}[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{\mathcal{R}}.$
(20)

By Items 1 and 2 and the definition of $(I_{1},I_{2},J)$-approximation,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{Q}}=[lg_{1}l^{-1}:lg_{2}l^{-1}]_{\mathcal{Q}}.$

By the definition of $\mathcal{R}$,
$(lg_{1}l^{-1})_{\mathcal{R}}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$. We
claim that $(lg_{2}l^{-1})_{\mathcal{R}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$. Indeed, if
$\mathcal{P}_{K^{*}}(L)\subseteq\mathcal{P}_{K^{*}}(I_{1}I_{2})$, then
$\mathcal{P}_{K^{*}}(L)$ and $\mathcal{R}$ are disjoint so
$(lg_{2}l^{-1})_{\mathcal{R}}\in(\operatorname{Spin}_{f_{s}}(O^{*}))_{\mathcal{R}}\subseteq\operatorname{Spin}_{f_{s}}(O^{*})[L^{2}]$
while if $L^{2}\subseteq J$, then
$(lg_{2}l^{-1})_{\mathcal{R}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[L^{2}]$ by the definitions of $\mathcal{R}$ and
$(I_{1},I_{2},J)$-approximation. Thus, Item 9 of Lemma 4.4.5 implies that

$[lg_{1}l^{-1}:lg_{2}l^{-1}]_{\mathcal{R}}=1.$

Since for every $1\leq i\leq 2$,
$(lg_{i}l^{-1})_{\mathcal{P}_{K^{*}}(I_{1}I_{2})\cup\mathcal{Q}_{1}\cup\mathcal{Q}_{2}}=h_{i}(b_{i})_{\mathcal{P}_{K^{*}}(I_{1}I_{2})\cup\mathcal{Q}_{1}\cup\mathcal{Q}_{2}}$,
Equation (20) becomes

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}):h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{2}}.$

∎

######  Lemma 4.5.4.

Under Setting 4.4.2, let $b_{1},c_{1}\in({\bf A}^{*})^{\times}$ and
$b_{2},c_{2}\in({\bf O}^{*})^{\times}$ and
$\mathcal{R}\subseteq\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(L)$
be such that:

  1. (1)

For every $P\in\mathcal{R}$, $-1\leq\operatorname{val}_{P}(b_{1})\leq 1$.

  2. (2)

$(c_{1},c_{2})$ is an $(O^{*},\prod_{P\in\mathcal{R}}P,L^{2})$-approximation
of $(b_{1},b_{2})$.

  3. (3)

$(c_{1})_{\mathcal{R}^{c}}\in({\bf O}^{*})^{\times}$.

Then $[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}}=1$.

###### Proof.

Denote
$\mathcal{R}_{1}:=\\{P\in\mathcal{R}\mid\operatorname{val}_{P}(b_{1})=0\\}$
and $\mathcal{R}_{2}:=\mathcal{R}\smallsetminus\mathcal{R}_{1}$. Item 2 of
Lemma 4.4.5 implies that
$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}_{1}}=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{R}_{1}}=1$.
Lemma 4.4.6 implies that
$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}_{2}}=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{R}_{2}}$.
Thus,
$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}}=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{R}}$.
Let $g_{1}$, $g_{2}$, and $l$ be as in as in the definition of an
$(O^{*},I,L^{2})$-approximation. Then,

$[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}}=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{R}}=[lg_{1}l^{-1}:lg_{2}l^{-1}]_{\mathcal{R}}\overset{\star}{=}$
$\overset{\star}{=}[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{\mathcal{P}(L)}[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{(\mathcal{R}\cup\mathcal{P}(L))^{c}}.$

Item 2 of Lemma 4.4.5 implies that
$[h_{1}(c_{1}):h_{2}(c_{2})]_{(\mathcal{R}\cup\mathcal{P}_{K^{*}}(L))^{c}}=1$.
Since $\mathcal{R}$ is disjoint from $\mathcal{P}_{K^{*}}(L)$, by the
definition of an $(O^{*},\prod_{P\in\mathcal{R}}P,L^{2})$-approximation and
Item 7 of Lemma 4.4.5, $[lg_{1}l^{-1}:lg_{2}^{-1}l^{-1}]_{\mathcal{P}(L)}=1$.
It follows that $[h_{1}(b_{1}):h_{2}(b_{2})]_{\mathcal{R}}=1$. ∎

######  Lemma 4.5.5.

Under Setting 4.4.2, let $a,b\in({\bf A}^{*})^{\times}$ and
$I\in\mathcal{P}_{K^{*}}$ be such that
$\mathcal{P}_{K^{*}}(I)\cap\mathcal{P}_{K^{*}}(L)=\emptyset$ and $(b,-1)$ is a
$(O^{*},I,L^{2})$-approximation of $(a,-1)$. Then, for
$\mathcal{Q}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(IJ)}(b_{1})$,

$[h_{1}(a):h_{2}(-1)]_{\mathcal{P}_{K^{*}}(I)}=[h_{1}(b):h_{2}(-1)]_{\mathcal{Q}}.$

###### Proof.

The proof is similar to the proofs of Lemmas 4.5.3 and 4.5.4 and is omitted. ∎

####  4.5.1 Weak approximation

######  Lemma 4.5.6.

Let $a_{1},a_{2}\in(K^{*})^{\times}$ and let $I\in\mathcal{I}_{K^{*}}$. There
exists a $(I,I,O^{*},b_{1},b_{2})$-approximation of $(a_{1},a_{2})$.

###### Proof.

Follows from Lemma 5.4.3 below and a coordinate-wise argument. ∎

######  Corollary 4.5.7.

Under Setting 4.4.2, for every $a_{1},a_{2}\in({\bf A}^{*})^{\times}$, there
exist $b_{1},b_{2}\in\big{(}{\bf A}^{*}\big{)}^{\times}$ such that
$\mathcal{Q}_{1}:=\\{P\in\mathcal{P}\mid\operatorname{val}_{P}(b_{1})\neq
0\\}$,
$\mathcal{Q}_{2}:=\\{P\in\mathcal{P}\mid\operatorname{val}_{P}(b_{2})\neq
0\\}$ and $\mathcal{P}_{K^{*}}(L)$ are pairwise disjoint, and

$[h_{1}(a_{1}),h_{2}(a_{2})]=[h_{1}(b_{1}),h_{2}(b_{2})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}),h_{2}(b_{2})]_{\mathcal{Q}_{2}}$

###### Proof.

Denote

$\mathcal{Q}:=\mathcal{P}_{K^{*}}(L)\cup\\{P\in\mathcal{P}_{K^{*}}\mid\operatorname{val}_{P}(a_{1})\neq
0\ \lor\ \operatorname{val}_{P}(a_{2})\neq 0\\}.$

By Items 3 and 6 of Lemma 4.4.5, there exists
$I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$ such that for every
$c_{1},c_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(a_{1}c_{1}^{-1}),h_{1}(a_{2}c_{2}^{-1})\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,

$[h_{1}(a_{1}):h_{2}(a_{2})]=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{Q}}.$

We can assume that $\mathcal{P}_{K^{*}}(I)=\mathcal{Q}$. By Lemma 4.5.6, there
exists a $(I,I,O^{*})$-approximation $(b_{1},b_{1})$ of $(a_{1},a_{2})$. By
Lemma 4.5.3,

$[h_{1}(a_{1}),h_{2}(a_{2})]=[h_{1}(b_{1}),h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}),h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{2}}.$

∎

####  4.5.2 Every Steinberg symbol is a product of powers of Steinberg symbols

######  Lemma 4.5.8.

Let $I,J\in\mathcal{I}_{K^{*}}$ such that
$\mathcal{P}_{K^{*}}(I)\cap\mathcal{P}_{K^{*}}(J)=\emptyset$, $a_{1}\in({\bf
A}^{*})^{\times}$ and $a_{2}\in({\bf O}^{*})^{\times}$. There exists an
$(O^{*},I,J)$-approximation $(b_{1},b_{2})$ of $(a_{1},a_{2})$, $c\in({\bf
O}^{*})^{\times}$ and
$\mathcal{R}\subseteq\mathcal{Q}_{1}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(IJ)}(b_{1})$
such that $b_{2}\in({\bf O}^{*})^{\times}$ and

$h_{2}(b_{2})_{\mathcal{Q}_{1}}=h_{2}(c^{2})_{\mathcal{Q}_{1}}h_{2}(-1)_{\mathcal{R}}.$

###### Proof.

Follows from Lemma 5.4.4 below and a coordinate-wise argument. ∎

######  Corollary 4.5.9.

Under Setting 4.4.2, let $Q\subseteq\mathcal{P}_{K^{*}}$ be a small internal
subset disjoint from $\mathcal{P}_{K^{*}}(L)$, $a_{1}\in({\bf
A}^{*})^{\times}$ and $a_{2}\in({\bf O}^{*})^{\times}$. There exist small
internal subsets
$\mathcal{R}\subseteq\mathcal{Q}_{1}\subseteq\mathcal{P}_{K^{*}}$ which are
disjoint from $\mathcal{Q}\cup\mathcal{P}_{K^{*}}(L)$ and $b_{1}\in({\bf
A}^{*})^{\times}$ and $c\in({\bf O}^{*})^{\times}$ such that

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(b_{1}):h_{2}(c^{2})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}):h_{2}(-1)]_{\mathcal{R}}.$

###### Proof.

By Item 9 of Lemma 4.4.5, there exists $I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$
such that for every $c_{1},c_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(c_{1}a_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ and
$h_{2}(c_{2}a_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(c_{1}):h_{2}(c_{2})]_{\mathcal{Q}}.$

We can assume that $\mathcal{P}_{K^{*}}(I)=\mathcal{Q}$.

Lemma 4.5.8 implies that there exist an $(O^{*},I,L^{2})$-approximation
$(b_{1},b_{2})$ of $(a_{1},a_{2})$, $c\in({\bf O}^{*})^{\times}$ and
$\mathcal{R}\subseteq\mathcal{Q}_{1}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(IL)}(a_{1})$
such that $b_{2}\in({\bf O}^{*})^{\times}$ and

$h_{2}(b_{2})_{\mathcal{Q}_{1}}=h_{2}(c^{2})_{\mathcal{Q}_{1}}h_{2}(-1)_{\mathcal{R}}.$

By the definiton on $(O^{*},I,L^{2})$-approximation,
$h_{1}(a_{1}b_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ and
$h_{2}(a_{2}b_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$. Since $b_{2}\in({\bf O}^{*})^{\times}$,
$\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K^{*}}(IL)}(b_{2})=\emptyset$.
Lemma 4.5.3 implies that

$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}=[h_{1}(a_{1}):h_{2}(b_{2}^{-1})]_{\mathcal{Q}_{1}}=[h_{1}(b_{1}):h_{2}(c^{-2})]_{\mathcal{Q}_{1}}[h_{1}(b_{1}):h_{2}(-1)]_{\mathcal{R}}.$

∎

######  Lemma 4.5.10.

Let $a\in({\bf A}^{*})^{\times}$ and $I,J\in\mathcal{I}_{K^{*}}$ such that
$\mathcal{P}_{K^{*}}(I)\cap\mathcal{P}_{K^{*}}(J)=\emptyset$. There exists an
$(O^{*},I,J)$-approximation $(b,-1)$ of $(a_{1},-1)$ and $c\in({\bf
O}^{*})^{\times}$ such that
$h_{2}(-1)_{\mathcal{R}}=h_{2}(c^{2})_{\mathcal{R}}$ where
$\mathcal{R}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\setminus\mathcal{P}_{K^{*}}(IJ)}(b)$.

###### Proof.

Follows from Lemma 5.4.5 below and a coordinate-wise argument. ∎

######  Corollary 4.5.11.

Under Setting 4.4.2, let $a\in({\bf A}^{*})^{\times}$ and
$I\in\mathcal{P}_{K^{*}}$ such that
$\mathcal{P}_{K^{*}}(I)\cap\mathcal{P}_{K^{*}}(L)=\emptyset$. Then there exist
small internal subsets $\mathcal{Q}\subseteq\mathcal{P}_{K^{*}}$ which is
disjoint from $\mathcal{Q}\cup\mathcal{P}_{K^{*}}(L)$, $b\in({\bf
A}^{*})^{\times}$ and $c\in({\bf O}^{*})^{\times}$ such that
$[h_{1}(a):h_{2}(-1)]_{\mathcal{P}_{K^{*}}(I)}=[h_{1}(b):h_{2}(c^{2})]_{\mathcal{Q}}$

###### Proof.

Follows from Lemmas 4.5.5 and 4.5.10. ∎

######  Proposition 4.5.12.

Under Setting 4.4.2, for every $a_{1},a_{2}\in({\bf A}^{*})^{\times}$,

$[h_{1}(a_{1}):h_{2}(a_{2})]\in\langle[h_{1}(b_{1}),h_{2}(b_{2})]^{16}\mid
a_{1}\in({\bf A}^{*})^{\times}\ \land\ a_{2}\in({\bf O}^{*})^{\times}\rangle.$

###### Proof.

Let $a_{1},a_{2}\in({\bf A}^{*})^{\times}$. Since
$[h_{1}(a_{1}):h_{2}(a_{2})]$ belongs to the abelian group $D$, it is enough
to prove that

$[h_{1}(a_{1}):h_{2}(a_{2})]\in H:=\langle[h_{1}(b_{1}),h_{2}(b_{2})]^{2}\mid
b_{1}\in({\bf A}^{*})^{\times}\ \land\ b_{2}\in({\bf O}^{*})^{\times}\rangle.$

By symmetry, Item 2 of Lemma 4.4.1 and Corollary 4.5.7 implies that it is
enough to prove that if $\mathcal{Q}$ is an internal subset of
$\mathcal{P}_{K^{*}}$ which is disjoint from $\mathcal{P}_{K^{*}}(L)$,
$a_{1}\in({\bf A}^{*})^{\times}$ and $a_{2}\in({\bf O}^{*})^{\times}$, then
$[h_{1}(a_{1}):h_{2}(a_{2})]_{\mathcal{Q}}\in H$.

Corollary 4.5.9 implies that it is enough to prove that if $\mathcal{Q}$ is an
internal subset of $\mathcal{P}_{K^{*}}$ which is disjoint from
$\mathcal{P}_{K^{*}}(L)$ and $a_{1}\in({\bf A}^{*})^{\times}$, then
$[h_{1}(a_{1}):h_{2}(-1)]_{\mathcal{Q}}\in H$. This is true by Corollary
4.5.11. ∎

####  4.5.3 Every Steinberg symbol is trivial

######  Lemma 4.5.13.

Let $I,J\in\mathcal{I}_{K^{*}}$ be coprime, $a_{1}\in\big{(}{\bf
A}^{*}\big{)}^{\times}$ and $a_{2}\in({\bf O}^{*})^{\times}$. There exist an
$(O^{*},I,J)$-approximation $(b_{1},b_{2})$ of $(a_{1}^{8},a_{2})$ such that
$b_{2}\in({\bf O}^{*})^{\times}$ for every
$P\in\mathcal{R}:=\operatorname{supp}_{\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K}^{*}(IJ)}(b_{1})$,
$\operatorname{val}_{P}(b_{1})=-1$, and an
$(O^{*},\prod_{P\in\mathcal{R}}P,J)$-approximation $(c_{1},c_{2})$ of
$(b_{1},b_{2}^{-2})$ such that such that
$h_{1}(c_{1})_{\mathcal{R}^{c}}\in\operatorname{Spin}_{f_{s}}({\bf O}^{*})$
and $c_{2}\in({\bf O}^{*})^{\times}$.

###### Proof.

Follows from Lemma 5.5.11 below and a coordinate-wise argument. ∎

######  Corollary 4.5.14.

Under Setting 4.4.2, let
$\mathcal{Q}\in\mathcal{P}_{K^{*}}\smallsetminus\mathcal{P}_{K}^{*}(L)$ be a
small internal subset, $a_{1}\in({\bf A}^{*})^{\times}$ and $a_{2}\in({\bf
O}^{*})^{\times}$. Then $([h_{1}(a_{1}),h_{2}(a_{2})]_{\mathcal{Q}})^{16}=1$.

###### Proof.

By Item 9 of Lemma 4.4.5, there exists $I\in\mathcal{I}_{K^{*}}(\mathcal{Q})$
such that for every $d_{1},d_{2}\in({\bf A}^{*})^{\times}$ such that
$h_{1}(a_{1}^{8}d_{1}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})$ and
$h_{2}(a_{2}d_{2}^{-1})_{\mathcal{Q}}\in\operatorname{Spin}_{f_{s}}({\bf
O}^{*})[I]$,