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\title{Definability in the group of infinitesimals of a compact Lie group}
\author{M. Bays \& Y. Peterzil}
\maketitle
\abstract{
We show that for $G$ a simple compact Lie group, the infinitesimal subgroup
$\Goo$ is bi-interpretable with a real closed convexly valued field.
We deduce that for $G$ an infinite definably compact group definable in an
o-minimal expansion of a field, $\Goo$ is bi-interpretable with the disjoint
union of a (possibly trivial) $\Q$-vector space and finitely many (possibly
zero) real closed valued fields.
We also describe the isomorphisms between such infinitesimal subgroups, and
along the way prove that every {\em definable} field in a real closed
convexly valued field $R$ is definably isomorphic to $R$.
}
\newcommand{\sub}{\subseteq}
\section{Introduction}
\label{s:intro}
Let $G$ be compact linear Lie group, by which we mean a compact closed Lie
subgroup of $\GL_d(\R)$ for some $d \in \N$, e.g.\ $G=SO_d(\R)$.
By Fact~\ref{f:chevalley}(i) below, any compact Lie group is isomorphic to a linear
Lie group.
Let $\Ru \succ \R$ be a real closed field properly extending the real field.
By Fact~\ref{f:chevalley}(ii), $G$ is the group of $\R$-points of an algebraic
subgroup of $\GL_d$ over $\R$, and we write $G(\Ru) \leq \GL_d(\Ru)$ for the
$\Ru$-points of this algebraic group.
Let $\st : \O \rightarrow \R$ be the standard part map,
the domain $\O = \bigcup_{n \in \N} [-n,n] \subseteq \Ru$ of which is a valuation ring
in $\Ru$.
Let
\[ \m = \st^{-1}(0) = \bigcap_{n \in \N_{>0}} [{ -\frac1n,\frac1n }] \subseteq \Ru \]
be the maximal ideal of $\O$.
A field equipped with a valuation ring, such as $(\Ru;+,\cdot,\O)$, is known
as a {\em valued} {\em field}. The complete first-order theory of $(\Ru;+,\cdot,\O)$
is the theory $\RCVF$ of non-trivially convexly valued real closed fields
\cite{cherlinDickmann}.
Since $G$ is compact, $\st : \O \rightarrow \R$ induces a totally defined homomorphism
$\st_G : G(\Ru) \twoheadrightarrow G$.
% Note: in terms of \U, \st is the ultralimit map,
% so totality follows from the general topological Bourbakian fact that
% ultrafilters converge on compact spaces.
The kernel $\ker(\st_G) \triangleleft G$ is the ``infinitesimal subgroup'' of $G$ in
$\Ru$. Our main results below describe definability in this group.
In fact, by Fact~\ref{f:DCLit}(iv) below, $\ker(\st_G)$ is precisely $\Goo(\Ru)$,
the set of $\Ru$-points of the smallest bounded-index $\bigwedge$-definable
subgroup $\Goo$ of the semialgebraic group $G$.
% Note that this will fall out anyway once we start looking at \Goo as the
% exponential image of the infinitesimals in the lie algebra.
In terms of matrices,
\begin{equation} \label{e:Goo}
\Goo(\Ru) = G(\Ru) \cap (I+\Mat_d(\m)) = \bigcap_{n \in \N_{>0}} ( G(\Ru)
\cap (I + \Mat_d([-\frac1n,\frac1n]) )
\end{equation}
(where we write $\Mat_d(X)$ for the set of matrices with entries in a set $X
\subseteq \Ru$). Note in particular that $\Goo(\Ru)$ is definable in the valued field
$(\Ru;+,\cdot,\O)$.
We adopt the convention that ``definable'' always means definable with
parameters: a {\em definable} {\em set} in a structure $\A=(A;\ldots )$ is a subset of a
Cartesian power $A^n$ defined by a first-order formula in the language of $\A$
with parameters from $A$.
Recall that an {\em interpretation} of a (one-sorted) structure $\A$ in a
structure $\B$ is a bijection of the universe of $\A$ with a definable set
in $\B$, or more generally with the quotient of a definable set by a definable
equivalence relation, such that the image of any definable set in $\A$ is
definable in $\B$. A {\em definition} of $\A$ in $\B$ is an interpretation whose
codomain is a definable set rather than a definable quotient; in fact, these
are the only interpretations which will arise in the main results of this
article. There is an obvious notion of composition of interpretations.
A pair of interpretations of $\A$ in $\B$ and of $\B$ in $\A$ form a
{\em bi-interpretation} if the composed interpretations of $\A$ in $\A$ and of
$\B$ in $\B$ are definable maps in $\A$ and $\B$ respectively.
If $\A$ and $\B$ are bi-interpretable,
then the structure induced by $\B$ on the image of $\A$ is precisely the
structure of $\A$,
i.e.\ the definable sets are the same whether viewed in $\A$ or in $\B$.
Indeed, if $(f,g)$ is a bi-interpretation, then given a subset $X \subseteq \A^n$, if
$X$ is $\A$-definable then $f(X)$ is $\B$-definable, and conversely if $f(X)$
is $\B$-definable then $g(f(X))$ and hence $X$ is $\A$-definable.
We adopt the following terminology throughout the paper.
A {\em linear} Lie group is a closed Lie subgroup of $\GL_d(\R)$ for some $d \in
\N$.
A Lie group is {\em simple} if it is connected and its Lie algebra is simple; a
Lie algebra is simple if it is non-abelian and has no proper non-trivial
ideal.
(A simple Lie group may have non-trivial discrete centre; however, the
underlying abstract group of a {\em centreless} simple Lie group is simple.)
Our first main result describes definability in $\Goo(\Ru)$ for $G$ simple:
\begin{theorem} \label{t:mainBiterp}
% arxiv's AutoTeX requires this for ref from section title:
\label{T:MAINBITERP}
Let $G$ be a simple compact linear Lie group.
Let $\Ru \succ \R$ be a proper real closed field extension of $\R$.
Then the inclusion $\iota : \Goo(\Ru) \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} G(\Ru)$, viewed as a definition
of the group $(\Goo(\Ru);*)$ in the valued field $(\Ru;+,\cdot,\O)$, can be
extended to a bi-interpretation: there is a definition $\theta$ of the
valued field $(\Ru;+,\cdot,\O)$ in the group $(\Goo(\Ru);*)$ such that the
pair $(\iota,\theta)$ form a bi-interpretation.
In particular, the $(\Goo(\Ru);*)$-definable subsets of powers
$(\Goo(\Ru))^n$ are precisely the $(\Ru;+,\cdot,\O)$-definable subsets.
\end{theorem}
\begin{remark}
The bi-interpretation of Theorem~\ref{t:mainBiterp} requires parameters. Indeed,
$\Goo(\Ru)$ has non-trivial inner automorphisms (this follows from
Lemma~\ref{l:Coo} below), which under a hypothetical parameter-free
bi-interpretation would induce definable non-trivial automorphisms of
$(\Ru;+,\cdot,\O)$.
However, no such automorphism exists. We sketch a proof of this, following a
% As we're presenting it now the method seems rather standard\ldots but we
% might as well thank MH anyway
method suggested by Martin Hils. By Fact~\ref{f:semialgebraic}(1),
%First note that definable closure in $(\Ru;+,\cdot,\O)$ is just field
%theoretic relative algebraic closure; this is
%\cite[Theorem~8.1(1)]{Mellor-RCVFEI}, and follows from the Cherlin-Dickman
%quantifier elimination in the language $(+,-,\cdot,0,1,<,|)$, where $x|y
%\Leftrightarrow (x\neq 0 \wedge \frac yx \in \O)$ for $\RCVF = \Th(\Ru;+,\cdot,\O)$.
%Now $(\Ru;+,\cdot,\O)$ is weakly o-minimal, so
if $\sigma$ is an $(\Ru;+,\cdot,\O)$-definable automorphism, then it agrees
on some infinite interval $I$ with an $(\Ru;+,\cdot)$-definable map $f$.
But then for $a,b,c,d \in I$ with $c\neq d$,
$\sigma(\frac{a-b}{c-d}) = \frac{\sigma(a)-\sigma(b)}{\sigma(c)-\sigma(d)} =
\frac{f(a)-f(b)}{f(c)-f(d)}$,
so $\sigma$ is an $(\Ru;+,\cdot)$-definable field automorphism, thus
$\sigma=\id$.
\end{remark}
\begin{remark}
For $\lieG$ a simple centreless compact Lie group, Nesin and Pillay
\cite{NP-compactLie} showed that the group itself, $(G;*)$, is
bi-interpretable with a real closed field. They interpret the field by
finding a copy of $(\SO_3;*)$ and using the geometry of its involutions. A
similar project is carried out in \cite{PPS-simpleAlgGrpsRCF} for definably
simple and semisimple groups in o-minimal structures. In the case of $\Goo$,
we also find a field by first finding a copy of $\SO_3^{00}$, but the
``global'' approach of considering involutions is not available; in fact
$\Goo$ is torsion-free. Instead, we work ``locally'', and obtain the field
by applying the o-minimal trichotomy theorem to a definable interval on a
curve within $\SO_3^{00}$. This kind of local approach was previously
mentioned in an ``added in proof'' remark at the end of
\cite{PPS-simpleAlgGrpsRCF} as an alternative method for the case of
$(G;*)$, but we have to take care to ensure that structure we apply
trichotomy to is definable both o-minimally and in $(\Goo;*)$.
\end{remark}
\begin{remark} \label{r:otherValns}
One might also consider ``smaller'' infinitesimal neighbourhoods
corresponding to larger valuation rings $\O' \supsetneq \O$;
Theorem~\ref{t:mainBiterp} holds for these too. More generally, if
$(\Ru';+,\cdot,\O') \vDash \RCVF$ extends $\R$ as an ordered field, then the
group $G(\Ru) \cap \Mat_n(\mu')$ is bi-interpretable as in
Theorem~\ref{t:mainBiterp} with $(\Ru';+,\cdot,\O')$. Indeed, the existence of
suitable parameters for the bi-interpretation is expressed by a sentence in
$\RCVF$ with parameters in $\R$, and since $\RCVF$ is complete we can apply
Theorem~\ref{t:mainBiterp} to deduce the result.
\end{remark}
We prove Theorem~\ref{t:mainBiterp} in \secref{simple}.
In \secref{borel-tits}, we deduce in the spirit of Borel-Tits a
characterisation of the group isomorphisms of groups of the form $\Goo$,
decomposing them as compositions of valued field isomorphisms and isomorphisms
induced by isomorphisms of Lie groups. In particular, this shows that simple
compact $G_1$ and $G_2$ have isomorphic infinitesimal subgroups if and only if
they have isomorphic Lie algebras. The key technical tool is
Theorem~\ref{t:defbleFieldsRCVF}, which shows that there are no unexpected fields
definable in $\RCVF = \Th(\Ru;+,\cdot,\O)$.
In \secref{DC}, we generalise Theorem~\ref{t:mainBiterp} to the setting of a
definably compact group $G$ definable in an o-minimal expansion of a field.
Here, to say that $G$ is \recalldefn{definably compact} means that any
definable continuous map $[0,1) \rightarrow G$ can be completed to a continuous map
$[0,1] \rightarrow G$; we refer to \cite{PS-defbleCompactness} for further details on
this notion.
Define the {\bf disjoint} {\bf union} of 1-sorted structures $M_i$ to be the structure
$(M_i)_i$ consisting of a sort for each $M_i$ equipped with its own structure,
with no further structure between the sorts.
\begin{theorem} \label{t:DCBiterp}
Let $(G;*)$ be an infinite definably compact group definable in a
sufficiently saturated o-minimal expansion $M$ of a field.
Then $(\Goo(M);*)$ is bi-interpretable with the disjoint union of a
(possibly trivial) divisible torsion-free abelian group and finitely many
(possibly zero) real closed convexly valued fields.
\end{theorem}
To indicate why this is the correct statement, let us note that it can not be
strengthened to bi-interpretability with a single real closed valued field as
in Theorem~\ref{t:mainBiterp}: one reason is that $G$ could be commutative, and then
$(\Goo;*)$ is just a divisible torsion free abelian group and thus does not
even define a field; another reason is that groups are orthogonal in their
direct product, so e.g.\ if $G = H\times H$ for a semialgebraic compact group
$H$ then, viewing $\Goo$ as a definable set in a valued field
$(\Ru;+,\cdot,\O)$ as above, the diagonal subgroup of $\Goo(\Ru) =
\Hoo(\Ru)\times\Hoo(\Ru)$ is $(\Ru;+,\cdot,\O)$-definable but not
$(\Goo(\Ru);*)$-definable.
Note that Theorem~\ref{t:DCBiterp} applies in particular to an arbitrary compact
linear Lie group, since by Fact~\ref{f:chevalley}(ii) any such group is definable in
the real field, and is definably compact.
\subsection{Acknowledgements}
We would like to thank Mohammed Bardestani, Alessandro Berarducci, Emmanuel
Breuillard, Itay Kaplan, and Martin Hils for useful discussion.
We would also like to thank the Institut Henri Poincar\'e and the organisers of
the trimester ``Model theory, combinatorics and valued fields'', where some of
this work was done.
We also thank the anonymous referee for suggestions which improved the
presentation of this paper substantially.
Greg Cherlin and Ali Nesin, on different occasions, have asked about the
abstract group structure of the infinitesimal subgroup of a compact simple Lie
group. This article can be seen as a partial answer to their questions.
Finally, we would like to mention the following question of Itay Kaplan which
initiated the discussions which led to our results (even though our results have in
the end almost nothing to do with the question): is any $\bigwedge$-definable field
in an NIP theory itself NIP as a pure field?
% any $\bigwedge$-definable field in o-minimal is definable?
% any $\bigwedge$-definable group in o-minimal is NIP?
\section{Preliminaries}
\label{s:prelims}
\subsection{Notation}
We consider $\Ru$ and $\R$ as fields and $\Goo(\Ru)$ as a group, thus
``$\Goo(\Ru)$-definable'' means definable in the pure group $(\Goo(\Ru);*)$,
and ``$\Ru$-definable'' means definable in the field $(\Ru;+,\cdot)$, while we
write ``$(\Ru;\O)$-definable'' for definability in the valued field
$(\Ru;+,\cdot,\O)$.
We use exponential notation for group conjugation, $g^h := hgh^{-1}$.
We write group commutators as $(a,b) := aba^{-1}b^{-1}$, reserving $[X,Y]$ for
the Lie bracket.
For $G$ a group, we write $(G,G)_1$ for the set of commutators in $G$,
$(G,G)_1 := \{ (g,h) : g,h \in G \}$,
and we write $(G,G)$ for the commutator subgroup, the subgroup generated by
$(G,G)_1$.
For $G$ a group and $A$ a subset, we write the centraliser of $A$ in $G$ as
$C_G(A) = \{ g \in G : \forall a \in A.\;(g,a) = e \}$,
and we write $Z(G)$ for the centre $Z(G)=C_G(G)$.
Similarly for $\g$ a Lie algebra and $A$ a subset, we write
the centraliser of $A$ in $\g$ as $C_\g(A) = \{ X \in \g : \forall Y \in A.\;
[X,Y] = 0 \}$.
We write $C_G(g)$ for $C_G(\{g\})$ and $C_\g(X)$ for $C_\g(\{X\})$.
\subsection{Compact Lie groups}
Proofs of the statements in the following Fact can be found in \cite{OV} as
Theorem~5.2.10 and Theorem~3.4.5 respectively.
\begin{fact}[Chevalley] \label{f:chevalley}
\begin{enumerate}[(i)]\item Any compact Lie group $\lieG$ is linear; that is, $\lieG$ is
isomorphic to a Lie subgroup of $\GL_d(\R)$ for some $d$.
\item Compact linear groups are algebraic; that is, any compact subgroup of
any $\GL_d(\R)$ is of the form $G(\R)$ for some algebraic subgroup $G \leq
\GL_d$ over $\R$.
\end{enumerate}
\end{fact}
\begin{lemma} \label{l:Coo}
Suppose $H$ is a connected closed Lie subgroup of a compact linear Lie group
$G$.
Then $C_{G(\Ru)}(H(\Ru)) = C_{G(\Ru)}(H^{00}(\Ru))$.
\end{lemma}
\begin{proof}
Since $G$ and $H$ are algebraic by Fact~\ref{f:chevalley}(ii), the conclusion can
be expressed as a first-order sentence in the complete theory $\RCVF_\R$ of
a non-trivially convexly valued real closed field extension
$(\Ru;+,\cdot,\O)$ of the trivially valued field $\R$, so we may assume
without loss that $\Ru$ is $\aleph_1$-saturated.
Suppose $x \in C_{G(\Ru)}(H^{00}(\Ru))$.
It follows from $\aleph_1$-saturation of $\Ru$ that $x \in
C_{G(\Ru)}(U(\Ru))$ for some $\R$-definable neighbourhood of the identity $U
\subseteq H$, since $H^{00}(\Ru)$ is the intersection of such (e.g.\ as in
\eqnref{Goo}).
Now we could argue from general results on o-minimal groups (see
\cite[Lemma~2.11]{P-groupsFieldsOMin}) and connectedness that
$C_{G(\Ru)}(U(\Ru)) = C_{G(\Ru)}(H(\Ru))$, but we can also argue directly as
follows.
$H$ is compact and connected, thus is generated in finitely many steps from
$U$. Since $\Ru$ is an elementary extension of $\R$, it follows that
$H(\Ru)$ is generated by $U(\Ru)$.
Hence $x \in C_{G(\Ru)}(H(\Ru))$.
\end{proof}
\subsection{$\SO(3)$}
We recall some elementary facts about the group of spatial rotations
$\SO(3)=\SO_3(\R)$, its universal cover $\Spin(3)$, and their common Lie
algebra $\so(3)$, as discussed in e.g.\ \cite[Chapter~6]{woit}.
Any rotation $g \in \SO(3)$ can be completely described as a planar rotation
$\alpha \in \SO(2)$ around an axis $L$, where $L$ is a ray from the origin in
$\R^3$. We can identify such a ray $L$ with the unique element of the unit
sphere $S^2$ which lies on the ray. Writing $\rho : \SO(2) \times S^2 \twoheadrightarrow
\SO(3)$ for the corresponding map, the only ambiguities in this description
are that $\rho(\alpha,L) = \rho(-\alpha,-L)$, and the trivial rotation is $e =
\rho(0,L)$ for any $L$. The centraliser in $\SO(3)$ of a non-trivial
non-involutary rotation $g=\rho(\alpha,L)$, $2\alpha \neq 0$, is the subgroup
$C_{\SO(3)}(g) = \rho(\SO(2),L) \cong \SO(2)$ of rotations around $L$. The
conjugation action of $\SO(3)$ on itself is by rotation of the axis:
$\rho(\alpha,L)^g = \rho(\alpha,gL)$, where $gL$ is the image of $L$ under the
canonical (matrix) left action of $g$ on $\R^3$. The map $\rho$ is continuous
and $\R$-definable.
This description transfers to non-standard rotations: given $\Ru \succ \R$,
$\rho$ extends to a map $\rho : \SO_2(\Ru) \times S^2(\Ru) \twoheadrightarrow \SO_3(\Ru)$.
The infinitesimal rotations are then the rotations about any (non-standard)
axis by an infinitesimal angle;
i.e.\ $\rho$ restricts to a map $\SO_2^{00}(\Ru) \times S^2(\Ru) \twoheadrightarrow
\SO_3^{00}(\Ru)$.
The universal group cover of $\SO(3)$ is denoted $\Spin(3)$; the corresponding
continuous covering homomorphism $\pi : \Spin(3) \rightarrow \SO(3)$ is a local
isomorphism with kernel $Z(\Spin(3)) \cong \pi_1(\SO(3)) \cong \Z/2\Z$. Since
$\Spin(3)$ is compact, by Fact~\ref{f:chevalley}(i) we may represent it as $\Spin(3)
= \Spin_3(\R)$ where $\Spin_3$ is a linear Lie group. Since $\pi$ is an
$\R$-definable local isomorphism, it induces an isomorphism of Lie algebras
and an isomorphism of infinitesimal subgroups $\Spin_3^{00}(\Ru) \cong
\SO_3^{00}(\Ru)$, with respect to which the action by conjugation of $g \in
\Spin_3(\Ru)$ on $\Spin_3^{00}(\Ru)$ agrees with the action by conjugation of
$\pi(g)$ on $\SO_3^{00}(\Ru)$.
The Lie algebra $\so(3) \cong \su(2)$ has $\R$-basis $\{H,U,V\}$ and bracket
relations
\begin{equation} \label{e:so3}
[U,V]=H,\; [H,U]=V,\; [V,H]=U .
\end{equation}
The adjoint action $\Ad$ of $g \in \SO(3)$ on $\so(3)$ is by the left matrix
action with respect to this basis, $\Ad_g(X) = gX$, and similarly for $g \in
\Spin(3)$ the adjoint representation is $\pi$, i.e.\ $\Ad_g(X) = \pi(g)X$.
\subsection{Lie theory in o-minimal structures}
\label{s:nsLie}
We recall briefly the Lie theory of a group $G$ definable in an o-minimal
expansion of a real closed field $R$; see \cite{PPS-defblySimple} for further
details, but in fact we apply it only to linear algebraic groups, for which it
agrees with the usual theory for such groups. The Lie algebra of $G$ is the
tangent space at the identity $\g = L(G) = T_e(G)$, a finite dimensional
$R$-vector space. For $h \in G(R)$, define $\Ad_h : \g \rightarrow \g$ to be the
differential of conjugation by $h$ at the identity, $\Ad_h := d_e(\cdot^h)$,
and define $\ad : \g \rightarrow \End_R(\g)$ as the differential of $\Ad : G \rightarrow
\Aut_R(\g)$ at the identity, $\ad := d_e(\Ad)$. Then the Lie bracket of $\g$
is defined as $[X,Y] := \ad_X(Y)$.
The statements above about the adjoint action of $\SO_3$ and $\Spin_3$ on
$\so_3$ transfer to $\Ru$: for $X \in \so_3(\Ru)$, we have that $\Ad_g(X) = gX$
for $g \in \SO_3(\Ru)$, and $\Ad_g(X) = \pi(g)X$ for $g \in \Spin_3(\Ru)$.
\section{Proof of Theorem~\ref{t:mainBiterp}}
\label{s:simple}
Let $G$ and $\Ru$ be as in Theorem~\ref{t:mainBiterp}, namely $G$ is a simple
compact linear Lie group and $\Ru$ is a proper real closed field
extension\footnote{
Readers familiar with model theory might be made more comfortable by an
assumption that $\Ru$ is (sufficiently) saturated. They may in fact freely
assume this, since the conclusion of the theorem can be seen to not depend on
the choice of $\Ru$.}
of $\R$.
% |this is true but we don't need it.
%Lemma:
% If Theorem~\ref{t:mainBiterp} holds for some proper real closed field extension
% $\Ru \succ \R$ then it holds for any proper extension $\Ru' \succ \R$.
% .
%Proof:
% The bi-interpretation of $(\Ru;+,\cdot,\O)$ with $(\Goo(\Ru);*)$ is given by
% $(\Ru;+,\cdot,\O)$-definable sets for the interpretations of the universes
% and the predicates and functions, and for the definable bijections. Now the
% fact that these various definable sets do constitute a bi-interpretation is
% an elementary property of the tuple $c$ of parameters defining these various
% sets, given by some $(+,\cdot,\O)$-formula $\phi(x)$ with parameters from
% $\R$.
%
% Now the theory RCVF of non-trivially valued real closed fields is complete
% and model complete,
% so $\Ru$ and $\Ru'$ are elementarily equivalent as valued fields with
% parameters for $\R$. So $\Ru' \vDash \exists x.\; \phi(x)$, so
% Theorem~\ref{t:mainBiterp} holds also for $\Ru'$.
% .
%
%So we {\bf assume} without loss that $\Ru$ is $\aleph_1$-saturated.
In this section, we write $\Goo$ for the group $\Goo(\Ru)$.
We first give an overview of the proof. We begin in \secref{SO300} by finding a
copy of $\SO(3)$ or $\Spin(3)$ in $G$ which is definable in such a way that
its infinitesimal subgroup is $\Goo$-definable. A reader who is interested
already in the case $G=\SO(3)$ may prefer to skip that section on a first
reading. In \secref{interval} we use the structure of $\SO_3^{00}$ to find in
it an interval on a centraliser which is definable both in the group and the
field. In \secref{trich} we see that the non-abelianity of $G$ endows this
interval with a rich enough structure to trigger the existence of a field by
the o-minimal trichotomy theorem. Finally, in \secref{Goo}, we use an adjoint
embedding to see the valuation on this field and obtain bi-interpretability.
\subsection{Finding an $\SO_3^{00}$}
\label{s:SO300}
In this subsection, we find a copy of $\so(3)$ in the Lie algebra of $G$ which
is the Lie algebra of a Lie subgroup $S \leq G$, and which moreover is defined
in such a way that the infinitesimal subgroup $\Soo \leq \Goo$ is
$\Goo$-definable.
Let $\g_0 := L(\lieG)$ be the Lie algebra of $\lieG$.
\begin{lemma} \label{l:so3}
There exist Lie subalgebras $\s \leq \s' \leq \g_0$ such that
\begin{enumerate}[(i)]\item $\s' = C_{\g_0}(C_{\g_0}(\s'))$
\item $\s = [\s',\s']$
\item $\s \cong \so(3)$.
\end{enumerate}
\end{lemma}
\begin{proof}
In this proof, and in this proof alone, we assume familiarity with the basic
theory and terminology of the root space decomposition of a semisimple Lie
algebra. This can be found in e.g.\ \cite[\S\S II.4, II.5]{Knapp}.
%We use some standard results on the root space decomposition of a simple
%compact Lie group. These can be found in e.g.\ \cite[\S\S II.4, II.5, IV.4,
%IV.5]{Knapp}.
We write $V^*$ for the dual space of a vector space $V$.
Let $\g := \g_0 \otimes_\R \C$ be the complexification of $\g_0$, and let
$\overline{\cdot} : \g \rightarrow \g$ be the corresponding complex conjugation.
Let $\h_0 \leq \g_0$ be a Cartan subalgebra of $\g_0$, meaning that the
complexification $\h := \h_0 \otimes_\R \C$ is a Cartan subalgebra of $\g$; we
can take $\h_0 := L(\lieT)$ where $\lieT$ is a maximal torus of $\lieG$
(where a torus of $\lieG$ is a Lie subgroup isomorphic to a power of the
circle group).
Now since $\lieG$ is simple, $\g$ is semisimple and so admits a root space
decomposition $\g = \h \oplus \bigoplus_{\alpha \in \Delta} \g_\alpha$ where each root
space $\g_\alpha$ is the 1-dimensional eigenspace of $\ad(\h)$ with
eigenvalue $\alpha \in \h^* \setminus \{0\}$, i.e.\ if $H \in \h$ and $X \in \g_\alpha$
then $[H,X] = \alpha(H)X$. The roots $\alpha$ span $\h^*$. If $\alpha \in
\Delta$ then $\C\alpha \cap \Delta = \{\alpha,-\alpha\}$.
We have $[\g_\alpha,\g_{-\alpha}] \leq \h$.
Since $\lieG$ is compact, each root $\alpha \in \Delta$ takes purely
imaginary values on $\h_0$, thus $i\alpha\negmedspace\restriction_{\h_0} \in \h_0^*$, and the
subalgebra $\s_{\alpha}$ of $\g_0$ generated by $\l_\alpha := \g_0 \cap
(\g_{\alpha} \oplus \g_{-\alpha})$ is isomorphic to $\so(3)$ (see
\cite[Proposition~26.4]{FultonHarris}, or \cite[(4.61)]{Knapp}).
Explicitly, if $E_\alpha \in \g_\alpha \setminus \{0\}$, so $\overline{E_\alpha} \in
\g_{-\alpha}$, then $\l_\alpha$ is spanned by $U_\alpha := iE_\alpha -
i\overline{E_\alpha}$ and $V_\alpha := E_\alpha + \overline{E_\alpha}$.
Then for $H \in \h_0$ we have
\begin{equation} \label{e:adl}
[H,U_\alpha] = i\alpha(H) V_\alpha \;\;\text{and}\;\; [H,V_\alpha] =
-i\alpha(H) U_\alpha ,
\end{equation}
and $[U_\alpha,V_\alpha] = 2i[E_\alpha,\overline{E_\alpha}] \in \s_\alpha
\cap \h_0$. Using that $\g_0$ admits an $\Ad(\lieG)$-invariant inner
product, one can argue (see the proof of \cite[(4.56)]{Knapp} for details)
that $\alpha([E_\alpha,\overline{E_\alpha}]) < 0$, and hence that after
renormalising $E_\alpha$, the $\R$-basis
$\{[U_\alpha,V_\alpha], U_\alpha, V_\alpha\}$
for $\s_\alpha$ satisfies the standard bracket relations \eqnref{so3} of
$\so(3)$.
Let $\alpha_0 \in \Delta$.
Let $\s := \s_{\alpha_0}$ and $\l := \l_{\alpha_0}$,
and let $\s' := \h_0 \oplus \l$.
It follows from the bracket relations above that $\s'$ is a subalgebra of
$\g_0$, and the commutator subalgebra $[\s',\s']$ is precisely $\s$.
%, and moreover this is equal to the set of commutators (rather than just to
%the subspace spanned by them), since this holds for $\s \cong \so(3)$.
% ^true, but we don't use it.
It remains to show that $\s' = C_{\g_0}(C_{\g_0}(\s'))$.
It follows from \eqnref{adl} and $C_{\g_0}(\h_0)=\h_0$ that
$C_{\g_0}(\s') = \h_0 \cap C_{\g_0}(\l) = \ker(\alpha_0\negmedspace\restriction_{\h_0})$.
Now for $\alpha' \in \Delta$ and $W \in \g_{\alpha'} \setminus \{0\}$, we have
$[W,\ker(\alpha_0\negmedspace\restriction_{\h_0})] = 0$ if and only if $\ker(\alpha'\negmedspace\restriction_{\h_0}) \supseteq
\ker(\alpha_0\negmedspace\restriction_{\h_0})$;
since $i\alpha'\negmedspace\restriction_{\h_0},i\alpha_0\negmedspace\restriction_{\h_0} \in \h_0^*$, this holds if and
only if $\alpha' \in \R\alpha_0 \cap \Delta = \{\alpha_0,-\alpha_0\}$.
Thus
$C_{\g_0}(C_{\g_0}(\s')) = \g_0 \cap (\h + \g_{\alpha_0} + \g_{-\alpha_0}) =
\s'$, as required.
\end{proof}
Before the next lemma we recall some facts and terminology from Lie theory.
An \recalldefn{integral subgroup} of $\lieG$
% Also known as an \underline{analytic subgroup} (e.g.\ in Knapp)
is a connected Lie group which is an abstract subgroup of $\lieG$ via an
inclusion map which is an immersion. An integral subgroup is a Lie subgroup
iff it is closed in $\lieG$ (see
\cite[Proposition~III.6.2.2]{Bourbaki-Lie}). The map $\lieH \mapsto L(\lieH)$ of
taking the Lie algebra is a bijection between integral subgroups of $\lieG$
and Lie subalgebras of $\g_0=L(\lieG)$ (see
\cite[Theorem~III.6.2.2]{Bourbaki-Lie}).
\begin{lemma} \label{l:spin3}
There exists a closed Lie subgroup $S \leq \lieG$ isomorphic to either
$\Spin(3)$ or $\SO(3)$, such that the infinitesimal subgroup $\Soo = S(\Ru)
\cap \Goo \cong \SO_3^{00}$ is $\Goo$-definable as a subgroup of $\Goo$.
\end{lemma}
\begin{proof}
Let $\s$ and $\s'$ be as in Lemma~\ref{l:so3}.
Let $\lieS$ and $\lieS'$ be the integral subgroups of $\lieG$ with Lie
algebras $\s$ and $\s'$, respectively.
Since $\s' = C_{\g_0}(C_{\g_0}(\s'))$, we have by
\cite[Proposition~III.9.3.3]{Bourbaki-Lie} that $\lieS' =
C_{\lieG}(C_{\lieG}(\lieS'))$.
In particular, $\lieS'$ is closed.
Since $[\s',\s']=\s$ and $\s$ is an ideal in $\s'$, we have by
\cite[Proposition~III.9.2.4]{Bourbaki-Lie} that the commutator subgroup
$(\lieS',\lieS')$ is equal to $\lieS$.
Now $\s$ is isomorphic to $\so(3)$ and $\Spin(3)$ is simply connected, thus
(by \cite[Theorem~III.6.3.3]{Bourbaki-Lie}) $\lieS$ is the image of a
non-singular homomorphism $\Spin(3) \rightarrow \lieG$ with central kernel,
and so since $\Spin(3)$ is compact and its centre is of order 2, $\lieS$ is
a closed Lie subgroup of $\lieG$ isomorphic either to $\Spin(3)$ or to
$\SO(3) \cong \Spin(3) / Z(\Spin(3))$.
%Considering the structure of these groups, we observe that
%$\lieS=(\lieS,\lieS)_1$; one way to see this is to note that given $g$ we
%can find an involution $\tau$ such that $g^\tau=g^{-1}$, so $(g,\tau) =
%g^\tau g^{-1} = g^{-2}$, and the squaring map is surjective on $\lieS$
%since it is surjective on $\SO(2)$.
%So since $\lieS \leq \lieS'$, we conclude that $\lieS=(\lieS',\lieS')_1$.
%Finally, we consider the same construction once more, at the level of
%$\Goo$.
Now $S$ is a closed Lie subgroup of $G$, and thus is a compact linear Lie
group, and its infinitesimal subgroup $\Soo$ is correspondingly the subgroup
$S(\Ru) \cap \Goo$ of $\Goo$. The same goes for the closed Lie subgroups
$S'$ and $S'' := C_G(S')$ of $G$.
\begin{claim}
$(S')^{00} = C_{\Goo}(C_{\Goo}(S'(\Ru)))$.
\end{claim}
\begin{proof}
%$(S')^{00}
%= S'(\Ru) \cap \Goo
%= C_{G(\Ru)}(C_{G(\Ru)}(S'(\Ru))) \cap \Goo
%= C_{\Goo}(C_{G(\Ru)}(S'(\Ru)))
%= C_{\Goo}(S''(\Ru))$, and by Lemma~\ref{l:Coo}
%$C_{\Goo}(S''(\Ru))
%= C_{\Goo}((S'')^{00})
%= C_{\Goo}(C_{G(\Ru)}(S'(\Ru)) \cap \Goo)
%= C_{\Goo}(C_{\Goo}(S'(\Ru)))$.
\begin{align*} (S')^{00}
&= S'(\Ru) \cap \Goo \\
&= C_{G(\Ru)}(C_{G(\Ru)}(S'(\Ru))) \cap \Goo \\
&= C_{\Goo}(C_{G(\Ru)}(S'(\Ru))) \\
&= C_{\Goo}(S''(\Ru)) \\
&= C_{\Goo}((S'')^{00}) & \text{(by Lemma~\ref{l:Coo})} \\
&= C_{\Goo}(C_{G(\Ru)}(S'(\Ru)) \cap \Goo) \\
&= C_{\Goo}(C_{\Goo}(S'(\Ru))) .\end{align*}
\end{proof}
By the Descending Chain Condition for definable groups in o-minimal
structures, \cite[Corollary~1.16]{PPS-defblySimple}, there exists a finite
set $X_0\subseteq C_{G^{00}}(S')\subseteq G^{00}$ such that
$C_G(C_{G^{00}}(S'))=C_G(X_0)$. Thus,
$$S^{00}=C_G(X_0)\cap G^{00}=C_{G^{00}}(X_0)$$ is $\Goo$-definable.
%Now for any $A \subseteq G(\Ru)$, we have $C_{G(\Ru)}(A) = C_{G(\Ru)}(A_0)$ for
%some finite $A_0 \subseteq A$; this can be seen as a matter of Noetherianity of
%linear algebraic groups, or as a special case of a general result for
%groups definable in o-minimal structures
%\cite[Corollary~1.16]{PPS-defblySimple}.
%Thus $(S')^{00} = C_{\Goo}(X_0)$ for some finite $X_0 \subseteq C_{\Goo}(S')$.
%Hence $(S')^{00}$ is $\Goo$-definable.
Meanwhile, for the commutator subgroup, we have
$((S')^{00},(S')^{00}) \subseteq (S',S')^{00} = S^{00}$.
Now $(\SO_3^{00},\SO_3^{00})_1 = \SO_3^{00}$;
in fact \cite{dAndreaMaffei} proves $(\Goo,\Goo)_1=\Goo$ for any compact
semisimple Lie group, but one can also see it directly in this case,
since $(\SO_3^{00},\SO_3^{00})_1$ is invariant under conjugation by
$\SO_3(\Ru)$ and can be seen to contain infinitesimal rotations of all
infinitesimal angles.
So since $(S')^{00} \supseteq S^{00} \cong \SO_3^{00}$,
we have $((S')^{00},(S')^{00})_1=S^{00}$.
Hence $\Soo$ is $\Goo$-definable.
Thus $S$ is as required.
\end{proof}
\subsection{Finding a group interval in $\Soo$}
\label{s:interval}
Let $S \leq G$ be as given by Lemma~\ref{l:spin3}. Now $S$ is isomorphic to $\SO_3$ or
$\Spin_3$ via an isomorphism which has compact graph and hence is
$\R$-definable, so let the $\R$-definable map $\pi : S \twoheadrightarrow \SO_3$ be this
isomorphism or its composition with the universal covering homomorphism
respectively. Then $\pi$ induces an isomorphism $\pi : S^{00} \xrightarrow{\cong}
\SO_3^{00}$.
From now on, we work in $\Ru$, and consider $S$, $G$, $\SO_2$, $\SO_3$, and
$\Spin_3$ as $\R$-definable (linear algebraic) groups rather than as Lie
groups; moreover, {\bf we write $S$ for $S(\Ru)$}, and similarly with $G$,
$\SO_2$, $\SO_3$, and $\Spin_3$.
Let $g \in \Soo \setminus \{e\}$. Then $C_S(g) = \pi^{-1}(C_{\SO_3}(\pi(g)))$, and
$C_{\SO_3}(\pi(g)) \cong \SO_2$ since $g$ is not torsion.
Thus the circular order on $\SO_2$ induces\footnote{
Explicitly, say $\pi(g) = \rho(\alpha,L)$; then if $\pi_{(1,2)} : \SO_2 \rightarrow
\Ru$ extracts the top-right matrix element,
then
$x_1 < x_2 \Leftrightarrow \pi_{(1,2)}(\beta_1) < \pi_{(1,2)}(\beta_2)$ where $\pi(x_i) =
\rho(\beta_i,L)$
defines an order on such a neighbourhood of the identity in $C_S(g)$.
}
an $\Ru$-definable linear order on a neighbourhood of the identity in $C_S(g)$
containing $C_{\Soo}(g)$, and the induced linear order on $C_{\Soo}(g)$ makes
it a linearly ordered abelian group.
Moreover, the order topology on $C_{\Soo}(g)$ coincides with the group
topology.
We may assume that $g$ is positive with respect to this ordering.
\begin{lemma} \label{l:groupInterval}
There exists an open interval $J \subseteq C_{\Soo}(g)$ containing $e$ such that
$J$ and the restriction to $J$ of the order on $C_{\Soo}(g)$ are both
$\Soo$-definable and $\Ru$-definable.
\end{lemma}
\begin{proof}
First, consider the $\Ru$-definable set
$I := g^Sg^S \cap C_S(g)$,
where $g^Sg^S := \{ g^{a_1}g^{a_2} : a_1,a_2 \in S \}$.
Since $\Soo$ is normal in $S$, in fact $I \subseteq C_{\Soo}(g)$.
Say a subset $X$ of a group is \recalldefn{symmetric} if it is closed
under inversion, i.e.\ $X^{-1}=X$.
\begin{claim}
$I$ is a closed symmetric interval in $C_{\Soo}(g)$.
\end{claim}
\begin{proof}
Recall that a definable set in an o-minimal structure is
\recalldefn{definably connected} if it is not the union of disjoint
open definable subsets.
$X := g^Sg^S$ is the image under a definable continuous map of the
definably connected closed bounded set $(g^S)\times(g^S)$, and hence
(\cite[1.3.6,6.1.10]{vdD-omin}) $X$ is closed and definably connected.
Now $X$ is invariant under conjugation by $S$. Thus $\pi(X) \subseteq \SO_3^{00}$
consists of the rotations around arbitrary axes by the elements of some
$\Ru$-definable set $\Theta \subseteq \SO_2^{00}$, i.e.\ $\pi(X) =
\rho(\Theta,S^2(\Ru))$. Since $\rho(\theta,L)^{-1} = \rho(\theta,-L)$, it
follows that $\pi(X)$ is symmetric, and hence $\Theta$ is symmetric.
Similarly, $\pi(g^S)$ is symmetric, and hence $e \in \pi(X)$ and so $e \in
\Theta$.
Recall that if $L$ is the axis of rotation of $\pi(g)$ (i.e.\ $\pi(g) =
\rho(\alpha,L)$ for some $\alpha$), then $\pi(C_S(g)) = \rho(\SO_2,L)$. So
then $\pi(I)=\rho(\Theta,L)$. Since $\pi(X)$ is closed and definably
connected, $\Theta$ is of the form $\Theta' \cup \Theta'^{-1}$ where
$\Theta'$ is a closed interval. Thus since $\Theta$ contains the identity,
$\Theta$ is itself a closed symmetric interval. So $I$ is a closed
symmetric interval in $C_{\Soo}(g)$.
\end{proof}
Say $I=[h^{-1},h]$.
Write the group operation on $C_{\Soo}(g)$ additively.
For $g_1,g_2 \in \Soo$, let $Y_{g_1,g_2} := g_1^{\Soo} g_2^{\Soo}\cap
C_{\Soo}(g)$ (an $\Soo$-definable set). Clearly, $Y_{g_1,g_2}\subset
I=[-h,h]$.
\begin{claim}
There exist\footnote{
Although this existence statement suffices for our purposes, in fact one
may calculate that we may take $h=g^2$ and $g_1=g=g_2$.
This can be seen by combining the proof of the claim with the following
observations on $\Spin_3$ considered as the group of unit quaternions.
Firstly, if $a,b \in \Spin_3(\R)$ have the same scalar part,
$\Re(a)=\Re(b)$, then $\Re(a*b) \geq \Re(a*a)$, with equality iff $a=b$.
% Indeed, let g = a + dk with d > 0,
% and h = a + b'i + c'j + d'k.
% Then Re(g*h) = a^2 - d'd \geq a^2 - d^2 = Re(h*h)
% because b'^2+c'^2+d'^2 = 1 - a^2 = d^2, so d' \leq d,
% and we have equality iff d=d'.
Secondly, conjugation in $\Spin_3$ preserves scalar part.
Finally, the order on $C_{\Soo}(g)$ agrees (up to inversion) with the
order on the scalar part.
}
$h'' < h$ and $g_1,g_2 \in \Soo$ such that the interval $(h'',h]$ is
contained in $Y_{g_1,g_2}$.
\end{claim}
\begin{proof}
Consider the map $f:S^2 \rightarrow S$ defined by $f(a_1,a_2) := g^{a_1}g^{a_2}$.
By definable choice in $\Ru$,
$f$ admits a $\Ru$-definable section over the set $I$,
and hence $f$ admits a continuous $\Ru$-definable section on an open
interval $\theta : (h',h) \rightarrow S^2$ for some $h' \in C_{\Soo}(g)$.
By definable compactness of $S^2$, $\theta$ extends to a continuous
$\Ru$-definable section $\theta : (h',h] \rightarrow S^2$.
Say $\theta(h) = (a_1,a_2) \in S^2$.
Let $g_i := g^{a_i}$ for $i=1,2$. Note that $g_i \in \Soo$.
So by continuity of $\theta$, for some $h'' < h$ we have
\[ (h'',h] \subseteq \theta^{-1}((\Soo a_1)\times(\Soo a_2)) \subseteq f((\Soo
a_1)\times(\Soo a_2)) = g_1^{\Soo}g_2^{\Soo} .\]
\end{proof}
Thus
\[ P := h - Y_{g_1,g_2} \subseteq h - I = [0,2h] \]
is an $\Soo$-definable subset of the non-negative part of $C_{\Soo}(g)$.
So set $p := h-h''>0$. Note that $[0,p) \subseteq P \subseteq [0,2h]$.
It is now easy to see that the open interval $(0,p)$ is equal to $P\cap
(p-P)$, thus is definable in $\Soo$.
So $J := (-p,p) = [0,p) \cup -[0,p)$ and its order are $\Soo$-definable,
since for $a,b \in J$ we have $a\geq b$ iff $a-b \in [0,p)$.
Finally, $J = (-p,p)$ and its order are also $\Ru$-definable as an interval
in the $\Ru$-definable order on an $\R$-definable neighbourhood of the
identity in $C_S(g)$. This ends the proof of Lemma~\ref{l:groupInterval}.
\end{proof}
\subsection{Defining the field}
\label{s:trich}
Let $J$ be as given by Lemma~\ref{l:groupInterval}. We now return to working with
the full simple compact group $G$, of which $J$ is a subset. Let $n :=
\dim(G)$.
Recall that $J$ consists of
a neighborhood of the identity in the one-dimensional real algebraic group group $C_S(g)$,
thus it is a one-dimensional smooth sub-manifold of $G$. For $h\in G$, the set $J^h$ is an open neighborhood of $e$ in the group $C_S(g)^h$. We let $T_e(J^h)$ denote its tangent space at $e$ with respect to the real closed field $R$.
\begin{lemma} \label{l:chart}
There exists an open neighbourhood $U \subseteq \Goo$ of the identity, an open
interval $e \in J' \subseteq J$, and a bijection
$\phi : J'^n \rightarrow U$
which is both $\Goo$-definable and $\Ru$-definable,
with $\phi(x,e,\ldots ,e) = x$.
\end{lemma}
\begin{proof}
Let $\g = T_e(G)$ be the Lie algebra of $G$ (as discussed in
\secref{nsLie}).
Consider the $\Ru$-subspace $V \leq \g(\Ru)$ generated by
\[ \bigcup_{h \in \Goo} T_e(J^h) = \bigcup_{h \in \Goo} \Ad_h(T_e(J)) .\]
Then $V$ is $\Ad_{\Goo}$-invariant.
Thus $\Ad$ restricts to $\Ad : \Goo \rightarrow \Aut_\Ru(V)$, and hence the
differential at the identity is a map $\ad = d_e \Ad : \g(\Ru) \rightarrow
\End_\Ru(V)$, i.e.\ it follows that $V$ is $\ad_{\g(\Ru)}$-invariant.
So $V$ is a non-trivial ideal in $\g(\Ru)$.
But $\g$ is simple, thus we have $V=\g(\Ru)$.
So say $h_1,h_2,\ldots ,h_n \in \Goo$ are such that $T_e(J^{h_i})$ span $T_eG$.
Conjugating by $h_1^{-1}$, we may assume $h_1=e$.
Define $\phi : J^n \rightarrow \Goo$ by
\[ \phi(x_1,\ldots ,x_n) := x_1^{h_1}x_2^{h_2}\ldots x_n^{h_n} ,\]
which is clearly both $\Goo$-definable and $\Ru$-definable.
Then the differential $d_{(e,\ldots ,e)}\phi : T_eJ^n \rightarrow T_eG(\Ru)$ is an
isomorphism.
Thus by the implicit function theorem (for the real closed field $\Ru$),
for some open interval $e \in J' \subseteq J$,
the restriction $\phi\negmedspace\restriction_{(J')^n}$ is a bijection with some open neighbourhood
$U$ of $e$,
as required.
%(Note we get a very small neighbourhood, valuatively speaking, depending
%on the valuations of $g'$ and the $h_i$. But that's ok.)
\end{proof}
Redefine $J$ to be $J'$ as provided by the lemma.
We shall consider the o-minimal structure obtained by expanding the interval
$(J;<)$ by the pullback of the group operation near $e$ via the chart $\phi$.
As we will now verify, it follows from the non-abelianity of $G$ that the
resulting structure on $J$ is ``rich'' in the sense of the o-minimal
trichotomy theorem \cite{PS-trichotomy} (see below), and so by that theorem it
defines a field. Let us first recall the relevant notions:
\begin{definition} Let $\mathcal M=( M;<,\cdots)$ be an o-minimal structure. A \emph{definable family of curves}
is given by a definable set $F\sub M^n\times T$, for some definable $T\sub M^k$, such that for every $t\in T$,
the set $F_t=\{a\in M^n:(a,t)\in F\}$ is of dimension $1$.
The family is called {\em normal} if for $t_1\neq t_2$, the set $F_{t_1}\cap F_{t_2}$ is finite. In this case,
the dimension of the family is taken to be $\dim T$.
A linearly ordered set $(I;<)$, together with a partial binary function $+$ and a constant $0$, is called {\em a group-interval} if $+$ is continuous, definable
in a neighborhood of $(0,0)$, associative and commutative when defined, order preserving in each coordinate, has $0$ as a neutral element and each element near $0$ has additive inverse in $I$.
%If $\CM$is a one-dimensional normal family of one-dimensional curves in $U$ defines a normal family of plane curves of dimension at least $2$ then it is called {\em rich}.
\end{definition}
We shall use Theorem 1.2 in \cite{PS-trichotomy}:
\begin{fact} \label{trich} Let $\mathcal I=(I;<,+,0,\cdots)$ be an
$\omega_1$-saturated o-minimal expansion of a group-interval.
Then one and only one of the following holds:
\begin{enumerate}
\item There exists an ordered vector space $\mathcal V$ over an ordered
division ring, such that $(I;<,+,0)$ is definably isomorphic to a
group-interval in $V$ and the isomorphism takes every $\mathcal I$-definable
set to a definable set in $\mathcal V$.
\item A real closed field is definable in $\mathcal I$, with its underlying set a subinterval of $I$ and its ordering compatible with $<$.
\end{enumerate}
\end{fact}
We now return to our interval $J\sub C_S(g)\sub G^{00}$ and to the definable bijection $\phi:J^n\to U$.
We let $\star:J^n\times J^n\dashrightarrow J^n $ be the partial function obtained as the pullback via $\phi$
of the group operation in $G$. Namely, for $a,b,c\in J^n$, $$a\star b=c\Leftrightarrow \phi(a)\cdot \phi(b)=\phi(c).$$
Since $\phi$ and the group operation on $U$ are definable in both $\Goo$ and $\Ru$, then so is $\star$.
Because $J$ is an open interval around the identity inside the one-dimensional group $C_S(g)\sub G$,
the restriction to $J$ of the group operation of $G$ makes $J$ into a group-interval, and we let $+$ denote this restriction to $J$. Note that the ordering on $J$ is definable using $+$ and therefore definable in both $\Goo$ and in $\mathcal R$.
%The conjugation operation of $G$ restricted to $\phi(J^n)$ induces via $\phi$
%a partial map $\conj : J^n \times J^n \dashrightarrow J^n$ defined by
%$a\conj b=\phi^{-1}(\phi(a)^{\phi(b)})$.
%This is defined on a neighbourhood of $(e,\ldots ,e)$,
%and is definable both in $\Goo$ and in $\Ru$.
%Recall that a definable family of curves in an o-minimal structure is {\em normal}
%if pairwise intersections are finite. By o-minimality, the equivalence
%relation of having infinite intersection is definable, thus a definable family
%can be {\em normalised} by quotienting the parameters by this relation. The
%{\em dimension} of a definable family of curves is the dimension of the parameter
%set after normalisation.
%For $I \subseteq J$, define $\conj\negmedspace\restriction_I : I^n \times I^n \dashrightarrow I^n$ to be the
%partially defined map whose graph is the intersection with $(I^n)^3$ of the
%graph of $\conj$.
%We write the restriction to $J$ of the group operation of $G$ additively.
\begin{lemma} \label{l:nonlinear}
The structure $\mathcal J=(J;<,+,\star)$ is an o-minimal expansion of a
group-interval, not of type (1) in the sense of Fact \ref{trich}. It is definable in both $\Goo$ and in $\mathcal R$.
\end{lemma}
\begin{proof}
The structure $\mathcal J$ is o-minimal since it is definable in the
o-minimal structure $\Ru$ and the ordered interval $(J;<)$ is definably
isomorphic via a projection map with an ordered interval in $(\Ru;<)$.
We want to show that $\mathcal J$ is not of type (1).
To simplify notation we denote below the group $C_S(g)$ by $H$. Because $G$ is a simple group, there are infinitely many distinct conjugates of the one-dimensional group $H$. More precisely, $\dim N_G(H)<\dim G$ and if $h_1,h_2\in G$ are not in the same right-coset of $N_G(H)$ then $H^{h_1}\cap H^{h_2}$ and hence also $J^{h_1}\cap I^{h_2}$ is finite. Since $\dim (N_G(H))<\dim G$ one can find infinitely many $h\in G$, arbitrarily close to $e$, which belong to different right-cosets of $N_G(H)$. In fact, by Definable Choice in o-minimal expansions of groups or group-intervals, we can find in $\mathcal J$ a definably connected one-dimensional set $C\sub J^n$ with $(e,\ldots,e)\in C$, such that no two elements of $\phi(C)$ belong to the same right-coset of $N_G(H)$.
The family $\{H^h\cap U:h\in \phi(C)\}$ is a one-dimensional normal family of definably connected curves in $U$, all containing $e$, and we want to ``pull it back'' to the structure $\mathcal J$. In order to do that we first note that by replacing $J$ by a subinterval $J_0\sub J$, and replacing $C$ by a possibly smaller definably connected set, we may assume that for every $h\in \phi(C)$ and every $g\in J_0$, each of $h, h^{-1}g$ and $ h^{-1}gh$ is inside $U$. Thus, the one-dimensional normal family of definably connected curves
$$\{\phi^{-1}(J_0^h):h\in \phi(C)\}$$
is definable in $\mathcal J$, and all of these curves contain the point $(e,\ldots,e)$
(=$\phi^{-1}(e))$.
Now, if $\mathcal J$ was of type (1) in Fact \ref{trich}, then up to a change of signature it would be a reduct of an ordered vector space. However, it easily follows from quantifier elimination in ordered vector spaces that in such structures there is no definable infinite normal family of one-dimensional definably connected sets, all going through the same point.
Hence, $\mathcal J$ is not of type (1).
\end{proof}
% |existence of the field is elementary so doesn't need saturation
%Recall that we assumed $\Ru$ to be $\aleph_1$-saturated, and hence
%$(J;+,<,\conj)$ is also $\aleph_1$-saturated.
By Fact ~\ref{trich},
there is a $\mathcal J$-definable real closed field $K$ on an open interval
in $J$ containing $e$. Thus $K$ with its field structure is also
$\Goo$-definable and $\Ru$-definable.
%Note: a neater way to apply trichotomy (via
%\cite[Theorem~1.2]{PS-trichotomy}) would be to say that any $\bigwedge$-definable
%subgroup of an ordered vector space is virtually abelian, because $J$ does
%get a $\bigwedge$-definable copy of $\Goo$. But I haven't found a reference for,
%nor simple proof of, that.}
\subsection{Obtaining the valuation, and bi-interpetability}
\label{s:Goo}
\newcommand{\GooK}{G_K^{00}}
Let $K \subseteq J$ be the definable real closed field obtained in the previous
subsection, considered as a pure field.
By \cite{OPP-groupsRings},
there is an $\Ru$-definable field isomorphism $\theta_R : \Ru \rightarrow K$.
Let $G_K := G^{\theta_R}(K)$ be the group of $K$-points of the $K$-definable
(linear algebraic) group obtained by applying $\theta_R$ to the parameters
defining $G$, so
$\theta_R$ induces an $\Ru$-definable group isomorphism $\theta_G : G \rightarrow
G_K$.
Let $\GooK := (G^{\theta_R})^{00}(K)$ be the corresponding infinitesimal
subgroup,
thus $\theta_R$ restricts to an isomorphism $\theta_G\negmedspace\restriction_\Goo : \Goo \rightarrow \GooK$.
Denote by $(\Ru;\Goo)$
the expansion of the field $\Ru$ by a predicate for $\Goo \leq G$.
\begin{lemma} \label{l:rcvfBiterp}
$(\Ru;\Goo)$ and $(\Ru;\O)$ have the same definable sets.
\end{lemma}
\begin{proof}
Since $G$ is defined over $\R$,
it admits a chart at the identity defined over $\R$,
that is, an $\R$-definable homeomorphism $\psi : I^n \rightarrow U$,
where $I \subseteq \Ru$ is an open interval around $0$,
and $U \subseteq G$ is an open neighbourhood of $e$,
and $\psi((0,\ldots ,0))=e$.
Then $\st_G(\psi(x)) = \psi(\st(x))$,
and so $\Goo = \psi(\m^n)$.
Thus $\Goo$ is definable in $(\Ru;\O)$, and conversely $\m^n$, and hence
$\m$, and so also $\O = \Ru \setminus \frac1\m$, are definable in $(\Ru;\Goo)$.
\end{proof}
\begin{lemma} \label{l:gookdef}
$\GooK \leq G_K$ is $\Goo$-definable,
and moreover $\theta_G\negmedspace\restriction_\Goo : \Goo \rightarrow \GooK$ is $\Goo$-definable.
\end{lemma}
\begin{proof}
%Since $G$ is semisimple, its centre $Z(G)$ is finite, so the quotient map
%induces an isomorphism $\Goo \cong (G/Z(G))^{00}$.
%So we may assume that $Z(G)$ is trivial.
Let $n := \dim(G)$.
Precisely as in \cite[3.2.2]{PPS-defblySimple},
translating by an element of $\Goo$ if necessary we may assume that the
group operation is $C_1$ for $K$ on a neighbourhood of the identity
according to the chart $\phi\negmedspace\restriction_{K^n}$ (where $\phi$ is the map from
Lemma~\ref{l:chart}),
and then the adjoint representation yields an $\Ru$-definable homomorphism
$\Ad : G \rightarrow \GL_n(K)$.
Since $\Goo$ defines $\phi$ and the field $K$ and the conjugation maps $x
\mapsto x^g$ for $g,x \in \Goo$, the restriction
$\Ad\negmedspace\restriction_{\Goo} : \Goo \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \GL_n(K)$
is $\Goo$-definable, and is an embedding since $G$ has finite centre, and $\Goo$ is torsion-free.
Define $\eta := \Ad \o \theta_G^{-1} : G_K \ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow} \GL_n(K)$.
So $\eta$ is $\Ru$-definable.
Since $K$ is $\Ru$-definably isomorphic to $\Ru$, the $\Ru$-definable
structure on $K$ is just the field structure.
Thus $\eta$ is also $K$-definable, and hence $\Goo$-definable.
So $\theta_G\negmedspace\restriction_{\Goo} = \eta^{-1} \o \Ad\negmedspace\restriction_{\Goo}$ is $\Goo$-definable.
% MB: Could add a commutative diagram, but actually I'm not sure it would be
% helpful.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t:mainBiterp}]
By Lemma~\ref{l:gookdef}, $\theta_R$ provides a definition of $(\Ru;\Goo)$ in
$\Goo$ with universe $K$.
This forms a bi-interpretation together with the tautological interpretation
of $\Goo$ in $(\Ru;\Goo)$;
indeed, the composed interpretations are $\theta_R$ and $\theta_G\negmedspace\restriction_{\Goo}$,
which are $\Ru$-definable and $\Goo$-definable respectively.
Combining this with Lemma~\ref{l:rcvfBiterp} concludes the proof of
Theorem~\ref{t:mainBiterp}.
\end{proof}
\section{Isomorphisms of infinitesimal subgroups}
\label{s:borel-tits}
Cartan \cite{cartan} and van der Waerden \cite{waerden} showed that any
abstract group isomorphism between compact semisimple Lie groups is
continuous. In a similar spirit, Theorem~\ref{t:borelTits} below shows that every
abstract group isomorphism of two infinitesimal subgroups of simple compact Lie
groups is, up to field isomorphisms, given by an algebraic map.
We preface the proof with two self-contained preliminary subsections. In
outline, the proof is as follows. We first prove in Theorem~\ref{t:defbleFieldsRCVF}
that given $R \vDash \RCVF$, any model of $\RCVF$ definable $R$ is definably
isomorphic to $R$. As in other cases of the ``model-theoretic Borel-Tits
phenomenon'', first described for $\ACF$ in \cite{poizat-BT}, it follows that
every abstract group isomorphism of the infinitesimal subgroups is the
composition of a valued field isomorphism with an $\RCVF$-definable group
isomorphism; we give a general form of this argument in
Lemma~\ref{l:abstractNonsenseBT}. Finally, in Section~\ref{s:borelTits} we deduce the
final statement by seeing that any $\RCVF$-definable group isomorphism of the
infinitesimal subgroups is induced by an algebraic isomorphism of the Lie
groups.
\subsection{Definable fields in $\RCVF$}
Here, we show that there are no unexpected definable fields in $R^n$ for $R \vDash
\RCVF$.
In this subsection we reserve the term `semialgebraic' for $R$-semialgebraic
sets.
Let $\Ru_v=\langle R;\O\rangle \vDash \RCVF$.
We say a point $a$ of an $\Ru_v$-definable set $X$ over $A$ is {\em generic}
over $A$ if $\trd(a/A)$ is maximal for points in $X(R')$ for $R'$ an
elementary extension of $R$. Such an $a$ exists if $R$ is
$(\aleph_0+|A|^+)$-saturated. This maximal transcendence degree is the
dimension $\dim(X)$ of $X$, which coincides
(\cite[Theorem~4.12]{MMS-weakOMin}) with the largest $d$ such that the image
of $X$ under a projection to $d$ co-ordinates has non-empty interior.
\begin{fact} \label{f:semialgebraic}\begin{enumerate}\item Let $X\subseteq
R^n$ be an $\Ru_v$-definable set over $A$, and $a\in X$ a generic element
over $A$.
Then there exists a semialgebraic neighborhood $U\subseteq R^n$ of $a$ (possibly defined over additional parameters, which may be taken to be independent of $Aa$) such that $U\cap X$ is semialgebraic.
\item Let $U\subseteq R^d$ be an open $\Ru_v$-definable set over $A$, and $a\in U$ generic over $A$. Let $F:U\to R$ be an $\Ru_v$-definable function. Then there exists a semialgebraic neighborhood $U$ of $a$, such that $F| U$ is semialgebraic and $C^1$ with respect to $R$, meaning that all partial derivatives of $F$ with respect to $R$ exist and are continuous on $U$.
\end{enumerate}
\end{fact}
\proof
\begin{enumerate}
\item
Permuting co-ordinates, we may assume $a=(b,c)$ where $b \in R^d$ is
generic over $A$ and $c \in R^{n-d}$ is in $\dcl(bA)$.
Let $\pi : R^n \rightarrow R^d$ be the projection to the first $d$ co-ordinates.
By \cite[Theorem~4.11]{MMS-weakOMin}, $X$ admits a decomposition
into finitely many disjoint $A$-definable cells each of which is the graph of a
definable function on an open subset of some $R^t$.
Definable closure in $\Ru_v$ coincides with definable closure in $R$
\cite[Theorem~8.1(1)]{Mellor-RCVFEI}.
Thus if $C$ is the cell containing $a$, then $C$ is the graph of a
semialgebraic function on a neighbourhood of $b$. We now claim that
locally near $a$, the set $X$ is equal to $C$.
Suppose for a contradiction that $C' \neq C$ is another cell in the
decomposition, and $a \in \cl(C')$, the topological closure of $C'$.
The cell decomposition of $X$ induces a cell decomposition of $\pi(X)\sub R^d$, and since $b$ is generic in $R^d$
over $A$, it must belong to the interior of $\pi(C)$. It follows that $\pi(C')=\pi(C)$, and so there exists $c'\neq c$ such that $(b,c')\in C'$.
By the inductive definition of a cell
and the genericity of $a$ in $X$, also $C'$ is the graph of a
semialgebraic function on a neighbourhood of $b$, thus in particular is
locally closed, contradicting $a \in \cl(C')$.
\item By (1), the graph of $F$ is a semialgebraic set in a neighborhood of
$(a,f(a))$, and since $a$ is generic in its domain, the function $F$ is
$C^1$ in a neighborhood of $a$.
\end{enumerate}
\qed
\begin{theorem}\label{t:defbleFieldsRCVF} Let $\Ru_v=\langle R;\O\rangle \vDash
\RCVF$.
\begin{enumerate}[(a)]\item
If $F \subseteq R^n$ is a definable field in $\Ru_v$ then it is $\Ru_v$-definably
isomorphic to either $R$ or its algebraic closure $R(\sqrt{-1})$.
\item
If $F \subseteq R^n$ is a definable non-trivially valued field in $\Ru_v$ then it
is $\Ru_v$-definably isomorphic to either $\Ru_v$ or its algebraic closure
$\Ru_v(\sqrt{-1})$.
\end{enumerate}
\end{theorem}
\proof
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
Passing to an elementary extension as necessary, we assume $R$ is
$(\aleph_0+|A|^+)$-saturated for any parameter set $A$ we consider.
We first prove (a). Let $d=\dim F$.
We first show that the additive group of $F$ can be endowed with the structure of a definable $C^1$ atlas (not necessarily finite). By that we mean: a definable family of subsets of $F$, $\{U_t:t\in T\}$, and a definable family of bijections $f_t:U_t\to V_t$, where each $V_t$ is an open subset of $R^{d}$, such that for every $s, t\in T$, the set $f_t(U_t\cap U_s)$ is open in $V_t$ and the transition maps $\sigma_{t,s}: f_t(U_t\cap U_s)\to f_s(U_t\cap U_s)$
are $C^1$ with respect to $R$. Moreover, the group operation and additive inverse are $C^1$-maps when read through the charts.
To see this we follow the strategy of the paper of Marikova, \cite{Marikova}. Without loss of generality $F$ is definable over $\emptyset$. We fix $g\in F$ generic and an open neighborhood $U\ni g$ as in Fact \ref{f:semialgebraic} (1). By the cell decomposition in real closed fields, we may assume that $U\cap F$ is a cell, so definably homeomorphic to some open subset $V$ of $R^{d}$. By replacing $U\cap F$ with $V$ we may assume that $U$ is an open subset of $R^d$, and $g$ is generic in $F$ over the parameters defining $U$.
\begin{claim} \label{fact2.10} The map $(x,y,z)\mapsto x-y+z$ is a $C^1$-map (as a map from $U^3$ into $U$) in some neighborhood of $(g,g,g)$.\end{claim}
\proof The proof is identical to \cite[Lemma 2.10]{Marikova}, with Fact \ref{f:semialgebraic} (2) above replacing Lemma 2.8 there.\qed
Thus, there exists $U_0\ni g$, such that the map $(x,y,z)\mapsto x-y+z$ is a $C^1$ map from $U_0^3$ into $U$.
We now consider the definable cover of $F$:
$$\U=\{h+U_0:h\in F\},$$ (with $+$ the $F$-addition) and the associated family of chart maps $f_h:h+U_0\to U_0$, $f_h(x)=x-h$.
Using Claim \ref{fact2.10}, it is not hard to see that $\U$ endows $\la F,+\ra$ with a definable $C^1$-atlas;
indeed, if $h+U_0 \cap h'+U_0 \neq \emptyset$, say $h+u_0 = h'+u_0'$,
then $h-h' = u_0' - u_0$,
so $\sigma_{h,h'}(u_0'') = h + u_0'' - h' = u_0'' - u_0 + u_0'$.
Similarly, the function $+$ is a $C^1$-map from $F^2$ into $F$ (where $F^2$ is endowed with the product atlas), and $x\mapsto -x$ is a $C^1$-map as well. Indeed, in \cite{Marikova} Marikova proves in exactly the same way that the same $\U$ endows the group with a topological group structure (using \cite[Lemma~2.10]{Marikova} in place of Claim \ref{fact2.10}).
By Fact \ref{f:semialgebraic} (2), every definable function from $F$ to $F$ is $C^1$ in a neighborhood of generic point of $F$. Thus, just as in \cite[Lemma 2.13]{Marikova}, we have:
\begin{fact}\label{endo} If $\alpha :F\to F$ is a definable endomorphism of $\la F,+\ra$ then $\alpha$ is a $C^1$-map.\end{fact}
For every $c\in F$, we consider the map $\lambda_c:F\to F$, defined by $\lambda_c(x)=c x$ (multiplication in $F$).
By fact \ref{endo} each $\lambda_c$ is a $C^1$-map and we consider its Jacobian matrix at $0$, with respect to $R$, denoted by $J_0(\lambda_c)$. This is a matrix in $M_d(R)$, and the map $c\mapsto J_0(\lambda_c)$ is $\Ru_v$-definable.
As was discussed in \cite[Lemma 4.3]{OPP-groupsRings}, it follows from the chain rule that the map $c\mapsto J_0(\lambda_c)$
is a ring homomorphism into $M_d(R)$ (note that we do not use here the uniqueness of solutions of ODE as in \cite{OPP-groupsRings}, thus we a-priori only obtain a ring homomorphism). However, since $F$ is a field the map is injective.
To summarize, we mapped $F$ isomorphically and definably onto an $\Ru_v$-definable field, call it $F_1$, of matrices inside $M_d(R)$. Notice that now the field operations are just the usual matrix operations, $1_{F_1}$ is the identity matrix, so in particular, all non-zero elements of $F_1$ are invertible matrices in $M_d(R)$. Our next goal is to show that $F_1$ is semialgebraic.
By Fact \ref{f:semialgebraic} (2), there exists some non-empty relatively open subset of $F_1$ which is semialgebraic.
By translating it to $0$ (using now the semialgebraic $F_1$-addition), we find such a neighborhood, call it $W\subseteq F_1$, of the $0$-matrix. But now, given any $a\in F$, by multiplying $a$ by an invertible matrix $b\in W$ sufficiently close to $0$, we obtain $ba\in W$. Thus, $F_1=\{a^{-1}b: a,b\in W\}$. Because $W$ is semialgebraic so is $F_1$.
Thus, we showed that $F$ is definably isomorphic in $\Ru_v$ to a semialgebraic field $F_1$. We now apply Theorem \cite[Theorem 1.1]{OPP-groupsRings} and conclude that $F_1$ is semialgebraically isomorphic to $R$ or to $R(\sqrt{-1})$.
Finally, we address (b).
This follows immediately from (a) once we observe that $\O$ is the only
definable valuation ring in $\Ru_v$. So suppose $\O'$ is another. By weak
o-minimality, $\O'$ is a finite union of convex sets, and then since it is a
subring with unity it is easy to see that $\O'$ is convex. Thus either $\O
\subseteq \O'$ or $\O' \subseteq \O$. Without loss of generality, $\O$ is the standard
valuation ring $\cup_{n \in \N} [-n,n]$, so $\O$ properly contains no
non-trivial convex valuation ring. Thus $\O \subseteq \O'$. But then if $v : R \rightarrow
\Gamma$ is the valuation induced by $\O$, then the image of the units of
$\O'$ is a definable subgroup $v((\O')^*) \leq \Gamma$. But $\Gamma$ is a pure
divisible ordered abelian group, and so has no non-trivial definable
subgroup. Hence $v((\O')^*) = \{0\}$, and hence $\O' = \O$.
\qed
\begin{remark}
The techniques we applied here will not readily adapt to handle imaginaries.
In the case of algebraically closed valued fields, a result of
\cite{HR-metastable} is that the only {\em interpretable} fields, up to
definable isomorphism, are the valued field and its residue field. It would
be natural to expect that, correspondingly, the only interpretable fields in
$\RCVF$ up to definable isomorphism are the valued field, its residue field,
and their algebraic closures.
\end{remark}
\subsection{Interpretations and general nonsense}
We address the ``Borel-Tits phenomenon'' associated with bi-interpretations
which require parameters. We spell out an abstract form of the argument given
by Poizat \cite{poizat-BT} in the case of algebraically closed fields. The
ideas in this subsection are well-known. For convenience of exposition, we
first give a name to the following key property.
\begin{definition} \label{d:self-recollecting}
Say a theory $T$ is \defn{self-recollecting} if any $\B' \models T$
interpreted in any $\B \models T$ is $\B$-definably isomorphic to $\B$.
%that is, if $\B,\B' \vdash T$ and $f : \terp{\B'}{\B}$ is an interpretation
%then $f$ is homotopic to an isomorphism $\theta : \B' \rightarrow \B$.
Say $T$ is \defn{self-recollecting for definitions} if this holds for
interpretations which are definitions (where recall a {\em definition} is an
interpretation which doesn't involve non-trivial quotients).
\end{definition}
\begin{examples}
$\ACF$ is self-recollecting by \cite{poizat-BT}, $\RCF$ is self-recollecting
by \cite{NP-compactLie}, and $\Th(\Q_p)$ is self-recollecting for
definitions by \cite{pillay-fieldsQp}. It follows directly from the
characterisation of interpretable fields in \cite{HR-metastable} that the
theory of non-trivially valued algebraically closed fields $\ACVF$ is
self-recollecting.
Theorem~\ref{t:defbleFieldsRCVF}(b) proves that $\RCVF$ is self-recollecting for
definitions, but we do not settle the question of whether it is
self-recollecting.
\end{examples}
We use the notation $\alpha : \terp{\A}{\B}$ to denote an interpretation of
$\A$ in $\B$, which recall we consider to be a map from $\A$ to some definable
quotient in $\B$. Note that any isomorphism is in particular an interpretation
(and even a definition). We denote composition of interpretations by
concatenation.
\begin{lemma} \label{l:abstractNonsenseBT}
Suppose $\A_i$ is a structure interpreted in a structure $\B_i$ for $i=1,2$,
and the interpretation of $\A_1$ in $\B_1$ can be completed to a
bi-interpretation.
Suppose further that $\Th(\B_1) = \Th(\B_2)$, and $T_B := \Th(\B_1)$ is
self-recollecting.
Suppose $\sigma : \A_1 \rightarrow \A_2$ is an isomorphism of structures.
Then there exist an isomorphism $\sigma' : \B_1 \xrightarrow{\cong} \B_2$
and a $\B_2$-definable isomorphism $\theta : \sigma'(\A_1) \rightarrow \A_2$
such that $\sigma = \theta(\sigma'\negmedspace\restriction_{\A_1})$.
If $T_B$ is only self-recollecting for definitions but the given
interpretations are definitions, then the same result holds.
\end{lemma}
\begin{proof}
Let $(f,g)$ be the bi-interpretation of $\A_1$ with $\B_1$,
and $\alpha : \terp{\A_2}{\B_2}$ the interpretation.
\[ \xymatrix{
\A_1 \biterpAr{d}{f} \ar[r]^\sigma & \A_2 \terpAr[d]^{\alpha} \\
\B_1 \biterpAr{u}{g} & \B_2 \\
} \]
%Following \cite{AZ-QFAwCat}, but allowing parameters, we say
%interpretations $\beta,\gamma : \terp{\A}{\B}$ are {\em homotopic}, written
%$\beta \sim \gamma$,
For interpretations $\beta,\gamma : \terp{\A}{\B}$, we write $\beta \sim
\gamma$ if
%$\{ (\beta(x),\gamma(x)) : x \in \A \}$ is a $\B$-definable set; in other
%words,
$\gamma \beta^{-1} : \beta(\A) \rightarrow \gamma(\A)$ is a $\B$-definable
isomorphism between the two copies of $\A$ in $\B$.\footnote{Interpretations
satisfying this condition are sometimes called {\em homotopic}.}
Now $\alpha \sigma g$ is an interpretation of $\B_1$ in $\B_2$,
and thus by self-recollecting
there is a $\B_2$-definable isomorphism $\tau : (\alpha\sigma g)(\B_1) \rightarrow
\B_2$. Let $\sigma' := \tau\alpha\sigma g : \B_1 \rightarrow \B_2$.
Then $\alpha \sigma g \sim \sigma'$, and so $\alpha \sigma gf \sim \sigma'
f$.
Now $gf$ is definable,
and it follows that $\alpha \sigma \sim \alpha \sigma gf$.
Thus $\alpha \sigma \sim \sigma' f$.
Then $\alpha' := \sigma' f \sigma^{-1} \sim \alpha$.
%So $\sigma$ is induced from $\sigma'$ according to $f$ and $\alpha'$, i.e.\
%the following diagram of interpretations commutes.
%\[ \xymatrix{
%\A_1 \ar[d]^f \ar[r]^\sigma & \A_2 \ar[d]^{\alpha'} \\
%\B_1 \ar[r]^{\sigma'} & \B_2 \\
%} \]
So $\theta := \alpha\alpha'^{-1}$ is a $\B_2$-definable isomorphism,
and $\theta \sigma' f = \alpha \sigma$. Since we view $\A_1$ and $\A_2$ in
$\B_2$ via $f$ and $\alpha$, this is as desired.
The proof in the case of definitions is identical.
\end{proof}
\subsection{Characterisation of isomorphisms of infinitesimal subgroups}
\label{s:borelTits}
\begin{lemma} \label{l:isomExtend}
If $G,H$ are compact connected centreless linear Lie groups,
and $\Ru \succ \R$ is a proper real closed field extension of $\R$,
and $\theta : \Goo(\Ru) \xrightarrow{\cong} \Hoo(\Ru)$ is an $(\Ru;\O)$-definable group
isomorphism,
then $\theta$ extends to an $\Ru$-definable algebraic isomorphism $G(\Ru)
\xrightarrow{\cong} H(\Ru)$.
\end{lemma}
\begin{proof}
We may assume $\Ru$ is $\aleph_0$-saturated.
Let $\Gamma \subseteq (G\times H)(\Ru)$ be the $\Ru$-Zariski closure of the graph
$\Gamma_\theta$ of $\theta$.
Since $\Gamma_\theta$ is an abstract subgroup, $\Gamma$ is (the set of
$\Ru$-points of) an algebraic subgroup over $\Ru$.
The image of the projection $\pi_G : \Gamma \rightarrow G(\Ru)$ contains
$\Goo(\Ru)$, which is Zariski-dense in $G$ since $G$ is connected, and thus
$\pi_G(\Gamma) = G(\Ru)$.
Let $k \leq \Ru$ be a finitely generated field over which $\theta$ is defined.
Since $\Gamma_\theta$ is Zariski dense in $\Gamma$, there exists
$(g,\theta(g)) \in \Gamma_\theta$ which is algebraically generic in $\Gamma$
over $k$ (i.e.\ of maximal transcendence degree);
indeed, the Zariski density implies that $\Gamma_\theta$ is contained in no
subvariety of $\Gamma$ over $k$ of lesser dimension, and so such a generic
exists by $\aleph_0$-saturation of $\Ru$.
But definable closure in $\RCVF$ agrees with field-theoretic algebraic
closure \cite[Theorem~8.1(1)]{Mellor-RCVFEI}, so $\trd(\theta(g)/k(g))=0$
and thus $\pi_G^{-1}(g)$ is finite.
Then also $\ker(\pi_G)$ is finite, and hence central.
So since $G$ is centreless, $\pi_G$ is an isomorphism.
Similarly, $\pi_H$ is an isomorphism.
So $\theta$ extends to the algebraic isomorphism $\pi_H \o \pi_G^{-1}$.
\end{proof}
\begin{theorem} \label{t:borelTits}
Suppose $G_1$ and $G_2$ are compact simple centreless linear Lie groups,
and $\Ru_i \succ \R$ is a proper real closed field extension of $\R$ for
$i=1,2$,
and $\sigma : G_1^{00}(\Ru_1) \xrightarrow{\cong} G_2^{00}(\Ru_2)$ is an abstract group
isomorphism.
Then there exist
a valued field isomorphism $\sigma' : (\Ru_1,\O_1) \xrightarrow{\cong} (\Ru_2,\O_2)$
and an $\Ru_2$-definable isomorphism $\theta : G_3(\Ru_2) \xrightarrow{\cong} G_2(\Ru_2)$,
where $G_3 = \sigma'(G_1)$,
such that $\sigma = \theta\negmedspace\restriction_{G_3^{00}} \o \sigma'\negmedspace\restriction_{G_1^{00}}$.
In particular, $\sigma$ extends to an abstract group isomorphism $G_1(\Ru_1)
\xrightarrow{\cong} G_2(\Ru_2)$.
\end{theorem}
\begin{proof}
By Theorem~\ref{t:mainBiterp}, Theorem~\ref{t:defbleFieldsRCVF}(b), and
Lemma~\ref{l:abstractNonsenseBT}, there exist an isomorphism $\sigma' :
(\Ru_1,\O_1) \xrightarrow{\cong} (\Ru_2,\O_2)$ and an $(\Ru_2,\O_2)$-definable isomorphism
$\theta' : \sigma'(G_1^{00}(\Ru_1)) \xrightarrow{\cong} G_2^{00}(\Ru_2)$
such that $\sigma = \theta' \o \sigma'\negmedspace\restriction_{G_1^{00}}$.
Then $\sigma'(G_1^{00}(\Ru_1)) = G_3^{00}(\Ru_2)$.
Thus by Lemma~\ref{l:isomExtend}, $\theta'$ extends to an $\Ru_2$-definable
algebraic isomorphism $\theta : G_3(\Ru_2) \xrightarrow{\cong} G_2(\Ru_2)$, as required.
\end{proof}
\begin{remark}
We have stated the results of this section in terms of $\Goo$, but it is
easy to see that they apply equally to other infinitesimal subgroups as in
Remark~\ref{r:otherValns}.
\end{remark}
\section{Infinitesimal subgroups of definably compact groups}
\label{s:DC}
In this section, we prove Theorem~\ref{t:DCBiterp} by combining Theorem~\ref{t:mainBiterp}
with results in the literature on definably compact groups and $\Goo$.
We work in a sufficiently saturated o-minimal expansion $M$ of a real closed
field, say $\kappa$-saturated where $\kappa$ is sufficiently large. (In fact
$\kappa=2^{\aleph_0}$ suffices for the arguments below; moreover, it follows
after the fact that Theorem~\ref{t:DCBiterp} holds with only $\kappa=\aleph_0$, but
we do not spell this out).
For $G$ a definable group, let $\Goo$ be the smallest $\bigwedge$-definable
(in the sense of $M$) subgroup of bounded index. Here, a
\recalldefn{$\bigwedge$-definable set} is a set defined by an infinite
conjunction of formulas over a common parameter set $A \subseteq M$ with $|A| <
\kappa$.
\begin{lemma} \label{l:oostruct}
Let $G$ and $H$ be definably compact definable groups.
\begin{enumerate}[(i)]\item If $\theta : G \twoheadrightarrow H$ is a definable surjective homomorphism, then
$\theta(\Goo) = \Hoo$.
\item If $H$ is a definable subgroup of $G$, then $\Hoo = \Goo \cap H$.
\item $\Goo$ is the unique $\bigwedge$-definable subgroup of bounded index
which is divisible and torsion-free.
\item $(G\times H)^{00} = \Goo \times \Hoo$
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(i)]\item $\theta^{-1}(\Hoo)$ resp.\ $\theta(\Goo)$ is a $\bigwedge$-definable
bounded index subgroup of $G$ resp.\ $H$.
\item \cite[Theorem~4.4]{B-spectra}.
\item \cite[Corollary~4.7]{B-spectra}.
% ^Technically, that paper and some of the papers it references assume full
% saturation. But one can see (c.f.\ commented out remark below) that the
% above properties go down from an elementary extension to our
% \kappa-saturated M. Probably the proofs directly go through assuming only
% sufficient saturation, but we have not checked this.
% (See also git branch `sat`.)
\item This follows directly from (iii).
\end{enumerate}
\end{proof}
Say a group $(G;*)$ is the \defn{definable internal direct product} of
its subgroups $H_1,\ldots ,H_n$ if each $H_i$ is $(G;*)$-definable and
$(h_1,h_2,\ldots ,h_n) \mapsto h_1h_2\ldots h_n$ is an isomorphism $\prod_i H_i \rightarrow G$.
The following lemma is an immediate consequence of the definition.
\begin{lemma} \label{l:biterpProd}
If a group $(G;*)$ is the definable internal direct product of subgroups
$H_1,\ldots ,H_n$,
then $(G;*)$ is bi-interpretable with the disjoint union of the $(H_i;*)$.
\end{lemma}
The following Fact extracts from the literature the key results we will need
on the structure of definably connected definably compact groups.
A definable group is \recalldefn{definably simple} if it contains no proper
non-trivial normal definable subgroup.
First recall that if $G\sub M^n$ is a definable group in an o-minimal
structure $M$ then, by \cite{P-groupsFieldsOMin}, it admits a topology with a
definable basis which makes it into a topological group. Moreover, this is the
unique topology which agrees with the ambient $M^n$-topology on a definable
subset of $G$ whose complement has smaller dimension. All topological notions
below (e.g.\ definable compactness) are with respect to this topology.
\begin{fact} \label{f:DCLit}
Let $G$ be a definably connected definably compact definable group.
Let $G' := (G,G)$ be the derived subgroup of $G$. Then:
\begin{enumerate}[(i)]\item $G'$ and $Z(G)$ are definable and definably compact.
\item $G$ is the product of its subgroups $G'$ and $Z(G)^0$,
and $Z(G)^0 \cap G'$ is finite.
%**(ii) The map $Z(G)^0 \times G' \rightarrow G; (a,g) \rightarrow ag$ is surjective with
%finite kernel $\{ (x,x^{-1}) : x \in Z(G)^0 \cap G' \}$.
\item $Z(G')$ is finite, and $G'/Z(G')$ is the direct product of finitely
many definably simple definably compact definable subgroups $H_i$.
\item If $G$ is definably simple, then there exists a compact real linear
Lie group $H$ and a real closed field $\Ru$ extending $\R$ such that
$(\Goo;*)$ is isomorphic to $(\Hoo;*)$, where $\Hoo := \Hoo(\Ru)$ is the
infinitesimal neighbourhood of the identity as defined in \secref{intro}.
\end{enumerate}
\end{fact}
\begin{proof}
\begin{enumerate}[(i)]\item By \cite[Corollary~6.4(i)]{HPP-central}, $G'$ is definable. Since
definable subgroups are closed, both groups are definably compact.
\item This is \cite[Corollary~6.4(ii)]{HPP-central}.
\item This is immediate from \cite[Corollary~6.4(i)]{HPP-central} and
\cite[Fact~1.2(3)]{HPP-central} (based on
\cite[Theorem~4.1]{PPS-defblySimple}).
\item This follows from the proof of
\cite[Proposition~3.6]{Pillay-conjecture}. Indeed, as discussed there (see
also \cite[Fact~1.2(1)]{HPP-central}), $G$ is definably isomorphic to $H(R)$
for $R$ a definable real closed field and $H$ a semialgebraic linear group
over a copy of the real field within $R$. Since $G$ is definably compact,
$H$ is compact (see \cite[Theorem~2.1]{PS-defbleCompactness}). Now {\em Case
II} in the proof of \cite[Proposition~3.6]{Pillay-conjecture} shows that the
smallest $M$-$\bigwedge$-definable subgroup $H^{00}(R)$ of $H$ is precisely the
infinitesimal neighbourhood $\st^{-1}(e)$, as required.
\end{enumerate}
\end{proof}
We now repeat the statement of Theorem~\ref{t:DCBiterp}, and prove it.
{
\renewcommand{\thetheorem}{\ref{t:DCBiterp}}
\begin{theorem}
Let $(G;*)$ be an infinite definably compact group definable in a
sufficiently saturated o-minimal expansion $M$ of a field.
Then $(\Goo(M);*)$ is bi-interpretable with the disjoint union of a
(possibly trivial) divisible torsion-free abelian group and finitely many
(possibly zero) real closed convexly valued fields.
\end{theorem}
\addtocounter{theorem}{-1}
}
\begin{proof}
It follows from Lemma~\ref{l:oostruct}(ii) that $\Goo = (G^0)^{00}$ where $G^0$ is
the smallest definable subgroup of finite index, so we may assume $G=G^0$,
and hence (by \cite[Lemma~2.12]{P-groupsFieldsOMin}) that $G$ is definably
connected.
Let $G' := (G,G)$ be the derived subgroup of $G$, and let $(\Goo)' :=
(\Goo,\Goo)$ be the derived subgroup of $\Goo$.
By Fact~\ref{f:DCLit}(i), $G'$ and $Z(G)$ are definable and definably compact, and
so Lemma~\ref{l:oostruct} applies to them.
Let $H := G'/Z(G')$. Let $H_i$ be as in Fact~\ref{f:DCLit}(iii), so $H = \prod_i
H_i$.
\begin{claim} \label{c:deriv00}
\begin{enumerate}[(i)]\item $(G')^{00} \cong H^{00}$ as groups.
\item $H^{00} = (\prod_i H_i)^{00}$ is the definable internal direct
product of the $H_i^{00}$.
\end{enumerate}
\end{claim}
\begin{proof}
\begin{enumerate}[(i)]\item
$(G')^{00}$ is torsion-free by Lemma~\ref{l:oostruct}(iii), and $Z(G')$ is
finite by Fact~\ref{f:DCLit}(iii), thus $Z(G') \cap (G')^{00} = \{e\}$. So by
Lemma~\ref{l:oostruct}(i), the quotient map induces such an isomorphism.
\item Given Lemma~\ref{l:oostruct}(iv), we need only show that $H_i^{00}$ is
$(\Hoo;*)$-definable.
By Lemma~\ref{l:Coo}, $C_H(H_i^{00}) = C_H(H_i)$, and since each $H_i$ is
centreless we have $H_i = \bigcap_{j \neq i} C_H(H_j)$, thus $H_i^{00} =
\bigcap_{j \neq i} C_{\Hoo}(H_j^{00})$ (using Lemma~\ref{l:oostruct}(iv) again).
\end{enumerate}
\end{proof}
\begin{claim} \label{c:GooProd}
\begin{enumerate}[(i)]\item $\Goo = Z(G)^{00}(G')^{00}$.
\item $(G')^{00} = (\Goo)'$.
\item $Z(G)^{00} = Z(\Goo)$.
\item $\Goo$ is the definable internal direct product of $Z(G)^{00}$ and
$(G')^{00}$.
\end{enumerate}
\end{claim}
\begin{proof}
\begin{enumerate}[(i)]\item
Since $Z(G)^{00}$ is central and each of $Z(G)^{00}$ and $(G')^{00}$ is
divisible and torsion-free, also $L := Z(G)^{00}(G')^{00}$ is divisible
and torsion-free. Now $G = Z(G)G'$ by Fact~\ref{f:DCLit}(ii), so any coset of
$L$ can be written as $zaL = (zZ(G)^{00})(a(G')^{00})$ with $z \in Z(G)$
and $a \in G'$, thus the index of $L$ in $G$ is bounded by the product of
the indices of $Z(G)^{00}$ in $Z(G)$ and of $(G')^{00}$ in $G'$. So $L$
is a bounded index subgroup. Thus we conclude by Lemma~\ref{l:oostruct}(iii).
\item
By Fact~\ref{f:DCLit}(iv) and \cite{dAndreaMaffei}, $H_i^{00} =
(H_i^{00},H_i^{00})_1$ for each $i$.
So by Claim~\ref{c:deriv00}, $(G')^{00} = ((G')^{00},(G')^{00})_1$.
Also $\Goo \cap G' = (G')^{00}$ by Lemma~\ref{l:oostruct}(ii),
and so $(\Goo,\Goo)_1 = (G')^{00}$, and then since this is a subgroup we
also have $(\Goo)' = (\Goo,\Goo)_1$.
\item
By (i) it suffices to see that $Z((G')^{00}) = \{e\}$. But indeed, as in
Lemma~\ref{l:Coo}, $Z((G')^{00}) \leq C_{G'}((G')^{00} = C_{G'}(G') = Z(G') =
\{e\}$. Alternatively, one can see this way that each $Z(H_i^{00}) =
\{e\}$, and apply Claim~\ref{c:deriv00}(ii).
\item
By Fact~\ref{f:DCLit}(ii), $Z(G)^0 \cap G'$ is finite. Hence also $Z(G)^{00}
\cap (G')^{00}$ is finite,
and thus by Lemma~\ref{l:oostruct}(iii) it is trivial.
Combining this with the previous items of this Claim, we conclude.
\end{enumerate}
\end{proof}
Now each $H_i^{00}$ is bi-interpretable with a model of RCVF by
Fact~\ref{f:DCLit}(iv) and Theorem~\ref{t:mainBiterp}, and $Z(G)^{00}$ is (by
Lemma~\ref{l:oostruct}(iii)) a divisible torsion free abelian group, so we
conclude by Claim~\ref{c:GooProd}(iv), Claim~\ref{c:deriv00}, and Lemma~\ref{l:biterpProd}.
\end{proof}
\begin{remark}
Since $M$ is an o-minimal expansion of a field, any $M$-definable real
closed field is $M$-definably isomorphic to $M$ as a field. Thus the valued
fields $R_i$ interpreted in the groups $H_i^{00}$ in the above proof are
$M$-definably isomorphic as fields.
However, the disjoint union structure clearly does not define any such
isomorphisms between the $R_i$, and hence nor does the group $(\Goo;*)$.
\end{remark}
%Remark:
% \renewcommand{\a}{\overline{a}}
% \renewcommand{\b}{\overline{b}}
% We assumed saturation of $M$ in Theorem~\ref{t:DCBiterp} for convenience of
% exposition, but it is not hard to see that the same result follows when $M$
% is just $\aleph_0$-saturated.
%
% Indeed, let $M' \succ M$ be a saturated elementary extension, and let $G$ as in
% the theorem be defined over $M$. Let $\a$ be a finite tuple from $M$ over
% which $G$ is defined. By a result of Shelah (\cite[Theorem~8.4]{simon}),
% also $\Goo$ is over $\a$, i.e.\ $\Goo \leq G(M')$ is the set of realisations
% $\Phi(M')$ of a partial type $\Phi$ with $\a$ as the only parameters. Then
% we define $\Goo(M) := \Phi(M) \leq G(M)$.
%
% Now Theorem~\ref{t:DCBiterp} establishes a bi-interpretation of $(\Goo;*)$ with a
% certain structure $X$.
% Let $\b$ be a finite tuple from $\Goo$ such that the data of the
% bi-interpretation is defined in $(\Goo;*)$ over $\b$; namely, such that the
% definable sets interpreting the structure $X$ in $\Goo$ are definable in
% $(\Goo;*)$ over $\b$, and similarly for the reverse interpretation (seen in
% $\Goo$ via the first interpretation), and similarly for the definable
% bijections which are the compositions of the interpretations. Note that it
% is part of the first-order type in $(\Goo;*)$ of $\b$ that this data defines
% a bi-interpretation with a structure elementarily equivalent to $X$.
%
% Now let $\b'$ be a realisation in $M$ of $\tp(\b/\a)$.
% Then by the saturation, $(M;\a,\b')$ is back-and-forth equivalent to
% $(M';\a,\b)$, and it follows directly that $(\Goo(M);*,\b')$ is
% back-and-forth equivalent to $(\Goo;*,\b)$, and so in particular
% elementarily equivalent. So the bi-interpretation over $\b$ transfers to a
% bi-interpretation over $\b'$ of $\Goo(M)$ with a structure elementary
% equivalent to $X$, i.e.\ the disjoint union of a divisible torsion-free
% abelian group and finitely many real closed convexly valued fields, as
% required.
% .
%# Local groups
%
%Consider an open neighbourhood $U$ of the identity of a compact real lie
%group equipped with the restriction to $U^3$ of the graph of the group
%operation.
%By "the same" construction as above, $U$ interprets a real closed field.
%TODO: details.
%TODO: if moreover $U$ is definable, then it is bi-interpretable with the real
%field.
%
%But if $U$ is over $\R$, which is what one might think is the interesting
%case, then actually the structure on $U$ defines the whole group $G$, so then
%Nesin-Pillay suffices. Sketch argument for that:
%
%Suppose we can find $V \subseteq U$ a definable neighbourhood and finitely many $u_i
%\in U$ such that $VV \subseteq U$ and $V^{u_i} \subseteq U$ and $VV$ is covered by the
%$u_iV$. This seems like it should be possible, taking $V$ to be an
%intersection of finitely many $(uU) \cap U$ (having made $U$ inversive),
%basically because of local compactness - but I don't see an actual argument
%for it. But assume such a $V$ and $u_i$ exist.
%Kobi suggests a better argument: take $(g^U)^k$ (i.e.\ multiplying it with its
%inverse k times). For sufficiently large $k$ we get a
%$U$-conjugation-invariant neighborhoood of $e$.
%Call a word in the $u_i$ a ``u-word''.
%Now say $n$ is such that $G$ is covered by the $wV$ for $w$ a u-word of
%length at most $n$. Then if $w_1,w_2$ have length at most $n+1$ and $v_i \in
%V$, then $w_1v_1w_2v_2 = w_1w_2v_1^{w_2}v_2$, and $U$ knows how to write
%$v_1^{w_2}v_2$ as $u_iv'$ for some $i$, and then it knows (using a parameter
%for each of the finitely many u-words of length at most $2n+3$) how to write
%$w_1w_2u_i$ as some $w'v''$ for $w'$ of length at most $n$, and finally it
%can write $v''v'$ as $u_{i'}v'''$, so we have $w_1v_1w_2v_2 =
%(w'u_{i'})v'''$ where $w'u_{i'}$ is a u-word in the $u_i$ of length at most
%$n+1$. So $V$ defines the multiplication on $G$ in the cover $(wV)_w$.
\bibliography{G00R}
\end{document}