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\renewcommand{\O}{\mathcal{O}}
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{\it Notes from a seminar in Münster on [Hrushovski-Rideau ``Valued Fields, 
Metastable Groups''], Nov 2019.}

Work in $\U \vDash  ACVF$ sufficiently saturated, in geometric language (sorts 
$K,\Gamma,k,S_n,T_n$). Definable means definable over a small subset of $\U$.
We write $\dcl$ resp.\ $\acl$ for $\dcleq$ resp.\ $\acleq$.

\section{Pure imaginaries}

\begin{definition}
  $e \in \Ueq$ is {\bf purely} {\bf imaginary} over $C \subseteq  \U$
  if $\acl(Ce) \cap K = \acl(C) \cap K$.
\end{definition}

\begin{lemma}
  $e$ is purely imaginary over $C$ iff
  $\dcl(Ce) \cap K \subseteq  \acl(C)$.
\end{lemma}
\begin{proof}
  Symmetric polynomials.

  STS $\acl(Ce) \cap K \subseteq  \acl(\dcl(Ce) \cap K)$.

  But if $a \in \acl(Ce) \cap K$,
  say $a=a_1,\ldots ,a_n$ are the conjugates over $Ce$,
  then the coefficients of $\prod_i (x-a_i)$
  are in $\dcl(Ce) \cap K$.
\end{proof}

\begin{definition}
  An $\infty$-definable set $D$ is {\bf purely} {\bf imaginary} if there is no 
  definable map $D \rightarrow  K$ with infinite image.

  Equivalently: for any $C$ over which $D$ is defined, any 
  $e \in D$ is purely imaginary over $C$.
\end{definition}

\begin{definition}
  A $\infty$-definable set $D$ is {\bf boundedly} {\bf imaginary} if any 
  definable map $D \rightarrow  \Gamma$ is bounded.

  Equivalently: for any $C$ over which $D$ is defined, any 
  $e \in D$ is boundedly imaginary over $C$, where
  $e \in \Ueq$ is {\bf boundedly} {\bf imaginary} over $C \subseteq  \U$
  if for every $\gamma \in \Gamma(Ce) := \Gamma \cap \dcl(Ce)$,
  $\tp(\gamma/C)$ is bounded (i.e.\ not $+\infty$ or $-\infty$,
  i.e.\ $\gamma$ is in the convex hull of $\Gamma(C)$).
\end{definition}

\begin{lemma} \label{l:boundedPure}
  Any boundedly imaginary $D$ is purely imaginary.
\end{lemma}
\begin{proof}
  If the image of $D \rightarrow  K$ is infinite, it contains a ball $B$.
  But for $b \in B$,
  $x \mapsto  v(x-b)$ is an unbounded map $B \rightarrow  \Gamma$.
\end{proof}

Define $\alpha\O := \{ x \in K : v(x) \geq  \alpha \}$
and $\beta\m := \{ x \in K : v(x) > \beta \}$.

\begin{lemma} \label{l:orth}
  Let $X$ be the set of closed or open balls of a fixed radius $\alpha$,
  i.e.\ $X = K / \alpha\O$ or $X = K / \alpha\m$.

  There is no definable finite correspondence $\Gamma \rightarrow  X$
  with infinite image,
  i.e.\ for any definable $Z \subseteq  \Gamma\times X$ with $\forall \gamma \in 
  \Gamma.\; |\pi_1^{-1}(\gamma) \cap Z| < \aleph_0$, $|\pi_2(Z)| < \aleph_0$.
\end{lemma}
\begin{proof}
  Else, by Swiss cheese decomposition applied to the union of the balls, 
  $\pi_2(Z)$ contains all but finitely many of the balls within a closed ball 
  $B$ of some radius $\gamma \leq  \alpha$, with $\gamma < \alpha$ in the case $X 
  = K / \alpha\O$.

  Then $Z$ induces a finite correspondence from $\Gamma$ onto $B / \gamma\m$, 
  and hence onto an infinite subset of $k$.
  By definable Skolem functions for $\Gamma$, this yields a definable partial 
  function $k \rightarrow  \Gamma$ with infinite image, contradicting strong minimality 
  of $k$.
\end{proof}

\begin{lemma} \label{l:ballsBounded}
  Let $\alpha \leq  0 \leq  \beta \in \Gamma$ and $n \geq  1$.
  Then $(\alpha\O / \beta\m)^n$ is boundedly imaginary, and hence purely 
  imaginary.
\end{lemma}
\begin{proof}
  \underline{$n=1$}: we may assume $\alpha = 0$. Suppose $f : \O / \beta\m \rightarrow  
  \Gamma$ is an unbounded $C$-definable map. Say it is unbounded above. Let 
  $\gamma > \Gamma(C)$.
  Let $g : \O \rightarrow  \O / \beta\m$ be the quotient map.
  Then $g^{-1}(f^{-1}(\gamma))$ is a boolean combination of balls each of 
  radius in $[0,\beta] \cap \Gamma(C\gamma)$.
  Now $\Gamma$ is a pure divisible ordered abelian group,
  so any definable map from $\Gamma$ to the bounded interval $[0,\beta]$ is 
  eventually constant,
  so $[0,\beta] \cap \Gamma(C\gamma) \subseteq  \Gamma(C)$.

  But the balls vary with $\gamma$ since $f$ is a function,
  so we contradict Lemma~\ref{l:orth}.

  \underline{$n=k+1$}:
  Given $f : (\alpha \O / \beta\m)^{k+1} \rightarrow  \Gamma$,
  inductively $y \mapsto  \sup_\x f(\x,y)$ defines a map
  $\alpha \O / \beta\m \rightarrow  \Gamma$, so this map is bounded, and hence so is $f$.
\end{proof}

\begin{lemma} \label{l:pureBalls}
  $e \in \Ueq$ is purely imaginary over $C$
  iff
  for some $\alpha \leq  0 \leq  \beta$ with $\alpha,\beta \in \Gamma(Ce)$,
    $e \in \dcl(\acl(C), \alpha\O / \beta\m)$.
\end{lemma}
\begin{proof}
  $\Leftarrow $:
    Immediate from Lemma~\ref{l:ballsBounded}.
    %Let $X := \{ \code{\O(1,a) + 2\beta\O(0,1)} : a \in K \} \subseteq  S_2$.
    %Then $X$ is $\beta$-definable
    %and intersection with $(1,K)$ gives a $\emptyset $-definable bijection
    %%$X \rightarrow  K / 2\beta\O$,
    %and the quotient map induces a $\beta$-definable surjection
    %$K / 2\beta\O \twoheadrightarrow  K / \beta\m \supseteq \alpha\O / \beta\m$.
    %So $\alpha\O / \beta\m \subseteq  \dcl(\beta S_2)$.
    %(Thanks to Martin Hils for explaining this trick of representing torsors 
    %as submodules.)

    %Then $e \in \dcl(\acl(C)\beta d)$ for some $d \in S_2^n$,
    %so $\acl(Ce) \cap K \subseteq  \acl(C\beta d) \cap K = \acl(C) \cap K$
    %since $\beta d$ is purely imaginary over $C$.

  $\Rightarrow $: 
    By the EI and the definition of purely imaginary,
    $e$ is interdefinable with a 
    finite tuple from $\bigcup_n S_n \cup \bigcup_n T_n \cup (\acl(C) \cap 
    K)$.

    Note that $K \ni x \mapsto  \min (\beta, v(x))$ induces a map
    $\alpha\O / \beta\m \twoheadrightarrow  [\alpha,\beta]$,
    so $[\alpha,\beta] \subseteq  \dcl(\alpha\O / \beta\m)$.
    Hence if $\alpha \leq  \alpha' < 0 < \beta' \leq  \beta$
    then $\alpha'\O / \beta'\m \subseteq  \dcl(\alpha\O / \beta\m)$.

    So it suffices to consider the case $e \in S_n$ or $e \in T_n$ for some $n$.

    Let $\Lambda \leq  K^n$ be a rank $n$ free $\O$-submodule,
    say with $\O$-basis $(\lambda_1,\ldots ,\lambda_n)$.
    
    Let $\Lambda^- := \{ \alpha : \Lambda \subseteq  (\alpha\O)^n \}$
    and $\Lambda^+ := \{ \beta : (\beta\m)^n \subseteq  \Lambda \}$.
    Note $\sup\Lambda^- = \min_{i,j} v((\lambda_i)_j) \in \Lambda^-$.

    Let $\alpha = \min \{0, \sup \Lambda^- \} \in \Lambda^-$,
    and $\beta = \max \{0, \inf \Lambda^+ \} \in \Lambda^+$.

    Then $\alpha \leq  0 \leq  \beta$ and $\alpha,\beta \in \Gamma(Ce)$
    and $(\beta\m)^n \subseteq  \Lambda \subseteq  (\alpha\O)^n$.
    then $S_n \ni \code{\Lambda} \in \dcl((\lambda_i / \beta\m)_i) \subseteq  \dcl(\alpha\O 
    / \beta\m)$.

    Also $\beta\m^n \subseteq  \m\Lambda$ (since $\m(\beta\m) = \beta\m$),
    so $T_n \supseteq \Lambda/\m\Lambda \subseteq  \dcl(\alpha\O / \beta\m)$.
    
    Since $\Lambda$ was arbitrary, we conclude.
\end{proof}

\begin{remark} \label{r:SnTnPure}
  It follows from the proof that each $S_n$ and $T_n$ is purely imaginary.
\end{remark}

\begin{lemma} \label{l:boundedBalls}
  Let $D$ be a $\infty$-definable set over $C$. TFAE:
  \begin{enumerate}[(1)]\item $D$ is boundedly imaginary.
  \item There exists a definable surjection $g : (\O / \beta\m)^n \twoheadrightarrow  D$.
  \item For some $\alpha \leq  0 \leq  \beta$ with $\alpha,\beta \in \Gamma(C)$,
  $D \subseteq  \dcl(\acl(C), \alpha\O / \beta\m)$.

  {\it NOTE: the paper has $C$ rather than $\acl(C)$, but I don't see how to get that.
  This statement is good enough for the application in Corollary~6.4.}
  \end{enumerate}
\end{lemma}
\begin{proof}
  (2) $\Rightarrow $ (1): by Lemma~\ref{l:ballsBounded}.

  (3) $\Rightarrow $ (2): by compactness, we get finitely many $g_i : (\alpha_i \O / \beta_i\m)^{n_i} \rightarrow  D$
  with $D = \bigcup_i \im g_i$.
  We can assume $\alpha_i = 0$ by multiplying,
  and we can assume $\beta_i = \max_i \beta_i =: \beta$,
  and then combine the $g_i$ into a surjection $g : (\O / \beta\m)^{\sum_i n_i} \rightarrow  D$.

  (1) $\Rightarrow $ (3): By compactness, it suffices to show that if $e \in D$,
  then such $\alpha,\beta \in \Gamma(C)$ exist with $e \in \dcl(C, \alpha\O / 
  \beta\m)$.

  Now $D$ is purely imaginary by Lemma~\ref{l:boundedPure},
  so by Lemma~\ref{l:pureBalls} we can find such $\alpha,\beta \in \Gamma(Ce)$.

  Say $\alpha = f(e)$, where $f$ is over $C$. Since $D$ is boundedly 
  imaginary, $\alpha \geq  \inf_{D \cap \dom f} f =: \alpha' \in \Gamma(C)$.
  Similarly we find $\beta \leq  \beta' \in \Gamma(C)$.
  
  Then $\alpha\O / \beta\m \subseteq  \dcl(\alpha'\O / \beta'\m)$,
  so $e \in \dcl(\acl(C), \alpha'\O / \beta'\m)$ as required.
\end{proof}

\begin{remark}
  Martin Hils remarks that in Lemma~\ref{l:pureBalls} we can equivalently ask just 
  for $\alpha,\beta \in \Gamma$ rather than require them in $\Gamma(Ce)$. From 
  this one can conclude that $\infty$-definable $D$ is purely imaginary iff 
  there is a definable surjection $f : (B^{\operatorname{open}})^n \twoheadrightarrow  D$ where 
  $B^{\operatorname{open}}$ is the set of all open balls (of all radii).

  Then it follows from this and Lemma~\ref{l:boundedBalls}, that if $D$ is 
  $\infty$-definable and purely resp.\ boundedly imaginary, it is contained in 
  a purely resp.\ boundedly imaginary definable set.
\end{remark}


\section{Redundant}

Alternative direct proof of Remark~\ref{r:SnTnPure}:

\begin{lemma}
  Any product of $S_n$'s and $T_n$'s is purely imaginary.

  So by the EI, $e \in \Ueq$ is purely imaginary over $C$ iff it is interdefinable with a 
  finite tuple from $\bigcup_n S_n(\U) \cup \bigcup_n T_n(\U) \cup (\acl(C) \cap K)$.
\end{lemma}
\begin{proof}
  Let $T$ be a completion of $\ACVF$.

  It suffices to find an uncountable model of $T$ in which each $S_n$ and 
  $T_n$ is countable,
  since the image of a definable map to $K$ with infinite image contains a 
  ball and so has the same cardinality as $K$.

  Thanks to Martin Hils for providing the following example.

  Let $L \vDash  T$ be countable with $\Gamma(L) = \Q$.
  (We can take $L$ to be an algebraic closure of $\Q(t)$, $\F_p(t)$, or $\Q$ 
  with the $p$-adic valuation.) 
  Consider $L$ as a normed field (with $\|x\| := 2^{-v(x)}$).
  
  \newcommand{\Lbar}{\bar{L}}

  Let $\Lbar$ be the completion of $L$.
  Fact: $\Lbar \vDash  \ACVF$.
  We have $|\Lbar| = 2^{\aleph_0}$,
  but $S_n(\Lbar) = \GL_n(\Lbar) / \GL_n(\O(\Lbar)) \leftarrow  \GL_n(L) / \GL_n(\O(L)) 
  = S_n(L)$ is a bijection
  since $\GL_n(\O(\Lbar))$ is an open neighbourhood of the identity
  and $\GL_n(\O(\Lbar)) \cap \GL_n(L) = \GL_n(\O(L))$.
  Similarly $k(\Lbar) = \O(\Lbar) / \m(\Lbar) \leftarrow  \O(L) / \m(L) = k(L)$ is a bijection.

  So $|S_n(\Lbar)| = \aleph_0 = |T_n(\Lbar)|$.
\end{proof}
\end{document}
