\subsection{Meromorphic Groups}
Let $\G$ be a meromorphic group in the sense of \cite{PillayScanlon}, i.e. a
0-definable group in the structure $\mathcal{A}$ of compact complex spaces.
By \cite[Fact~2.10]{PillayScanlon}, $\G$ can be uniquely identified with a
compact complex Lie group.

Considering $\G$ with its induced structure, it is a finite Morley rank group.
Suppose $\G$ is commutative and rigid. By the classification in
\cite{PillayScanlon} and the fact that any commutative complex linear
algebraic group is a product of copies of $\G_m$ and $\G_a$, there is a
definable exact sequence of Lie groups
    \[ 0 -> \G_m^n -> \G -> T -> 0 \]
where $T$ is a complex torus.  It is also shown in \cite{PillayScanlon} that
$\G$ is definable in a K\"{a}hler space; The latter may be considered in a
countable language by \cite{MoosaKahler}, so we may consider the language of
$\G$ to be the induced countable language; let $T=\Th(\G)$. 

Now let $U_\G$ be the analytic universal cover of the Lie group $\G$ as a
structure in the language of $^T$, where we interpret $\rho$ as the Lie
exponential $T_0(\G) -->> \G$, with $\rho_n(x) = \rho(x/n)$, and interpret
$^H$ for $H<=G^n$ as the $\C$-subspace $T_0(H)$ of $T_0(\G^n)=T_0(G)^n$.

\begin{proposition}
    $U_\G|=^T$.
\end{proposition}
\begin{proof}
  We show that $U_\G$ satisfies the axioms $^T'$ above.
  These are immediate from the definitions and basic properties of Lie groups.
  For \axref{ax:proj}, note that the connected component in the model
  theoretic sense is the connected component of the identity in the analytic
  sense.
\end{proof}

So by \thmref{thm:classification}, $U_\G$ is the unique kernel-preserving
extension of its restriction to the prime model $\G_0$ of $\G$, which is a
countable structure.

\begin{question}
  Could the Kummer theory of Lemma \ref{lem:kummer} apply here?
  Concretely: is $\pi^{-1}(\G_0)$ atomic over $\Lambda$? 
\end{question}