\documentclass[a4paper]{article}

\usepackage{amsmath}
\newcommand{\acl}{\operatorname{acl}}
\begin{document}
%\bibliographystyle{alpha}

Building on previous work on the model theory of belles paires and their
imaginaries, this paper gives a characterisation of interpretable and
definable groups in belles paires under certain hypotheses. The strongest
result is for pairs of algebraically closed fields; it states that any group
interpretable in a pair $F<K$ of algebraically closed fields is isogenous to
an interpretable group $G$ for which there are exact sequences
\[ 1 \rightarrow N(K) \rightarrow G(K) \rightarrow H(F) \rightarrow 1 \]
\[ 1 \rightarrow N'(F) \rightarrow V(K) \rightarrow N(K) \rightarrow 1 ,\]
where $H$,$V$ and $N'$ are algebraic groups, and $H$ and $N'$ are over
$F$. If the group is actually definable rather than merely interpretable, $N$
can itself be taken to be an algebraic group.

The proof is conducted in the generality of belles paires of a strongly
minimal theory with infinite $\acl(\emptyset)$, and the version for definable
groups is proven in the greater generality of belles paires of an arbitrary
stable theory with elimination of imaginaries and NFCP.

The paper is clear and concise, and was a pleasure to read. The proofs are
elegant applications of geometric stability techniques to pre-existing
analyses of forking and imaginaries in pairs.

%\bibliography{rigidity}

\end{document}
