Compactifications of pseudofinite and pseudo-amenable groups

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing's work, the Jordan-Schur Theorem, and a (relatively) more recent result of Kazhdan \cite{Kazh} on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudo-amenable groups to compact Lie groups. Together with the stabilizer theorems of \cite{HruAG,MOS}, we obtain a uniform (but non-quantitative) analogue of Bogolyubov's Lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.