Compactifications of pseudofinite and pseudo-amenable groups

TL;DR

This paper simplifies and extends results on the structure of pseudofinite and pseudo-amenable groups' compactifications, connecting classical theorems with modern logic and group theory to analyze their properties and homomorphisms.

Contribution

It provides simplified proofs of existing results, extends the scope to amenable groups, and develops a continuous logic framework for studying definable homomorphisms.

Findings

Definable compactifications of pseudofinite groups have abelian connected components.

A continuous logic framework for pseudo-amenable groups is established.

A uniform analogue of Bogolyubov's Lemma for amenable groups is obtained.

Abstract

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.