Search for: All records

We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) 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 (1982) 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 pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), 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. 

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups.