Remarks on compactifications of pseudofinite groups

TL;DR

This paper investigates the properties of the Bohr compactification in pseudofinite groups, showing triviality in ultraproducts of finite simple groups and a commutative-by-profinite structure in definable cases.

Contribution

It provides new insights into the structure of Bohr compactifications in pseudofinite groups, addressing a question posed by Boris Zilber.

Findings

Bohr compactification of ultraproducts of finite simple groups is trivial.

Definable Bohr compactification of pseudofinite groups is commutative-by-profinite.

Abstract

We discuss the Bohr compactification of a pseudofinite group, motivated by a question of Boris Zilber. Basically referring to results in the literature we point out (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the "definable" Bohr compactification of any pseudofinite group G, relative to a nonstandard model of set theory in which it is definable, is commutative-by-profinite.