Some model theory of profinite groups

Abstract

The main purpose of these notes is to give more background and details for the results obtained in [13] which rely heavily on deep results by Lazard, Lubotzky, Mann, du Sautoy and others. At the center is Lazard's purely group theoretic characterization of $p$-adic analytic groups given in [8] (see Section 4.2 below). By Lazard's result a compact topological group is a $p$-adic analytic group if and only if it has an open uniformly powerful pro-$p$ subgroup (see Section 3).