A profinite analogue of Lasserre's theorem

Abstract

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of finite presentability.

Solves Problem 1 in `Finite axiomatizability for profinite groups', Proc. London Math. Soc.(3) 123, (2021)