21.120 (2026)

A pro-$p$ group is (relatively) strictly finitely presented if it is the pro-$p$ completion of a group that is finitely presented (respectively, finitely presented in some finitely-based variety of groups). A pro-$p$ group is finitely axiomatizable if it is determined up to isomorphism by a single sentence in the first-order language of group theory.
$\qquad$ a) Does there exist a (relatively) strictly finitely presented pro-$p$ group that is not finitely axiomatizable in the class of all pro-$p$ groups?
$\qquad$ b) In particular, is every finitely generated free pro-$p$ group finitely axiomatizable?

See (A. Nies, K. Tent, D. Segal, Proc. London Math. Soc. (3), 123 (2021), 597–635; D. Segal, Preprint, 2025, https://arxiv.org/abs/2505.04816).