Profinite groups with NIP theory and $p$-adic analytic groups

TL;DR

This paper characterizes profinite groups with NIP theories as those containing a finite index normal subgroup that is a product of finitely many compact p-adic analytic groups, linking model theory with p-adic group structure.

Contribution

It establishes a precise model-theoretic characterization of profinite groups with NIP theories in terms of their p-adic analytic components and normal subgroups.

Findings

NIP theory corresponds to groups with a finite index product of p-adic analytic groups.

Any NIP profinite group has an open prosoluble normal subgroup.

NIP can be weakened to NTP2 in this context.

Abstract

We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of {\em all} open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact $p$-adic analytic groups, for distinct primes $p$. In fact, the condition NIP can here be weakened to NTP${}_2$. We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.