Profinite groups with NIP theory and p-adic analytic groups

We consider profinite groups as 2‐sorted first‐order structures, with a group sort, and a second sort that acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of 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 that 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