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arXiv · 1603.02179

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

Dugald Macpherson·Katrin Tent
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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.

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Dugald Macpherson, Katrin Tent. 2016-03-07. Profinite groups with NIP theory and $p$-adic analytic groups. https://doi.org/10.1112/blms%2Fbdw064

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