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Preprint Number 1831

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1831. Tim Clausen
Dp-minimal profinite groups and valuations on the integers

Submission date: 20 August 2020


We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup A such that either A is a direct product of countably many copies of 𝔽_p for some prime p, or A is of the form A ≅ ∏_p ℤ_p^{α_p} × A_p where α_p < ω and A_p is a finite abelian p-group for each prime p. Moreover, we show that if A is of this form, then there is a fundamental system of open subgroups such that the expansion of A by this family of subgroups is dp-minimal. Our main ingredient is a quantifier elimination result for a class of valued abelian groups. We also apply it to (ℤ,+) and we show that if we expand (ℤ,+) by any chain of subgroups (B_i)_{i<ω}, we obtain a dp-minimal structure. This structure is distal if and only if the size of the quotients B_i/B_{i+1} is bounded.

Mathematics Subject Classification: 03C60

Keywords and phrases:

Full text arXiv 2008.08797: pdf, ps.

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