On groups interpretable in various valued fields

Abstract

We study infinite groups interpretable in three families of valued fields: V-minimal, power bounded T-convex, and p-adically closed fields. We show that every such group G has unbounded exponent and that if G is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field K, its residue field \({\textbf {k}}\) (when infinite), its value group \(\Gamma \), or \(K/\mathcal {O}\), where \(\mathcal {O}\) is the valuation ring. Our work uses and extends techniques developed in Halevi et al. (Adv Math 404:108408, 2022) to circumvent elimination of imaginaries.