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 $\mathbf{k}$ (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 [11] to circumvent elimination of imaginaries.