On groups interpretable in various valued fields

TL;DR

This paper investigates infinite groups interpretable in various valued fields, establishing their unbounded exponent and structural properties, especially in the dp-minimal case where they are almost abelian, and links them to distinguished sorts within the valued field structure.

Contribution

It introduces a novel approach to analyze interpretable groups in valued fields, showing their unbounded exponent and structural constraints, and associates them with distinguished sorts.

Findings

Infinite interpretable groups have unbounded exponent.

Dp-minimal interpretable groups are abelian-by-finite.

Associates interpretable groups with distinguished sorts like the valued field, residue field, or value group.

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.