The infinitesimal subgroup of interpretable groups in some dp-minimal valued fields

TL;DR

This paper investigates the structure of infinitesimal subgroups within interpretable groups in certain dp-minimal valued fields, revealing their properties and potential links to model-theoretic ranks and elimination of imaginaries.

Contribution

It introduces a new type-definable infinitesimal subgroup associated with interpretable groups in dp-minimal valued fields and analyzes its properties and interactions.

Findings

The subgroup $ u(G)$ is type-definable and generated by four commuting infinitesimal subgroups.

The subgroup $ u(G)$ behaves well under direct products and definable subgroups.

Connections between the dp-rank of $ u(G)$ and elimination of imaginaries are discussed.

Abstract

We continue our local analysis of groups interpretable in various dp-minimal valued fields, as introduced in [8]. We associate with every infinite group $G$ interpretable in those fields an infinite type-definable infinitesimal subgroup $\nu(G)$, generated by the four infinitesimal subgroups $\nu_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $\Gamma$ and $K/\mathcal{O}$. To show that $\nu(G)$ is type-definable, we show that the resulting subgroups $\nu_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $\nu(G)$. Among others, we show that $\nu(G_1\times G_2)=\nu(G_1)\times \nu(G_2)$ and that if $G_1\le G$ is a definable subgroup then $\nu(G_1)$ is relatively definable in $\nu(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(\nu(G))$ and elimination of imaginaries.