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

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 $ν(G)$, generated by the four infinitesimal subgroups $ν_D(G)$ associated with the distinguished sorts $K$, $\textbf{k}$, $Γ$ and $K/\mathcal{O}$. To show that $ν(G)$ is type-definable, we show that the resulting subgroups $ν_D(G)$ commute with each other as $D$ ranges over the four distinguished sorts. We then study the basic properties of $ν(G)$. Among others, we show that $ν(G_1\times G_2)=ν(G_1)\times ν(G_2)$ and that if $G_1\le G$ is a definable subgroup then $ν(G_1)$ is relatively definable in $ν(G)$. We also discuss possible connections between $\mathrm{dp\text{-}rk}(ν(G))$ and elimination of imaginaries.