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 $\nu (G)$, generated by the four infinitesimal subgroups ${\nu }_D(G)$ associated with the distinguished sorts $K$, $\text{k}$, $\Gamma$ 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 $\mathrm{dp}\text{-}\mathrm{rk}(\nu (G))$ and elimination of imaginaries.