Semisimple groups interpretable in various valued fields

TL;DR

This paper classifies infinite interpretable semisimple groups in certain valued fields, showing they are essentially products of linear groups over the field and residue field, based on their interaction with key sorts.

Contribution

It provides a structural classification of interpretable semisimple groups in valued fields, linking them to linear groups over distinguished sorts.

Findings

Semisimple groups are, up to finite index, products of $K$-linear and $ extbf{k}$-linear groups.

The analysis involves studying interactions with valued field, residue field, value group, and $0$-balls.

The classification applies to power bounded $T$-convex, $V$-minimal, and $p$-adically closed fields.

Abstract

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $\Gamma$, and the closed $0$-balls $K/\mathcal{O}$.