On simple groups definable in some valued fields
Jakub Gismatullin, Immanuel Halupczok, Dugald Macpherson
TL;DR
The paper classifies definably simple groups definable in 1-$h$-minimal henselian valued fields. It constructs a definable dwsd Lie group structure on such a group $G$, derives a Jacobian/adjoint representation into $\mathrm{SL}_d(K)$, and analyzes its Lie algebra to constrain $G$ to arise from a semisimple, almost $K$-simple isotropic linear algebraic group $\mathbf{H}$ with $G^*$ positioned between $\mathbf{H}(K)^+$ and $\mathbf{H}(K)$ (or is definably isomorphic to $\mathbf{H}(K)$ modulo center in positive characteristic). The proof leverages boundedness results (Prasad–Tits-type) and a detailed boundedness analysis to exclude pathological cases, thereby reducing the definable group to algebraic data. The results illuminate how definable groups in valued-field settings relate to linear algebraic groups, with consequences for tame geometric contexts and interpretability in valued-field frameworks. The work also connects to broader model-theoretic developments on interpretable groups and uses the adjoint representation to bridge model-theoretic definability with classical algebraic structure.
Abstract
We prove that non-abelian definable, definably simple groups in 1-h-minimal henselian valued fields are essentially already linear algebraic groups. Here, the group is assumed to live in the home sort. We have a similar result in pure algebraically closed valued fields of positive characteristic, under the additional assumption that the definable group is a subgroup of a linear algebraic group.