Semisimple groups interpretable in various valued 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 $K/\mathcal {O}$ .