Semisimple groups interpretable in various valued fields

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, where is a K-linear group and is a -linear group. The analysis is carried out by studying the interaction of G with four distinguished sorts: the valued field K, the residue field, the value group, and the closed -balls.