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 $K/\mathcal {O}$.

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

We survey the history of Shelah’s conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.

A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p, there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.

Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.