Abstract: We continue the study ofn-dependent groups, fields and related structures, largely motivated by the conjecture that everyn-dependent field is dependent. We provide evidence toward this conjecture by showing that every infiniten-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters inn-dependent groups, generalizing Shelah’s absoluteness of$G^{00}$in dependent theories and relative absoluteness of$G^{00}$in$2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly$2$-dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are$2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show thatn-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for$2$-dependence and use it to deduce$2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theoryTby a generic predicate is dependent if and only if it isn-dependent for somen, if and only if the algebraic closure inTis disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.

Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that $k$-dependence of the theory is preserved, for all $k \in \mathbb{N}$, and that NTP$_2$ is preserved. We apply this result to obtain first examples of strictly $k$-dependent groups (with no additional structure).