Mekler's construction and generalized stability

Abstract

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).

v.2 many minor corrections and presentation improvements throughout the article, more details were added in some of the proofs; Remarks 2.12, 2.13 and Problem 5.8 are new; accepted to the Israel Journal of Mathematics