Mekler's construction and generalized stability

TL;DR

This paper explores how Mekler's construction preserves various model-theoretic properties, including stability, simplicity, and k-dependence, when interpreting structures as groups, and provides new examples of k-dependent groups.

Contribution

It proves that Mekler's construction preserves k-dependence and NTP2, and introduces the first examples of strictly k-dependent groups without extra structure.

Findings

k-dependence is preserved for all k

NTP2 is preserved under Mekler's construction

First examples of strictly k-dependent groups without additional structure

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).