Mekler’s construction and Murphy’s law for2-nilpotent groups

Mekler's construction is a powerful technique for building purely algebraic structures from combinatorial ones.Its power lies in the fact that it allows various model-theoretic tameness properties of the combinatorial structure to transfer to the algebraic one.In this paper, we push this ideology much further, describing a broad class of properties that transfer through Mekler's construction.This technique subsumes many well-known results and opens avenues for many more.As a straightforward application of our methods, we obtain transfer principles for stably embedded pairs of Mekler groups and construct strictly NFOP k pure groups for all k >2 .We also answer a question of Chernikov and Hempel on transfer of burden.