Mekler's Construction and Murphy's Law for 2-Nilpotent Groups

Abstract

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 (1) obtain transfer principles for stably embedded pairs of Mekler groups and (2) construct strictly ${\mathsf{NFOP}}_k$ pure groups for all $k\in {\mathbb{N}}_{>2}$. We also answer a question of Chernikov and Hempel on transfer of burden.

arXiv comment50 pages. Corrections following referee suggestions. Examples have been added, and the exposition in Sec. 2 has been improved. Prop. 3.4 (now Cor. 3.6) and Def. 3.12 (now 3.13) have been corrected. The proof of Prop. 3.13 (now 3.17) has been expanded

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• Some properties of analytic difference fields
• On n-dependence
• Indiscernibles, EM-types, and Ramsey Classes of Trees
• The domination monoid in henselian valued fields