Submission date: 29 March 2024

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 the first examples of strictly NFOP_k pure groups for all k ∈ ℕ_{>2}. We also answer a question of Chernikov and Hempel on the transfer of burden.

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Full text arXiv 2403.20270: pdf, ps.