Mekler's construction and Murphy's law for 2-nilpotent groups

Blaise Boissonneau, Aris Papadopoulos and Pierre Touchard

Vol. 341 (2026), No. 2, 219–274

DOI: 10.2140/pjm.2026.341.219

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

Keywords

Mekler's construction, nilpotent groups of class 2, model theory, relative quantifier elimination, Shelah's classification

Mathematical Subject Classification

Primary: 03C10, 03C45, 03C60

Secondary: 03C50, 20F18

Milestones

Received: 20 January 2025

Revised: 20 November 2025

Accepted: 21 November 2025

Published: 23 March 2026