Mekler's Construction and Murphy's Law for 2-Nilpotent Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918406777733120 |
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| author | Boissonneau, Blaise Papadopoulos, Aris Touchard, Pierre |
| author_facet | Boissonneau, Blaise Papadopoulos, Aris Touchard, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20270 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mekler's Construction and Murphy's Law for 2-Nilpotent Groups Boissonneau, Blaise Papadopoulos, Aris Touchard, Pierre Logic Combinatorics Group Theory 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. |
| title | Mekler's Construction and Murphy's Law for 2-Nilpotent Groups |
| topic | Logic Combinatorics Group Theory |
| url | https://arxiv.org/abs/2403.20270 |