Abelian groups from random hypergraphs

Andrew Newman

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arXiv · 2111.10641

Abelian groups from random hypergraphs

Andrew Newman

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Abstract

For a $k$-uniform hypergraph $\mathcal{H}$ on vertex set $\{1, ..., n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős--Rényi random $k$-uniform hypergraph, $\text{coker}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = ω(1/n^{k - 1})$, $\text{coker}(M(\mathcal{H}))$ is torsion-free.

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Andrew Newman. 2021-11-20. Abelian groups from random hypergraphs. https://arxiv.org/abs/2111.10641

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