Abelian groups from random hypergraphs

TL;DR

This paper studies the structure of abelian groups derived from random hypergraphs by analyzing the cokernel of associated incidence matrices, showing that for certain probabilities, these groups are torsion-free.

Contribution

It introduces a model for random abelian groups via hypergraph incidence matrices and proves torsion-freeness under specific probabilistic conditions.

Findings

For p = ω(1/n^{k-1}), the cokernel is torsion-free.

The model connects random hypergraphs to algebraic structures.

Supports conjectures in random simplicial complex theory.

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 Erd\H{o}s--R\'{e}nyi 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 $p = \omega(1/n^{k - 1})$, $\text{coker}(M(\mathcal{H}))$ is torsion-free.