Abelian groups from random hypergraphs

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 $p=\omega (1/n^{k-1})$, $\text{coker}(M(\mathcal{H}))$ is torsion-free.

10 pages