The fundamental group of random 2-complexes

TL;DR

This paper investigates the properties of random 2-complexes, identifying the threshold for simple connectivity, and demonstrates that below this threshold, the fundamental group is hyperbolic, with several independent topological results.

Contribution

It establishes the threshold for simple connectivity in random 2-complexes and shows hyperbolicity of the fundamental group below this threshold, introducing new classification and isoperimetric results.

Findings

Threshold for simple connectivity at p = n^{-1/2}

Fundamental group is hyperbolic when p = O(n^{-1/2 -psilon})

Classified homotopy types of sparse 2-complexes

Abstract

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\epsilon}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.