The fundamental group of random 2-complexes

Abstract

We study Linial-Meshulam random 2 2 -complexes Y ( n , p ) Y(n,p) , which are 2 2 -dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n − 1 / 2 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 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 − ϵ p = O( n^{-1/2 -\epsilon } ), the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2 2 -dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.