The fundamental group of random 2-complexes

////////////////////////////////////////////

We study Linial-Meshulam random $2$-complexes, which are two-dimensional analogues of Erdős–Rényi random graphs. We find the threshold for simple connectivity to be roughly $p = n^{-1/2}$. (The exponent is sharp.) 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.