  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.
