The Fundamental Group of Random 2-complexes

In this article we find the threshold for simple connectivity of the random 2dimensional simplicial complexes Y (n, p) introduced by Linial and Meshulam [10] to be roughly p = n−1/2. One motivation for this is continuing the thread of probabilistic topology initiated by Linial and Meshulam [10], and even earlier by Erdős and Rényi [3]. (Other recent work concerning the topology of random simplicial complexes can be found in [8, 9, 11, 14].) Another motivation for this study is the connection to the random groups studied in geometric group theory [12]. In face we must use geometric group theory techniques to show that in the sparse regime the fundamental group is hyperbolic on the way to showing that it is nontrivial; in particular, we apply Gromov’s localto-global principle for linear isoperimetric inequalities. Erdős and Rényi initiated the now vast subject of random graphs with their edge-independent model G(n, p) [3].