Large random simplicial complexes, II; The fundamental group

We study random simplicial complexes in the multi-parameter model focusing mainly on the properties of the fundamental groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups never have odd-prime torsion and their geometric and cohomological dimensions are either 0, 1, 2 or ∞. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical, with probability tending to one.