Fundamental groups of clique complexes of random graphs

We study fundamental groups of clique complexes associated to random Erdős–Rényi graphs (Formula presented.). We establish thresholds for a number of properties of fundamental groups of these complexes (Formula presented.). In particular, if (Formula presented.), then we show that (Formula presented.) asymptotically almost surely (a.a.s.), where (Formula presented.) and (Formula presented.) denote the geometric dimension and cohomological dimension correspondingly. It is known that the fundamental group (Formula presented.) is trivial for (Formula presented.). We prove that for (Formula presented.) the fundamental group (Formula presented.) has 2-torsion but has no (Formula presented.) -torsion for any given prime (Formula presented.). We also prove that aspherical subcomplexes of the random clique complex (Formula presented.) satisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, a.a.s.