Freeness of the random fundamental group
Abstract
Let Y(n,p) denote the probability space of random 2-dimensional simplicial complexes in the Linial–Meshulam model, and let Y∼Y(n,p) denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber and Kappeler, it is shown that for p=o(1/n) with high probability π1(Y) is free. Following that, a paper of Costa and Farber shows that for values of p which satisfy 3/n<p≪n−46/47 with high probability, π1(Y) is not free. Here, we improve on both these results to show that there are explicit constants γ2<c2<3, so that for p<γ2/n with high probability Y has free fundamental group and that for p>c2/n with high probability Y has fundamental group which either is not free or is trivial.