Freeness of the random fundamental group
Abstract
Let denote the probability space of random 2-dimensional simplicial complexes in the Linial–Meshulam model, and let denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber and Kappeler, it is shown that for with high probability is free. Following that, a paper of Costa and Farber shows that for values of which satisfy with high probability, is not free. Here, we improve on both these results to show that there are explicit constants , so that for with high probability has free fundamental group and that for with high probability has fundamental group which either is not free or is trivial.