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https://doi.org/10.1142/S1793525319500468Cited by:3 (Source: Crossref)

Let Y(n,p) denote the probability space of random 2-dimensional simplicial complexes in the Linial–Meshulam model, and let YY(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<pn46/47 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.

AMSC: 60C05, 20J06