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