Freeness of the random fundamental group. (English) Zbl 1434.60051
Summary: Let \(Y(n,p)\) denote the probability space of random 2-dimensional simplicial complexes in the Linial-Meshulam model, and let \(Y \sim Y(n,p)\) denote a random complex chosen according to this distribution. In a paper of D. Cohen et al. [Discrete Comput. Geom. 47, No. 1, 117–149 (2012; Zbl 1237.55009)], it is shown that for \(p=o(1/n)\) with high probability \(\pi_1(Y)\) is free. Following that, a paper of A. E. Costa and M. Farber [Isr. J. Math. 209, Part 2, 883–927 (2015; Zbl 1330.55016)] shows that for values of \(p\) which satisfy \(3/n < p \ll n^{- 4 6 / 4 7}\) with high probability, \( \pi_1(Y)\) is not free. Here, we improve on both these results to show that there are explicit constants \(\gamma_2 < c_2 < 3\), so that for \(p < \gamma_2 /n\) with high probability \(Y\) has free fundamental group and that for \(p> c_2 / n\) with high probability \(Y\) has fundamental group which either is not free or is trivial.
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