On freeness of the random fundamental group

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 Cohen, Costa, Farber, and Kappeler, it is shown that for $p = o(1/n)$ with high probability $\pi_1(Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ which satisfy $3/n < p \ll n^{-46/47}$, with high probability $\pi_1(Y)$ is not free. Here we improve on both of 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.