  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.

Algebraic Topology;Combinatorics;Group Theory;Probability