Tangle free permutations and the Putman-Wieland property of Random covers

Let $\Sigma^p_g$ denote a surface of genus $g$ and with $p$ punctures. Our main result is that the fraction of degree $n$ covers of $\Sigma^p_g$ which have the Putman-Wieland property tends to $1$ as $n\to \infty$. In addition, we show that the monodromy of a random cover of $\Sigma^p_g$ is asymptotically almost surely tangle free.