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

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