Tangle Free Permutations and the Putman–Wieland Property of Random Covers

Abstract

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 $\Sigma ^{p}_{g}$ that 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.