Applications of the algebraic geometry of the Putman-Wieland conjecture

We give two applications of our prior work toward the Putman-Wieland conjecture. First, we deduce a strengthening of a result of Marković-Tošić on virtual mapping class group actions on the homology of covers. Second, let $g\ge 2$ and let ${\mathrm{\Sigma }}_{g^{\mathrm{\prime }},n^{\mathrm{\prime }}}\to {\mathrm{\Sigma }}_{g,n}$ be a finite $H$-cover of topological surfaces. We show the virtual action of the mapping class group of ${\mathrm{\Sigma }}_{g,n+1}$ on an $H$-isotypic component of $H^1({\mathrm{\Sigma }}_{g^{\mathrm{\prime }}})$ has non-unitary image.