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⩾2$g\geqslant 2$ and let Σg′,n′→Σg,n$\Sigma _{g^{\prime },n^{\prime }}\rightarrow \Sigma _{g, n}$ be a finite H$H$ ‐cover of topological surfaces. We show the virtual action of the mapping class group of Σg,n+1$\Sigma _{g,n+1}$ on an H$H$ ‐isotypic component of H1(Σg′)$H^1(\Sigma _{g^{\prime }})$ has nonunitary image.