Applications of the algebraic geometry of the Putman-Wieland conjecture

Abstract

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 ${\Sigma }_{g^{\prime },n^{\prime }}\to {\Sigma }_{g,n}$ be a finite $H$-cover of topological surfaces. We show the virtual action of the mapping class group of ${\Sigma }_{g,n+1}$ on an $H$-isotypic component of $H^1({\Sigma }_{g^{\prime }})$ has non-unitary image.

arXiv comment21 pages, comments welcome! Typo fixes; updated in light of arXiv:2209.12958