Applications of the algebraic geometry of the Putman-Wieland conjecture

TL;DR

This paper explores algebraic geometry methods to advance the understanding of the Putman-Wieland conjecture, providing new results on mapping class group actions on surface homology and their properties.

Contribution

It offers two key applications: a strengthened result on virtual mapping class group actions and a demonstration of non-unitary images in specific surface covers.

Findings

Strengthened the result of Marković-Tošić on virtual mapping class group actions.

Showed the virtual action of the mapping class group has non-unitary image in certain surface covers.

Provided new insights into the algebraic geometry aspects 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 Markovi\'c-To\v{s}i\'c on virtual mapping class group actions on the homology of covers. Second, let $g\geq 2$ and let $\Sigma_{g',n'}\to \Sigma_{g, n}$ 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 $H^1(\Sigma_{g'})$ has non-unitary image.