5 problems

Let $\Sigma$ be the surface under consideration, let $\widetilde{\Sigma}\to\Sigma$ be a finite characteristic cover, and let $v\in {\rm H}_1(\widetilde{\Sigma},\mathbb{Q})$ be a no…

Let $\Sigma$ be the surface under consideration and let $\phi\in\operatorname{Mod}(\Sigma)$ be pseudo-Anosov. A finite cover $\widetilde{\Sigma}\to\Sigma$ is allowed to carry a lif…

Let $S_g$ be a surface of genus $g\geq 2$, and let a Prym representation be the linear representation of $\operatorname{Mod}(S_g)$ on the first homology of a finite characteristic…

Let $p:\widetilde{\Sigma}\to\Sigma$ be a finite-sheeted regular, possibly branched cover of compact surfaces with boundary and deck group $D$. Let $\mathcal{K}$ be the groupoid def…

Let $S$ be a surface and let $H$ be a nonelementary subgroup of its mapping class group. A cover $\widetilde{S}$ to which $H$ lifts is a cover on which every element of $H$ admits…