Putman–Wieland conjecture on finite-index subgroups of surface groups

Let $\Sigma^p_g$ be a smooth surface of genus $g\ge 2$ with $p\ge 0$ punctures, let $\star\in\Sigma^p_g$ be a basepoint, and let $K<\pi_1(\Sigma^p_g,\star)$ be a finite-index subgroup. Let $\pi:(\Sigma',\star')\to(\Sigma^p_g,\star)$ be the corresponding pointed cover, let $\widehat{\Sigma}$ be the closed surface obtained by filling in the punctures of $\Sigma'$, and let $\Gamma_K<\operatorname{Mod}^{p+1}_g$ be the finite-index subgroup preserving $K$ under the standard action on $\pi_1(\Sigma^p_g,\star)$. The subgroup $K$ has the Putman–Wieland property if, for every nonzero vector $v\in H^1(\widehat{\Sigma},\mathbb{Q})$, its $\Gamma_K$-orbit is infinite.

Putman–Wieland conjecture. For every $g\ge 2$, every $p\ge 0$, and every finite-index subgroup $K<\pi_1(\Sigma^p_g,\star)$, the subgroup $K$ has the Putman–Wieland property.

The conjecture concerns the dynamics of mapping class groups on the first cohomology of compactified finite covers of punctured surfaces. It was disproved in genus $g=2$ by Marković, so the conjecture is refuted.