Quotients of surface groups and homology of finite covers via quantum representations

Abstract

We prove that for each sufficiently complicated orientable surface $S$, there exists an infinite image linear representation $\rho$ of $\pi_1(S)$ such that if $\gamma\in\pi_1(S)$ is freely homotopic to a simple closed curve on $S$, then $\rho(\gamma)$ has finite order. Furthermore, we prove that given a sufficiently complicated orientable surface $S$, there exists a regular finite cover $S'\to S$ such that $H_1(S',\mathbb{Z})$ is not generated by lifts of simple closed curves on $S$, and we give a lower bound estimate on the index of the subgroup generated by lifts of simple closed curves. We thus answer two questions posed by Looijenga, and independently by Kent, Kisin, March\'e, and McMullen. The construction of these representations and covers relies on quantum $\text{SO}(3)$ representations of mapping class groups.