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

TL;DR

This paper constructs infinite image linear representations of surface groups and finite covers with specific homological properties using quantum SO(3) representations, answering longstanding questions in the field.

Contribution

It introduces new quantum representation techniques to produce surface group quotients and homology of covers with prescribed properties, advancing understanding of surface group structures.

Findings

Existence of infinite image linear representations with finite order images for simple closed curves.

Construction of finite covers with non-generated homology by lifts of simple closed curves.

Lower bounds on the index of subgroups generated by lifts of simple closed curves.

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.