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 $S^{\prime }\to S$ such that $H_1(S^{\prime },\mathbb{Z})$ 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, Marché, and McMullen. The construction of these representations and covers relies on quantum $\text{SO}(3)$ representations of mapping class groups.

22 pages, revised version. To appear in Inventiones mathematicae

Cited by in corpus (13)

• Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_n)
• Quotients of the mapping class group by power subgroups
• Representations of surface groups with finite mapping class group orbits
• Surface group representations in ${\mathrm{SL}}_2(\mathbb{C})$ with finite mapping class orbits
• The $p$-curvature conjecture and monodromy about simple closed loops
• Profinite completions of Burnside-type quotients of surface groups
• Curves with prescribed symmetry and associated representations of mapping class groups
• A remark on the homology of finite coverings of a surface
• Simple closed curves, non-kernel homology and Magnus embedding
• The Extrinsic Primitive Torsion Problem
• Mapping class groups of surfaces of genus $\ge 3$ do not virtually surject to $\mathbb{Z}$
• Linear representations of hyperelliptic mapping class groups
• On TQFT representations of mapping class groups with boundary