The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel

Abstract

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $\Sigma_{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(\Sigma_{g,n}),$ the matrix $\rho_r(f)$ has infinite order for any large $r,$ where $\rho_r$ is the $\mathrm{SO}(3)$-WRT quantum representation of the mapping class group $\mathrm{Mod}(\Sigma_{g,n})$ at a primitive $r$-th root of unity. We prove this conjecture for prime $r$ and any $f\in [J_2(\Sigma_{g,n}),J_2(\Sigma_{g,n})],$ where $J_2(\Sigma_{g,n})$ is the Johnson kernel.