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

TL;DR

This paper proves the Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel at prime levels, showing that certain quantum representations have infinite order for pseudo-Anosov elements.

Contribution

It establishes the conjecture for prime levels and elements in the derived subgroup of the Johnson kernel, advancing understanding of quantum representations of mapping class groups.

Findings

Proved the conjecture for prime r and elements in [J_2, J_2].

Showed that the matrix rho_r(f) has infinite order for these cases.

Extended the validity of the conjecture to a new class of mapping class group elements.

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.