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Published online by Cambridge University Press: 17 September 2025
In this paper, we study the kernels of the $\mathrm{SO}(3)$-Witten–Reshetikhin–Turaev quantum representations
$\rho_p$ of mapping class groups of closed orientable surfaces
$\Sigma_g$ of genus g. We investigate the question whether the kernel of
$\rho_p$ for p prime is exactly the subgroup generated by pth powers of Dehn twists. We show that if
$g\geq 3$ and
$p\geq 5$ then
$\mathrm{Ker} \rho_p$ is contained in the subgroup generated by pth powers of Dehn twists and separating twists, and if
$g\geq 6$ and p is a large enough prime then
$\mathrm{Ker} \rho_p$ is contained in the subgroup generated by the commutator subgroup of the Johnson subgroup and by pth powers of Dehn twists.