On the density and surjectivity of $\mathbf{SO(3)}$-Witten-Reshetikhin-Turaev quantum representations

In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations $\rho_{p,g,\underline{\lambda}}\colon\mathrm{PMod} (\Sigma_{g,n})\longrightarrow \mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$ of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface $\Sigma_{g,n}$ of genus $g\geq 3$, any number $n\geq 0$ of punctures, and any coloration $\underline{\lambda}$ of the punctures, $\rho_{p,g,\underline{\lambda}}$ has dense image in the projective unitary group $\mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$, extending a landmark result of Larsen and Wang. Moreover, we show that the representations $\rho_{p,g,\underline{\lambda}}$ are surjective modulo any unramified maximal ideal of $\mathbb{Z}[\zeta_p]$, establishing an effective version of strong approximation for these representations. We also give several applications of our main results to residual finite simpleness of $\mathrm{PMod}(\Sigma_{g,n})$ (answering a question of Masbaum and Reid); to subnormal cores of some subgroups of $\mathrm{PMod}(\Sigma_{g,n})$; to realizability of congruence classes of quantum invariants; to embedding obstructions between $3$-manifolds; and to homological stability for mapping class groups with coefficients in $\mathrm{SO}(3)$-quantum representations.