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

Abstract

In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations ${\rho }_{p,g,\underline{\lambda }}:\mathrm{PMod}({\Sigma }_{g,n})⟶{\mathrm{PSU}}_{d_{p,g,\underline{\lambda }}}$ of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface ${\Sigma }_{g,n}$ of genus $g\ge 3$, any number $n\ge 0$ 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.

79 pages, 4 figures