AI 中文总结
研究曲面映射类群的$\mathrm{SO}(3)$-量子表示，证明其在射影酉群中有稠密像且模$\mathbb{Z}[\zeta_p]$的非分歧极大理想是满射，给出这些结果在剩余有限单性、子群次正规核等多方面的应用。
AI 中文摘要
本文建立了曲面映射类群在单位根处的$\mathrm{SO}(3)$-量子表示$\rho_{p,g,\underline{\lambda}}\colon\mathrm{PMod} (\Sigma_{g,n})\longrightarrow \mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$的几个新的基本性质。对于$g\geq 3$的曲面$\Sigma_{g,n}$、任意数量的穿孔$n\geq 0$以及穿孔的任意着色$\underline{\lambda}$，证明了$\rho_{p,g,\underline{\lambda}}$在射影酉群$\mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$中有稠密像，扩展了Larsen和Wang的里程碑式结果。还表明$\rho_{p,g,\underline{\lambda}}$模$\mathbb{Z}[\zeta_p]$的任何非分歧极大理想是满射的，建立了这些表示的有效强逼近版本。此外，给出了主要结果在$\mathrm{PMod}(\Sigma_{g,n})$的剩余有限单性、$\mathrm{PMod}(\Sigma_{g,n})$某些子群的次正规核、量子不变量同余类的可实现性、3-流形之间的嵌入障碍以及具有$\mathrm{SO}(3)$-量子表示系数的映射类群的同调稳定性等方面的应用。
英文摘要
In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations $ρ_{p,g,\underlineλ}\colon\mathrm{PMod} (Σ_{g,n})\longrightarrow \mathrm{PSU}_{d_{p,g,\underlineλ}}$ of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface $Σ_{g,n}$ of genus $g\geq 3$, any number $n\geq 0$ of punctures, and any coloration $\underlineλ$ of the punctures, $ρ_{p,g,\underlineλ}$ has dense image in the projective unitary group $\mathrm{PSU}_{d_{p,g,\underlineλ}}$, extending a landmark result of Larsen and Wang. Moreover, we show that the representations $ρ_{p,g,\underlineλ}$ are surjective modulo any unramified maximal ideal of $\mathbb{Z}[ζ_p]$, establishing an effective version of strong approximation for these representations. We also give several applications of our main results to residual finite simpleness of $\mathrm{PMod}(Σ_{g,n})$ (answering a question of Masbaum and Reid); to subnormal cores of some subgroups of $\mathrm{PMod}(Σ_{g,n})$; 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.
Comments79 pages, 4 figures
