映射类群在有理完整性下的穿孔球模上的动力学
Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy
摘要：
设$M$是一个四孔球面，$Γ$是$M$固定边界$\partial M$的映射类群.群$Γ$作用在$M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$上，$M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$是$M$上的平坦$SL(2,C)$ -联络的完全可约$SL(2,C)$ -规范等价类空间，而$B$在$\partial M$上是固定的。设$B \in (-2,2)^4$和$M_B$是$M_B(SL(2,C))$的实点的紧分支。这些点对应于SU（2）-表示或$SL(2,R)$ -表示。$Γ$ -作用保持$M_B$，我们研究了$Γ$ -作用在$M_B$上的拓扑动力学，并证明了对于一个稠密的完整集$B \in (-2,2)^4$，$Γ$ -轨道在$M_B$中是稠密的。我们还产生了一类表示$ρ\in \Hom_B^+(pi_1(M),SL(2,R))$，使得$[ρ]$的$Γ$ -轨道在$M_B(SL(2,R))$的紧分支中是有限的，但$ρ(π_1(M))$在$SL(2,R)$中是稠密的.
英文摘要：
Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing the boundary $\partial M$. The group $Γ$ acts on $M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$ which is the space of completely reducible $SL(2,C)$-gauge equivalence classes of flat $SL(2,C)$-connections on $M$ with fixed holonomy $B$ on $\partial M$. Let $B \in (-2,2)^4$ and $M_B$ be the compact component of the real points of $M_B(SL(2,C))$. These points correspond to SU(2)-representations or $SL(2,R)$-representations. The $Γ$-action preserves $M_B$ and we study the topological dynamics of the $Γ$-action on $M_B$ and show that for a dense set of holonomy $B \in (-2,2)^4$, the $Γ$-orbits are dense in $M_B$. We also produce a class of representations $ρ\in \Hom_B^+(pi_1(M),SL(2,R))$ such that the $Γ$-orbit of $[ρ]$ is finite in the compact component of $M_B(SL(2,R))$, but $ρ(π_1(M))$ is dense in $SL(2,R)$.
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