Algebraic Mapping Class Group Rigidity

Let $g, n \geq 0$ and $Σ= Σ_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $Σ$ has a rigid relative automorphism group, whose elements fix each monodromies along punctures, and is a finite extension of the mapping class group. The exceptional isomorphism between the $\mathrm{SL}(2, \mathbb{C})$-character variety and moduli of points on complex $3$-sphere provides a new description of the mapping class group of certain $Σ$.