Algebraic Mapping Class Group Rigidity

Abstract

Let $g,n\ge 0$ and $\hat{I}\pounds =\hat{I}{\pounds }_{g,n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The ${\mathrm{SL}}_2$-character variety of $\hat{I}\pounds$ 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 $\hat{I}\pounds$.

18 pages, 3 figures