Finite mapping class group orbits on character varieties

Finite mapping class group orbits are both a rare and fascinating phenomenon of dynamics on character varieties. The first non-trivial examples go back to Fuchs and Picard. You won't find them in (higher) Teichmüller spaces; you'll have to explore the "other" components to stumble across interesting examples. This talk will focus on the most recent developments about representations of punctured spheres. We'll see how a result by Corlette--Simpson tells us exactly where to look for finite orbits and how it can be used to complete the classification of finite orbits for representations into SL(2,C), confirming Tykhyy's Conjecture. All of this is joint work with Samuel Bronstein.