The aim of this talk is to answer the Nielsen Realisation problem: Can every finite subgroup of the mapping class group can be realised as a subgroup of the isometry group of some hyperbolic surface? In other words, does every finite subgroup fix a point in the Teichmüller space of the surface?
The usual Fenchel-Nielsen coordinates can be thought of as fixing a pants decomposition and keeping track of the length of the boundary of these together with the amount on ‘twist’ while glueing. Shear coordinates on the other hand, instead of using pair of pants, use ideal triangles as the basic pieces. As ideal triangles are unique up to isometry, only the gluing data needs to be tracked in this case. We shall see a convexity result concerning the length of simple closed curves with respect to these coordinates. This result leads to a positive answer for the Nielsen Realisation problem. I’ll be mainly following the paper by Bestvina, Bromberg, Fujiwara, and Souto (AMJ, 2013). Some technical results will be assumed. Familiarity with Fenchel-Nielsen coordinates will be helpful.