• bounded euler class
• deformation spaces
• milnor-wood inequality
• semi-conjugacy
• surface groups

We cover some topics on rigidity for actions of surface groups on the circle.
Group actions on the circle are classified up to semi-conjugacy by their bounded Euler class. For actions of surface groups there is a weaker invariant, the Euler number which also carries some information. The prototype of the results we are intreseted in is a classical theorem by Goldman that ensures that representations into PSL(2,R) with maximal Euler number (with respect to the bound given by the Milnor-Wood inequality) are faithful and have discrete image.
The same holds in the topological setting by theorems of Matsumoto, Iozzi and Burger.
A representation is called geometric if it is faithful and has discrete image. Following the work of K. Mann and S. Matsumoto we will prove that the deformation space of a geometric representation is trivial meaning that it consists of a single semi-conjugacy class.