Actions of surface groups on the circle

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