Groups and Dynamics Seminar

We investigate the spaces of representations of surface groups into PSL(2, R). For a closed surface, by the classic result of Goldman, the Euler class together with the Milnor-Wood inequality provide a complete classification of the connected components of the representation spaces. However, describing these components becomes more subtle when considering the space of type-preserving representations for punctured surfaces. In this talk, I will introduce two invariants of type-preserving representations that characterize the connected components. As an application, I will show the existence of totally hyperbolic representations, giving a negative answer to a question of Bowditch.