CONNECTED COMPONENTS OF PGL(2, ℝ)-REPRESENTATION SPACES OF NON-ORIENTABLE SURFACES

The Teichmüller space of a surface naturally embeds as a connected component in the moduli space of representations from the fundamental group of the surface into the group of isometries of the hyperbolic plane. We present invariants that distinguish all the connected components of the space of representations. This allows us to compute the number of connected components of these spaces both in the orientable and in the non-orientable case.