Cross Ratios, Anosov Representations and the Energy Functional on Teichmuller Space

We study Hitchin representations and maximal symplectic representations of surface groups, which can be both thought of as generalisations of Fuchsian representations. We show that the corresponding energy functionals are proper on Teichmuller space. We also prove that the mapping class group acts properly on the corresponding moduli spaces. These two results follows from the fact these representations are well displacing which is a consequence they are associated to cross ratios. We state some applications.