Abstract: For some non-compact semisimple Lie groups G there are connected components of the character variety Hom(\pi_1(S),G)/G that only consist of conjugacy classes of injective homomorphisms with discrete image, the higher rank Teichmüller spaces (HRTS). After introducing and motivating the study of HRTS, I will discuss joint work with Sambarino and Wienhard in which we study how the orbit growth rate of the associated actions on the Riemannian symmetric space G/K varies on the different HRTS.