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. 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.