Growth rates of discrete subgroups

For the image of a Hitchin representation Gamma in PSL(d,R) Potrie and Sambarino showed that the entropy (or critical exponent) with respect to each simple roots equals 1. Due to Edwards and Oh this in particular implies that the representation of PSL(d,R) in $L^2(\Gamma PSL(d,R))$ is tempered which means that the quotient $\Gamma PSL(d,R)$ is indistinguishable from PSL(d,R) from a representation theoretical standpoint. In this talk I will present a characterization of the temperedness of $L^2(\Gamma G)$ for a general discrete subgroups Gamma in a semisimple Lie group G in terms of bounds on Quint's growth indicator function and the critical exponents. Furthermore, I will explain that these bounds hold for images of Borel Anosov representations.