The limit cone and bounds on the growth indicator function

Given a real semisimple Lie group $G$ with finite center and a discrete subgroup $\Gamma \subset G$ whose limit cone is disjoint from two facets of the Weyl chamber we show that Quint's growth indicator function $\psi_\Gamma$ is bounded by the half sum of positive roots $\rho$, i.e. it has slow growth, implying that the representation $L^2(\Gamma \backslash G)$ is tempered. In particular, this holds for each $I$-Anosov subgroup provided that $I$ contains at least two distinct simple roots that are not interchanged by the opposition involution.