The limit cone and bounds on the growth indicator function

Abstract

Given a real semisimple Lie group $G$ with finite center and a discrete subgroup $\hat{I}“\subset G$ whose limit cone is disjoint from two facets of the Weyl chamber we show that Quint's growth indicator function $\ddot{I}{ˆ}_{\hat{I}}“$ is bounded by the half sum of positive roots $\ddot{I}$, i.e. it has slow growth, implying that the representation $L^2(\hat{I}“\backslash G)$ 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.

22 pages, 1 figure