The limit cone and bounds on the growth indicator function

TL;DR

This paper establishes bounds on the growth indicator function for certain discrete subgroups of semisimple Lie groups, showing it is bounded by the half sum of positive roots, which implies temperedness of associated representations.

Contribution

It proves that for specific discrete subgroups with limit cones disjoint from certain facets, the growth indicator function is bounded by , extending to I-Anosov subgroups with particular properties.

Findings

Growth indicator function is bounded by  for certain subgroups.

Such subgroups have tempered $L^2$ representations.

Applicable to I-Anosov subgroups with specific simple roots.

Abstract

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.