Zariski dense non-tempered subgroups in higher rank of nearly optimal growth

TL;DR

This paper constructs examples of Zariski-dense, non-tempered, discrete subgroups in higher rank Lie groups with nearly optimal growth, expanding understanding of non-lattice subgroups and their properties.

Contribution

It provides the first explicit construction of Zariski-dense, non-tempered subgroups in higher rank Lie groups with near-maximal growth indicators.

Findings

Existence of Zariski-dense, non-tempered subgroups in higher rank Lie groups.

Construction of open sets of representations yielding such subgroups.

These subgroups have growth indicators close to the theoretical maximum.

Abstract

We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $\Gamma_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(\Gamma_0\backslash G)$ is non-tempered. More precisely, let $n\ge 3$ and let $\Gamma$ be the fundamental group of a closed hyperbolic $n$-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset $\mathcal O$ of $\operatorname{Hom}(\Gamma, \operatorname{SO}(n,2))$ such that for any $\sigma\in \mathcal O$, the subgroup $\sigma(\Gamma)$ is a Zariski-dense and non-tempered Anosov subgroup of $\operatorname{SO}(n,2)$. In addition, the growth indicator of $\sigma(\Gamma)$ is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property $(T)$ of $\operatorname{SO}(n,2)$.