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

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 $\operatorname{SO}(n,2)$.