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 $\mathrm{Hom}(\Gamma ,\mathrm{SO}(n,2))$ such that for any $\sigma \in \mathcal{O}$, the subgroup $\sigma (\Gamma )$ is a Zariski-dense and non-tempered Anosov subgroup of $\mathrm{SO}(n,2)$. 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 $\mathrm{SO}(n,2)$.

23 pages, To appear in Crelle's journal (the bicentennial volume)