Homological dimension of discrete subgroups in higher rank simple Lie groups

Abstract

We investigate the homological dimensions of discrete subgroups of non-compact simple Lie groups using a flow recently introduced by the authors. Using new estimates on the critical index for a broad class of discrete subgroups, we prove that the homological dimension of non-lattice, discrete, Zariski-dense subgroups is bounded above by $n-\alpha r$ where $n$ is the dimension of the associated symmetric space, $r$ is the real rank, and $\alpha \geq \frac{1}{8}$. Under the assumption that the discrete group has regular limit cone, our bound improves to $\frac{5}{6}n+1$. Additionally, we make some conjectures on the possible homological dimensions for non-lattice discrete subgroups and pose a few questions in the more general semisimple setting.