Quaternions and Some Global Properties of Hyperbolic 5-Manifolds

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension 5. The result implies improved universal lower bounds for the volume $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$ . The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$ of quaternionic $2\,\times \,2$ -matrices with Dieudonné determinant $\Delta$ equal to 1 and isolation properties of $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$ .