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 vol5(M) and, for M compact, new estimates relating the injectivity radius and the diameter of M with vol5(M). The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group PSΔ L(2, ℍ) of quaternionic 2 × 2-matrices with Dieudonné determinant Δ equal to 1 and isolation properties of PSΔ L(2, ℍ).