Integral Congruence Two Hyperbolic 5-manifolds

Abstract. In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ 2 of the group Γ of positive units of the Lorentzian quadratic form x 1 + · · ·+ x 5 − x 6 . We also show that Γ 2 is a reflection group with respect to a 5-dimensional right-angled convex polytope in H. As an application, we construct a hyperbolic 5-manifold of smallest known volume 7ζ(3)/4.