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 $H^5/\Gamma$ where $\Gamma$ is a torsion-free subgroup of minimal index of the congruence two subgroup ${\Gamma }_2^5$ of the group ${\Gamma }^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that ${\Gamma }_2^5$ is a reflection group with respect to a 5-dimensional right-angled convex polytope in $H^5$. As an application, we construct a hyperbolic 5-manifold of smallest known volume $7\zeta (3)/4$.

21 pages, 2 figures, LaTeX