Maximal Holonomy of Almost Bieberbach Groups for Heis5

Abstract

Let Heis2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis2n+1 with index bounded above by a universal constant In+1, i.e., In+1 is the maximal order of the holonomy groups. We prove that I3 = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.