Constructing hyperbolic manifolds which bound geometrically

Let H denote hyperbolic n-space, that is the unique connected simply connected Riemannian manifold of constant curvature −1. By a hyperbolic n-orbifold we shall mean a quotient H/Γ where Γ is a discrete group of isometries of H. If a hyperbolic n-manifold M is the totally geodesic boundary of a hyperbolic (n+1)-manifold W , we will say that M bounds geometrically. It was shown in [11] that if a closed orientable hyperbolic M4k−1 bounds geometrically, then η(M4k−1) ∈ Z. Closed hyperbolic 3-manifolds with integral eta are fairly rare – for example, of the 11, 000 or so manifolds in the census of small volume closed hyperbolic 3-manifolds, computations involving Snap (see [3]) rule out all but 41. (We refer the reader to [24] which contains the list of manifolds in the census with Chern-Simons invariant zero, as well as which of these have integral eta.) Hyperbolic 3-manifolds with totally geodesic boundary are fairly easily constructed given the Hyperbolization Theorem of Thurston [20], but to the authors’ knowledge, there was only one known prior example of a closed hyperbolic n-manifold (with n ≥ 3) which bounded geometrically, a somewhat ad hoc construction which appears in [18], based on a hyperbolic 4-manifold example due to Davis [4]. The difficulty is that almost nothing is known about hyperbolic manifolds in dimensions ≥ 4; some constructions exist (see [5], [6], [7]) but they do not appear to be sufficient to address this problem. This paper ameliorates this situation somewhat by providing a construction of examples in all dimensions. We show