Flat manifolds appearing as cusps of hyperbolic manifolds.

Abstract

In 1982, Hamrick and Royster proved that each compact, flat Riemannian manifold, F, occurs as the boundary of some compact manifold, M. One can ask about the possibility of obtaining such a result subject to conditions on the geometry of the manifold M. In particular, one can ask the following questions: (1) Do all diffeomorphism classes of compact, flat n-manifolds appear as ends of complete, finite-volume, hyperbolic (n + 1)-manifolds? (2) Are the flat structures which are induced on the cross-sections of the ends of complete, hyperbolic (n + 1)-manifolds of finite volume dense in the moduli spaces of flat n-manifolds? (3) Do all diffeomorphism classes of compact, flat n-manifolds bound hyperbolic (n + 1)-manifolds? (4) Are the flat structures which occur on the only boundary component of a compact hyperbolic (n + 1)-manifold dense in the moduli space of a flat n-manifold? We address these four questions in low dimensions, and obtain the following results. We show that all four question have affirmative answers for the 2-torus, and we give affirmative answers to the first three questions for the Klein bottle. We also give affirmative answers to the first two questions for all compact, flat 3-manifolds and for certain (but not all) compact, flat 4-manifolds.