Nonexistence of Cusp Cross-section of One-cusped Complete Complex Hyperbolic Manifolds Ii

Long and Reid have shown that some compact flat 3-manifold cannot be diffeomorphic to a cusp cross-section of any complete finite volume 1-cusped hyperbolic 4-manifold. Similar to the flat case, we give a negative answer that there exists a 3-dimensional closed Heisenberg infranilmanifold whose diffeomorphism class cannot be arisen as a cusp cross-section of any complete finite volume 1-cusped complex hyperbolic 2-manifold. This is obtained from the formula by the characteristic numbers of bounded domains related to the Burns-Epstein invariant on strictly pseudo-convex CR-manifolds [1],[3]. This paper is a sequel of our paper[11]. Introduction We shall consider whether every Heisenberg infranilmanifold can be arisen, up to diffeomorphism, as a cusp cross-section of a complete finite volume 1cusped complex hyperbolic manifold. Long and Reid considered the problem that every compact Riemannian flat manifold is diffeomorphic to a cusp cross-section of a complete finite volume 1-cusped hyperbolic manifold. They have shown it is false for some compact flat 3-manifold [15]. We shall give a negative answer similarly to the flat case. Theorem. Any 3-dimensional closed Heisenberg infranilmanifold with nontrivial holonomy cannot be diffeomorphic to a cusp cross-section of any complete finite volume 1-cusped complex hyperbolic 2-manifold. McReynolds informed us that W. Neumann and A. Reid have obtained the similar result. 2. Heisenberg infranilmaniold Let 〈z,w〉 = z̄1 ·w1+ z̄2 ·w2+ · · ·+ z̄n ·wn be the Hermitian inner product defined on C. The Heisenberg nilpotent Lie group N is the product R×C with group law: (2.1) (a, z) · (b, w) = (a+ b− Im〈z,w〉, z + w). Date: February 2, 2008. 1991 Mathematics Subject Classification. 53C55, 57S25, 51M10.