Nonexistence of cusp cross-section of one-cusped complete complex hyperbolic manifolds II

2006/03/31 by Yoshinobu Kamishima, Kamishima, Yoshinobu
Mathematics · #51M10 #53C55 #57S25 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/0603726

openalex publication_date 2006/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

D. D. Long and A. W. Reid have shown that some compact flat 3-manifold cannot be diffeomorphic to a cusp cross-section of any complete finite volume 1-cusped real hyperbolic 4-manifold. This note concerns the complex hyperbolic case. We give a negative answer that there exists a 3-dimensional closed Heisenberg infranilmanifold which cannot be diffeomorphic to a cusp cross-section of any complete finite volume 1-cusped complex hyperbolic 2-manifold.

Related

• Nil 3-manifolds and cusps of complex hyperbolic surfaces
• Cusp types of arithmetic hyperbolic manifolds
• Cusp cross-section phenomena for arithmetic hyperbolic manifolds
• Some cusp-transitive hyperbolic 4-manifolds
• Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds