Complex hyperbolic hyperplane complements

Abstract

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n>1 the fundamental group of M-S satisfies Mostow-type Rigidity, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M-S is biautomatic and satisfies Strong Tits Alternative.

37 pages, to appear in Math. Annalen

References in corpus (2)

• Algorithmic Properties of Relatively Hyperbolic Groups
• The conjugacy problem for relatively hyperbolic groups

Cited by in corpus (2)

• On finite volume, negatively curved manifolds
• CAT(0) and CAT(-1) fillings of hyperbolic manifolds