Complex hyperbolic hyperplane complements

TL;DR

This paper investigates the geometric and algebraic properties of spaces derived from complex hyperbolic manifolds by removing certain submanifolds, revealing their fundamental groups have rich hyperbolic-like features and satisfy numerous conjectures and properties.

Contribution

It establishes that the fundamental groups of these hyperplane complements are relatively hyperbolic and possess various geometric group theory properties, extending understanding of their structure.

Findings

Fundamental groups are relatively hyperbolic.

Spaces admit complete finite volume negatively curved metrics.

Groups satisfy Borel, Baum-Connes, and other conjectures.

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.