Rigidity and relative hyperbolicity of real hyperbolic hyperplane complements

TL;DR

This paper investigates the geometric and algebraic properties of spaces formed by removing a codimension-two submanifold from finite volume real hyperbolic manifolds, revealing their fundamental groups' complex hyperbolic features.

Contribution

It establishes that these fundamental groups are relatively hyperbolic and possess various rigidity and geometric properties, expanding understanding of hyperbolic manifold complements.

Findings

Fundamental groups are relatively hyperbolic and co-Hopf.

They are biautomatic and residually hyperbolic.

They satisfy Mostow-type Rigidity and Baum-Connes conjecture.

Abstract

For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not K\"ahler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in real Lie groups that are isomorphic to relatively hyperbolic groups.