  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.
