Topological nonrigidity of nonuniform lattices

We say that an arbitrary manifold (M, ∂M) is topologically rigid relative to its ends if it satisfies the following condition: If (N , ∂N ) is any other manifold with a compact subset C ⊂ N for which a proper homotopy equivalence h : (N , ∂N ) → (M, ∂M) is a homeomorphism on ∂N∪(N\C), then there is a compact subset K ⊂ N and a proper homotopy ht : (N , ∂N ) → (M, ∂M) from h to a homeomorphism such that ht and h agree on ∂N∪(N\K ) for all t ∈ [0, 1]. We say that a manifold M without boundary is properly rigid or absolutely topologically rigid if we eliminate the requirements that h be a homeomorphism on ∂N ∪ (N\C) and agree with ht on ∂N ∪ (N\K ) for all t ∈ [0, 1]. Along the lines of the classical Borel conjecture that all closed aspherical manifolds are topologically rigid, Farrell and Jones [8] provide the following important theorem: