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 requirement that h is a homeomorphism on ∂N ∪ (N\C) and agrees 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 [FJ] provide the following important theorem.