Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups

This paper continues the work announced in [EFW1] and begun in [EFW2]. For a more detailed introduction, we refer the reader to those papers. As discussed in those papers, all our theorems stated above are proved using a new technique, which we call coarse differentiation. Even though quasi-isometries have no local structure and conventional derivatives do not make sense, we essentially construct a “coarse derivative” that models the large scale behavior of the quasi-isometry. From this point of view, the coarse derivatives of maps studied here are constructed in [EFW2] and this paper consists entirely of a coarse analysis of coarsely differentiable maps. We now state the main results whose proofs are begun in [EFW2] and finished here. The group Sol ∼= RnR with R acting on R via the diagonal matrix with entries e and e−z/2. As matrices, Sol can be written as :