  In this paper, we prove that certain spaces are not quasi-isometric to Cayley
graphs of finitely generated groups. In particular, we answer a question of
Woess and prove a conjecture of Diestel and Leader by showing that certain
homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely
generated group.
  This paper is the first in a sequence of papers proving results announced in
[EFW0]. In particular, this paper contains many steps in the proofs of
quasi-isometric rigidity of lattices in Sol and of the quasi-isometry
classification of lamplighter groups. The proofs of those results are completed
in [EFW1].
  The method used here is based on the idea of "coarse differentiation"
introduced in [EFW0].
