Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs

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 our 2007 article "Quasi-isometries and rigidity of solvable groups." 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 lamp-lighter groups. The proofs of those results are completed in "Coarse differentiation of quasi-isometries II; Rigidity for lattices in Sol and Lamplighter groups." The method used here is based on the idea of coarse differentiation introduced in our 2007 article.