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 nitely 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 nitely generated group. This paper is the rst 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 classication of lamplighter groups. The proofs of those results are completed in \Coarse dierentiation of quasi-isometries II; Rigidity for lattices in Sol and Lamplighter groups." The method used here is based on the idea of coarse dierentiation introduced in our 2007 article.

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