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

Abstract

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 lamplighter 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.

Exotic quasi-conformally homogeneous surfaces

“…Eskin, Fisher and Whyte [4] recently established this conjecture. In this section we give a brief description of the DiestelLeader graphs (see or Woess [8] for more detailed descriptions).…”

“…We choose X to be a Diestel-Leader graph DL(m, n) with m = n. These graphs have transitive groups of automorphisms, but Eskin, Fisher and Whyte [4] recently showed that they are not quasi-isometric to the Cayley graph of any group. The proof is completed by the observation that any surface which is a quasiconformal deformation of a regular cover of a closed orbifold is quasi-isometric to the Cayley graph of the deck transformation group.…”

Coarse differentiation and quasi-isometries of a class of solvable Lie groups II

“…The following result is stated in Eskin, Fisher and Whyte [4]. Its proof is completed by Dymarz in [2].…”

“…In this last section, we remove the condition that standard maps are only defined for a subset of relative large measure and align the translational part of the standard maps by adopting the procedure used to achieve this in Eskin, Fisher and Whyte [4].…”

“…The setup to the proof of Theorem 5.1.1 follows the same sequence of steps as the analogue result in Section 6.1 of Eskin, Fisher and Whyte [4].…”