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 $\mathrm{Sol}$ 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 $\mathrm{Sol}$ and Lamplighter groups.” The method used here is based on the idea of coarse differentiation introduced in our 2007 article.

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Authors

Alex Eskin

Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514

David Fisher

Department of Mathematics, Indiana University - Bloomington, 831 E. 3rd Street, Bloomington, IN 47401

Kevin Whyte

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045