Geodesic
Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs
Alex Eskin1 ; David Fisher2 ; Kevin Whyte3
1 Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514
2 Department of Mathematics, Indiana University - Bloomington, 831 E. 3rd Street, Bloomington, IN 47401
3 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045
Annals of mathematics, Tome 176 (2012) no. 1, pp. 221-260.

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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.
DOI : 10.4007/annals.2012.176.1.3

Alex Eskin 1 ; David Fisher 2 ; Kevin Whyte 3

1 Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514
2 Department of Mathematics, Indiana University - Bloomington, 831 E. 3rd Street, Bloomington, IN 47401
3 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045 
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Alex Eskin; David Fisher; Kevin Whyte. Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs. Annals of mathematics, Tome 176 (2012) no. 1, pp. 221-260. doi : 10.4007/annals.2012.176.1.3. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.3/

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