Abstract
In this paper, we prove that certain spaces are not quasiisometric 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 quasiisometric 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 “Quasiisometries and rigidity of solvable groups.” In particular, this paper contains many steps in the proofs of quasiisometric rigidity of lattices in $\mathrm{Sol}$ and of the quasiisometry classification of lamplighter groups. The proofs of those results are completed in “Coarse differentiation of quasiisometries 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.

[BJLPS] S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, "Affine approximation of Lipschitz functions and nonlinear quotients," Geom. Funct. Anal., vol. 9, iss. 6, pp. 10921127, 1999.
@article {BJLPS, MRKEY = {1736929},
AUTHOR = {Bates, S. and Johnson, W. B. and Lindenstrauss, J. and Preiss, D. and Schechtman, G.},
TITLE = {Affine approximation of {L}ipschitz functions and nonlinear quotients},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {9},
YEAR = {1999},
NUMBER = {6},
PAGES = {10921127},
ISSN = {1016443X},
CODEN = {GFANFB},
MRCLASS = {46B20 (41A65 46B03 46G05 46T20)},
MRNUMBER = {1736929},
MRREVIEWER = {S. Cobza{\c{s}}},
DOI = {10.1007/s000390050108},
ZBLNUMBER = {0954.46014},
} 
[BLPS] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, "Groupinvariant percolation on graphs," Geom. Funct. Anal., vol. 9, iss. 1, pp. 2966, 1999.
@article {BLPS, MRKEY = {1675890},
AUTHOR = {Benjamini, I. and Lyons, R. and Peres, Y. and Schramm, O.},
TITLE = {Groupinvariant percolation on graphs},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {9},
YEAR = {1999},
NUMBER = {1},
PAGES = {2966},
ISSN = {1016443X},
CODEN = {GFANFB},
MRCLASS = {60K35 (60B99)},
MRNUMBER = {1675890},
MRREVIEWER = {Olle H{ä}ggstr{ö}m},
DOI = {10.1007/s000390050080},
ZBLNUMBER = {0924.43002},
} 
[BL] Y. Binyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Providence, RI: Amer. Math. Soc., 2000, vol. 48.
@book{BL,
author={Binyamini, Y. and Lindenstrauss, J.},
TITLE={Geometric Nonlinear Functional Analysis},
SERIES={Amer. Math. Soc. Colloq. Publ.},
VOLUME={48},
YEAR={2000},
ADDRESS={Providence, RI},
PUBLISHER={Amer. Math. Soc.},
MRNUMBER = {1727673},
ZBLNUMBER = {0946.46002},
} 
[Bourgain] J. Bourgain, "Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms," in Geometrical Aspects of Functional Analysis (1985/86), New York: SpringerVerlag, 1987, vol. 1267, pp. 157167.
@incollection {Bourgain, MRKEY = {0907692},
AUTHOR = {Bourgain, J.},
TITLE = {Remarks on the extension of {L}ipschitz maps defined on discrete sets and uniform homeomorphisms},
BOOKTITLE = {Geometrical Aspects of Functional Analysis (1985/86)},
SERIES = {Lecture Notes in Math.},
VOLUME = {1267},
PAGES = {157167},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1987},
MRCLASS = {46B20 (46B25 46E30)},
MRNUMBER = {0907692},
MRREVIEWER = {G. Schechtman},
DOI = {10.1007/BFb0078143},
ZBLNUMBER = {0633.46018},
} 
[Co] D. Cooper, "Appendix to Bilipschitz homeomorphisms of selfsimilar Cantor ÊŠÊŠÊŠÊŠÊŠÊŠÊŠÊŠsets (\citeFM1)," Invent. Math., vol. 131, iss. 2, pp. 419451, 1998.
@article{Co,
author={Cooper, D.},
TITLE = {Appendix to Bilipschitz homeomorphisms of selfsimilar Cantor ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊsets (\cite{FM1})},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {131},
YEAR = {1998},
NUMBER = {2},
PAGES = {419451},
ISSN = {00209910},
CODEN = {INVMBH},
MRCLASS = {57M07 (20F32 57M20)},
ZBLNUMBER = {0937.22003},
DOI = {10.1007/s002220050210},
MRNUMBER = {1608595},
} 
[dlH] P. de la Harpe, Topics in Geometric Group Theory, Chicago, IL: University of Chicago Press, 2000.
@book {dlH, MRKEY = {1786869},
AUTHOR = {de la Harpe, Pierre},
TITLE = {Topics in Geometric Group Theory},
SERIES = {Chicago Lectures in Math.},
PUBLISHER = {University of Chicago Press},
ADDRESS = {Chicago, IL},
YEAR = {2000},
PAGES = {vi+310},
ISBN = {0226317196; 0226317218},
MRCLASS = {20F65 (20F69 57M07)},
MRNUMBER = {1786869},
MRREVIEWER = {Lee Mosher},
ZBLNUMBER = {0965.20025},
} 
[DL] R. Diestel and I. Leader, "A conjecture concerning a limit of nonCayley graphs," J. Algebraic Combin., vol. 14, iss. 1, pp. 1725, 2001.
@article {DL, MRKEY = {1856226},
AUTHOR = {Diestel, Reinhard and Leader, Imre},
TITLE = {A conjecture concerning a limit of non{C}ayley graphs},
JOURNAL = {J. Algebraic Combin.},
FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
VOLUME = {14},
YEAR = {2001},
NUMBER = {1},
PAGES = {1725},
ISSN = {09259899},
CODEN = {JAOME7},
MRCLASS = {05C25 (20F65)},
MRNUMBER = {1856226},
MRREVIEWER = {Wolfgang Woess},
DOI = {10.1023/A:1011257718029},
ZBLNUMBER = {0985.05020},
} 
[ET] G. Elek and G. Tardos, "On roughly transitive amenable graphs and harmonic Dirichlet functions," Proc. Amer. Math. Soc., vol. 128, iss. 8, pp. 24792485, 2000.
@article {ET, MRKEY = {1657731},
AUTHOR = {Elek, G{á}bor and Tardos, G{á}bor},
TITLE = {On roughly transitive amenable graphs and harmonic {D}irichlet functions},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {128},
YEAR = {2000},
NUMBER = {8},
PAGES = {24792485},
ISSN = {00029939},
CODEN = {PAMYAR},
MRCLASS = {31C20 (94C15)},
MRNUMBER = {1657731},
MRREVIEWER = {Paolo M. Soardi},
DOI = {10.1090/S0002993900052886},
ZBLNUMBER = {0948.58017},
} 
[E] A. Eskin, "Quasiisometric rigidity of nonuniform lattices in higher rank symmetric spaces," J. Amer. Math. Soc., vol. 11, iss. 2, pp. 321361, 1998.
@article {E, MRKEY = {1475886},
AUTHOR = {Eskin, Alex},
TITLE = {Quasiisometric rigidity of nonuniform lattices in higher rank symmetric spaces},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {11},
YEAR = {1998},
NUMBER = {2},
PAGES = {321361},
ISSN = {08940347},
MRCLASS = {22E40 (20F32 53C35)},
MRNUMBER = {1475886},
MRREVIEWER = {Lee Mosher},
DOI = {10.1090/S0894034798002562},
ZBLNUMBER = {0885.22017},
} 
[EF] A. Eskin and B. Farb, "Quasiflats and rigidity in higher rank symmetric spaces," J. Amer. Math. Soc., vol. 10, iss. 3, pp. 653692, 1997.
@article {EF, MRKEY = {1434399},
AUTHOR = {Eskin, Alex and Farb, Benson},
TITLE = {Quasiflats and rigidity in higher rank symmetric spaces},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {10},
YEAR = {1997},
NUMBER = {3},
PAGES = {653692},
ISSN = {08940347},
MRCLASS = {22E40 (53C35)},
MRNUMBER = {1434399},
MRREVIEWER = {Alexander Starkov},
DOI = {10.1090/S0894034797002385},
ZBLNUMBER = {0893.22004},
} 
[EFW0] A. Eskin, D. Fisher, and K. Whyte, "Quasiisometries and rigidity of solvable groups," Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 927947, 2007.
@article {EFW0, MRKEY = {2402598},
AUTHOR = {Eskin, Alex and Fisher, David and Whyte, Kevin},
TITLE = {Quasiisometries and rigidity of solvable groups},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {3},
YEAR = {2007},
NUMBER = {4, part 1},
PAGES = {927947},
ISSN = {15588599},
MRCLASS = {20F65 (22E25)},
MRNUMBER = {2402598},
MRREVIEWER = {Bachir Bekka},
ZBLNUMBER = {1167.22007},
} 
[EFW1] A. Eskin, D. Fisher, and K. Whyte, Coarse differentiation of quasiisometries II; Rigidity for lattices in $\mathrm{Sol}$ and Lamplighter groups.
@misc{EFW1,
author = {Eskin, Alex and Fisher, David and Whyte, Kevin},
TITLE = {Coarse differentiation of quasiisometries {II}; {R}igidity for lattices in $\mathrm{Sol}$ and {L}amplighter groups},
NOTE={in preparation},
ARXIV={0706.0940},
} 
[FS] B. Farb and R. Schwartz, "The largescale geometry of Hilbert modular groups," J. Differential Geom., vol. 44, iss. 3, pp. 435478, 1996.
@article {FS, MRKEY = {1431001},
AUTHOR = {Farb, Benson and Schwartz, Richard},
TITLE = {The largescale geometry of {H}ilbert modular groups},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {44},
YEAR = {1996},
NUMBER = {3},
PAGES = {435478},
ISSN = {0022040X},
CODEN = {JDGEAS},
MRCLASS = {22E40 (11F41 22E46)},
MRNUMBER = {1431001},
MRREVIEWER = {Alexander Starkov},
URL = {http://projecteuclid.org/euclid.jdg/1214459217},
ZBLNUMBER = {0871.11035},
} 
[F] B. Farb, "The quasiisometry classification of lattices in semisimple Lie groups," Math. Res. Lett., vol. 4, iss. 5, pp. 705717, 1997.
@article {F, MRKEY = {1484701},
AUTHOR = {Farb, Benson},
TITLE = {The quasiisometry classification of lattices in semisimple {L}ie groups},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {4},
YEAR = {1997},
NUMBER = {5},
PAGES = {705717},
ISSN = {10732780},
MRCLASS = {22E40 (20F32 2202)},
MRNUMBER = {1484701},
MRREVIEWER = {Lee Mosher},
ZBLNUMBER = {0889.22010},
} 
[FM1] B. Farb and L. Mosher, "A rigidity theorem for the solvable BaumslagSolitar groups," Invent. Math., vol. 131, iss. 2, pp. 419451, 1998.
@article {FM1, MRKEY = {1608595},
AUTHOR = {Farb, Benson and Mosher, Lee},
TITLE = {A rigidity theorem for the solvable {B}aumslag{S}olitar groups},
NOTE = {with an appendix by Daryl Cooper},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {131},
YEAR = {1998},
NUMBER = {2},
PAGES = {419451},
ISSN = {00209910},
CODEN = {INVMBH},
MRCLASS = {57M07 (20F32 57M20)},
MRNUMBER = {1608595},
MRREVIEWER = {Raul Quiroga},
DOI = {10.1007/s002220050210},
ZBLNUMBER = {0937.22003},
} 
[FM2] B. Farb and L. Mosher, "Quasiisometric rigidity for the solvable BaumslagSolitar groups. II," Invent. Math., vol. 137, iss. 3, pp. 613649, 1999.
@article {FM2, MRKEY = {1709862},
AUTHOR = {Farb, Benson and Mosher, Lee},
TITLE = {Quasiisometric rigidity for the solvable {B}aumslag{S}olitar groups. {II}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {137},
YEAR = {1999},
NUMBER = {3},
PAGES = {613649},
ISSN = {00209910},
CODEN = {INVMBH},
MRCLASS = {20F65 (57M60)},
MRNUMBER = {1709862},
MRREVIEWER = {Raul Quiroga},
DOI = {10.1007/s002220050337},
ZBLNUMBER = {0931.20035},
} 
[FM3] B. Farb and L. Mosher, "On the asymptotic geometry of abelianbycyclic groups," Acta Math., vol. 184, iss. 2, pp. 145202, 2000.
@article {FM3, MRKEY = {1768110},
AUTHOR = {Farb, Benson and Mosher, Lee},
TITLE = {On the asymptotic geometry of abelianbycyclic groups},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {184},
YEAR = {2000},
NUMBER = {2},
PAGES = {145202},
ISSN = {00015962},
CODEN = {ACMAA8},
MRCLASS = {20F69 (20F65 57M27)},
MRNUMBER = {1768110},
MRREVIEWER = {Ilya Kapovich},
DOI = {10.1007/BF02392628},
ZBLNUMBER = {0982.20026},
} 
[FM4] B. Farb and L. Mosher, "Problems on the geometry of finitely generated solvable groups," in Crystallographic Groups and their Generalizations, Providence, RI: Amer. Math. Soc., 2000, vol. 262, pp. 121134.
@incollection {FM4, MRKEY = {1796128},
AUTHOR = {Farb, Benson and Mosher, Lee},
TITLE = {Problems on the geometry of finitely generated solvable groups},
BOOKTITLE = {Crystallographic Groups and their Generalizations},
VENUE={{K}ortrijk, 1999},
SERIES = {Contemp. Math.},
VOLUME = {262},
PAGES = {121134},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2000},
MRCLASS = {20F69 (20F16 22E25 57M07)},
MRNUMBER = {1796128},
MRREVIEWER = {Ilya Kapovich},
ZBLNUMBER = {0983.20038},
} 
[Gr1] M. Gromov, "Groups of polynomial growth and expanding maps," Inst. Hautes Études Sci. Publ. Math., vol. 53, pp. 5373, 1981.
@article {Gr1, MRKEY = {0623534},
AUTHOR = {Gromov, Mikhael},
TITLE = {Groups of polynomial growth and expanding maps},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {53},
YEAR = {1981},
PAGES = {5373},
ISSN = {00738301},
CODEN = {PMIHA6},
MRCLASS = {53C