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.

  • [BJLPS] Go to document 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. 1092-1127, 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 = {1092--1127},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {46B20 (41A65 46B03 46G05 46T20)},
      MRNUMBER = {1736929},
      MRREVIEWER = {S. Cobza{\c{s}}},
      DOI = {10.1007/s000390050108},
      ZBLNUMBER = {0954.46014},
      }
  • [BLPS] Go to document I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, "Group-invariant percolation on graphs," Geom. Funct. Anal., vol. 9, iss. 1, pp. 29-66, 1999.
    @article {BLPS, MRKEY = {1675890},
      AUTHOR = {Benjamini, I. and Lyons, R. and Peres, Y. and Schramm, O.},
      TITLE = {Group-invariant percolation on graphs},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {9},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {29--66},
      ISSN = {1016-443X},
      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] Go to document 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: Springer-Verlag, 1987, vol. 1267, pp. 157-167.
    @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 = {157--167},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {46B20 (46B25 46E30)},
      MRNUMBER = {0907692},
      MRREVIEWER = {G. Schechtman},
      DOI = {10.1007/BFb0078143},
      ZBLNUMBER = {0633.46018},
     }
  • [Co] Go to document D. Cooper, "Appendix to Bilipschitz homeomorphisms of self-similar Cantor ʊʊʊʊʊʊʊʊsets (\citeFM1)," Invent. Math., vol. 131, iss. 2, pp. 419-451, 1998.
    @article{Co,
      author={Cooper, D.},
      TITLE = {Appendix to Bilipschitz homeomorphisms of self-similar Cantor sets (\cite{FM1})},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {131},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {419--451},
      ISSN = {0020-9910},
      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 = {0-226-31719-6; 0-226-31721-8},
      MRCLASS = {20F65 (20F69 57M07)},
      MRNUMBER = {1786869},
      MRREVIEWER = {Lee Mosher},
      ZBLNUMBER = {0965.20025},
      }
  • [DL] Go to document R. Diestel and I. Leader, "A conjecture concerning a limit of non-Cayley graphs," J. Algebraic Combin., vol. 14, iss. 1, pp. 17-25, 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 = {17--25},
      ISSN = {0925-9899},
      CODEN = {JAOME7},
      MRCLASS = {05C25 (20F65)},
      MRNUMBER = {1856226},
      MRREVIEWER = {Wolfgang Woess},
      DOI = {10.1023/A:1011257718029},
      ZBLNUMBER = {0985.05020},
      }
  • [ET] Go to document G. Elek and G. Tardos, "On roughly transitive amenable graphs and harmonic Dirichlet functions," Proc. Amer. Math. Soc., vol. 128, iss. 8, pp. 2479-2485, 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 = {2479--2485},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {31C20 (94C15)},
      MRNUMBER = {1657731},
      MRREVIEWER = {Paolo M. Soardi},
      DOI = {10.1090/S0002-9939-00-05288-6},
      ZBLNUMBER = {0948.58017},
      }
  • [E] Go to document A. Eskin, "Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces," J. Amer. Math. Soc., vol. 11, iss. 2, pp. 321-361, 1998.
    @article {E, MRKEY = {1475886},
      AUTHOR = {Eskin, Alex},
      TITLE = {Quasi-isometric 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 = {321--361},
      ISSN = {0894-0347},
      MRCLASS = {22E40 (20F32 53C35)},
      MRNUMBER = {1475886},
      MRREVIEWER = {Lee Mosher},
      DOI = {10.1090/S0894-0347-98-00256-2},
      ZBLNUMBER = {0885.22017},
      }
  • [EF] Go to document A. Eskin and B. Farb, "Quasi-flats and rigidity in higher rank symmetric spaces," J. Amer. Math. Soc., vol. 10, iss. 3, pp. 653-692, 1997.
    @article {EF, MRKEY = {1434399},
      AUTHOR = {Eskin, Alex and Farb, Benson},
      TITLE = {Quasi-flats 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 = {653--692},
      ISSN = {0894-0347},
      MRCLASS = {22E40 (53C35)},
      MRNUMBER = {1434399},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.1090/S0894-0347-97-00238-5},
      ZBLNUMBER = {0893.22004},
      }
  • [EFW0] A. Eskin, D. Fisher, and K. Whyte, "Quasi-isometries and rigidity of solvable groups," Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 927-947, 2007.
    @article {EFW0, MRKEY = {2402598},
      AUTHOR = {Eskin, Alex and Fisher, David and Whyte, Kevin},
      TITLE = {Quasi-isometries 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 = {927--947},
      ISSN = {1558-8599},
      MRCLASS = {20F65 (22E25)},
      MRNUMBER = {2402598},
      MRREVIEWER = {Bachir Bekka},
      ZBLNUMBER = {1167.22007},
      }
  • [EFW1] A. Eskin, D. Fisher, and K. Whyte, Coarse differentiation of quasi-isometries 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 quasi-isometries {II}; {R}igidity for lattices in $\mathrm{Sol}$ and {L}amplighter groups},
      NOTE={in preparation},
      ARXIV={0706.0940},
     }
  • [FS] Go to document B. Farb and R. Schwartz, "The large-scale geometry of Hilbert modular groups," J. Differential Geom., vol. 44, iss. 3, pp. 435-478, 1996.
    @article {FS, MRKEY = {1431001},
      AUTHOR = {Farb, Benson and Schwartz, Richard},
      TITLE = {The large-scale geometry of {H}ilbert modular groups},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {44},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {435--478},
      ISSN = {0022-040X},
      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 quasi-isometry classification of lattices in semisimple Lie groups," Math. Res. Lett., vol. 4, iss. 5, pp. 705-717, 1997.
    @article {F, MRKEY = {1484701},
      AUTHOR = {Farb, Benson},
      TITLE = {The quasi-isometry classification of lattices in semisimple {L}ie groups},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {4},
      YEAR = {1997},
      NUMBER = {5},
      PAGES = {705--717},
      ISSN = {1073-2780},
      MRCLASS = {22E40 (20F32 22-02)},
      MRNUMBER = {1484701},
      MRREVIEWER = {Lee Mosher},
      ZBLNUMBER = {0889.22010},
      }
  • [FM1] Go to document B. Farb and L. Mosher, "A rigidity theorem for the solvable Baumslag-Solitar groups," Invent. Math., vol. 131, iss. 2, pp. 419-451, 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 = {419--451},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57M07 (20F32 57M20)},
      MRNUMBER = {1608595},
      MRREVIEWER = {Raul Quiroga},
      DOI = {10.1007/s002220050210},
      ZBLNUMBER = {0937.22003},
      }
  • [FM2] Go to document B. Farb and L. Mosher, "Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II," Invent. Math., vol. 137, iss. 3, pp. 613-649, 1999.
    @article {FM2, MRKEY = {1709862},
      AUTHOR = {Farb, Benson and Mosher, Lee},
      TITLE = {Quasi-isometric rigidity for the solvable {B}aumslag-{S}olitar groups. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {137},
      YEAR = {1999},
      NUMBER = {3},
      PAGES = {613--649},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20F65 (57M60)},
      MRNUMBER = {1709862},
      MRREVIEWER = {Raul Quiroga},
      DOI = {10.1007/s002220050337},
      ZBLNUMBER = {0931.20035},
      }
  • [FM3] Go to document B. Farb and L. Mosher, "On the asymptotic geometry of abelian-by-cyclic groups," Acta Math., vol. 184, iss. 2, pp. 145-202, 2000.
    @article {FM3, MRKEY = {1768110},
      AUTHOR = {Farb, Benson and Mosher, Lee},
      TITLE = {On the asymptotic geometry of abelian-by-cyclic groups},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {184},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {145--202},
      ISSN = {0001-5962},
      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. 121-134.
    @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 = {121--134},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2000},
      MRCLASS = {20F69 (20F16 22E25 57M07)},
      MRNUMBER = {1796128},
      MRREVIEWER = {Ilya Kapovich},
      ZBLNUMBER = {0983.20038},
      }
  • [Gr1] Go to document M. Gromov, "Groups of polynomial growth and expanding maps," Inst. Hautes Études Sci. Publ. Math., vol. 53, pp. 53-73, 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 = {53--73},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {53C20 (22E40 58F15)},
      MRNUMBER = {0623534},
      MRREVIEWER = {J. A. Wolf},
      DOI = {10.1007/BF02698687},
      ZBLNUMBER = {0474.20018},
      }
  • [Gr2] M. Gromov, "Infinite groups as geometric objects," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Warsaw, 1984, pp. 385-392.
    @inproceedings {Gr2, MRKEY = {0804694},
      AUTHOR = {Gromov, Mikhael},
      TITLE = {Infinite groups as geometric objects},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      VENUE={{W}arsaw, 1983},
      PAGES = {385--392},
      PUBLISHER = {PWN},
      ADDRESS = {Warsaw},
      YEAR = {1984},
      MRCLASS = {57S30 (20F32 53C99)},
      MRNUMBER = {0804694},
      MRREVIEWER = {J{ó}zef Dodziuk},
      ZBLNUMBER = {0586.20016},
      }
  • [Gr3] M. Gromov, "Asymptotic invariants of infinite groups," in Geometric Group Theory, Vol. 2, Cambridge: Cambridge Univ. Press, 1993, vol. 182, pp. 1-295.
    @incollection {Gr3, MRKEY = {1253544},
      AUTHOR = {Gromov, Mikhael},
      TITLE = {Asymptotic invariants of infinite groups},
      BOOKTITLE = {Geometric Group Theory, {V}ol. 2},
      VENUE={{S}ussex, 1991},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {182},
      PAGES = {1--295},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1993},
      MRCLASS = {20F32 (57M07)},
      MRNUMBER = {1253544},
      ZBLNUMBER = {0841.20039},
     }
  • [He] J. Heinonen, Lectures on Analysis on Metric Spaces, New York: Springer-Verlag, 2001.
    @book {He, MRKEY = {1800917},
      AUTHOR = {Heinonen, Juha},
      TITLE = {Lectures on Analysis on Metric Spaces},
      SERIES = {Universitext},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2001},
      PAGES = {x+140},
      ISBN = {0-387-95104-0},
      MRCLASS = {30C65 (28A75 28A78 46E35)},
      MRNUMBER = {1800917},
      MRREVIEWER = {Christopher Bishop},
      ZBLNUMBER = {0985.46008},
      }
  • [H] Go to document A. Hinkkanen, "Uniformly quasisymmetric groups," Proc. London Math. Soc., vol. 51, iss. 2, pp. 318-338, 1985.
    @article {H, MRKEY = {0794115},
      AUTHOR = {Hinkkanen, A.},
      TITLE = {Uniformly quasisymmetric groups},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {51},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {318--338},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {30C60 (26A30)},
      MRNUMBER = {0794115},
      MRREVIEWER = {H. Renggli},
      DOI = {10.1112/plms/s3-51.2.318},
      ZBLNUMBER = {0574.30022},
      }
  • [JLS] Go to document W. B. Johnson, J. Lindenstrauss, and G. Schechtman, "Banach spaces determined by their uniform structures," Geom. Funct. Anal., vol. 6, iss. 3, pp. 430-470, 1996.
    @article {JLS, MRKEY = {1392325},
      AUTHOR = {Johnson, W. B. and Lindenstrauss, J. and Schechtman, G.},
      TITLE = {Banach spaces determined by their uniform structures},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {6},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {430--470},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {46Bxx},
      MRNUMBER = {1392325},
      MRREVIEWER = {Gilles Godefroy},
      DOI = {10.1007/BF02249259},
      ZBLNUMBER = {0864.46008},
      }
  • [KL] Go to document B. Kleiner and B. Leeb, "Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings," Inst. Hautes Études Sci. Publ. Math., vol. 86, pp. 115-197, 1997.
    @article {KL, MRKEY = {1608566},
      AUTHOR = {Kleiner, Bruce and Leeb, Bernhard},
      TITLE = {Rigidity of quasi-isometries for symmetric spaces and {E}uclidean buildings},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      YEAR = {1997},
      VOLUME = {86},
      PAGES = {115--197},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {53C35 (20E42 20F32 53C23 57M07)},
      MRNUMBER = {1608566},
      MRREVIEWER = {Lee Mosher},
      DOI = {10.1007/BF02698902},
      ZBLNUMBER = {0910.53035},
      }
  • [K] B. Kleiner, Personal communication.
    @misc{K,
      author={Kleiner, B.},
      TITLE={personal communication},
     }
  • [Matusek] Go to document J. Matouvsek, "On embedding trees into uniformly convex Banach spaces," Israel J. Math., vol. 114, pp. 221-237, 1999.
    @article {Matusek, MRKEY = {1738681},
      AUTHOR = {Matou{š}ek, Ji{\v{r}}{\'ı}},
      TITLE = {On embedding trees into uniformly convex {B}anach spaces},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {114},
      YEAR = {1999},
      PAGES = {221--237},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {46B20 (05C12 46B07)},
      MRNUMBER = {1738681},
      MRREVIEWER = {G. Schechtman},
      DOI = {10.1007/BF02785579},
      ZBLNUMBER = {0948.46011},
      }
  • [MN] R. Moeller, Letter to W. Woess, 2011.
    @misc{MN,
      author={Moeller, R.},
      TITLE={letter to {W}. {W}oess},
      YEAR={2011},
     }
  • [MSW] Go to document L. Mosher, M. Sageev, and K. Whyte, "Quasi-actions on trees. I. Bounded valence," Ann. of Math., vol. 158, iss. 1, pp. 115-164, 2003.
    @article {MSW, MRKEY = {1998479},
      AUTHOR = {Mosher, Lee and Sageev, Michah and Whyte, Kevin},
      TITLE = {Quasi-actions on trees. {I}. {B}ounded valence},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {158},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {115--164},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20F65 (20E08 57M07)},
      MRNUMBER = {1998479},
      MRREVIEWER = {Gilbert Levitt},
      DOI = {10.4007/annals.2003.158.115},
      ZBLNUMBER = {1038.20016},
     }
  • [P] Go to document P. Pansu, "Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un," Ann. of Math., vol. 129, iss. 1, pp. 1-60, 1989.
    @article {P, MRKEY = {0979599},
      AUTHOR = {Pansu, Pierre},
      TITLE = {Métriques de {C}arnot-{C}arathéodory et quasiisométries des espaces symétriques de rang un},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {129},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {1--60},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {53C20 (22E40)},
      MRNUMBER = {0979599},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      DOI = {10.2307/1971484},
      ZBLNUMBER = {0678.53042},
      }
  • [P2] P. Pansu, "Dimension conforme et sphère à l’infini des variétés à courbure négative," Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 14, iss. 2, pp. 177-212, 1989.
    @article {P2, MRKEY = {1024425},
      AUTHOR = {Pansu, Pierre},
      TITLE = {Dimension conforme et sphère à l'infini des variétés à courbure négative},
      JOURNAL = {Ann. Acad. Sci. Fenn. Ser. A I Math.},
      FJOURNAL = {Annales Academiae Scientiarum Fennicae. Series A I. Mathematica},
      VOLUME = {14},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {177--212},
      ISSN = {0066-1953},
      CODEN = {AAFMAT},
      MRCLASS = {53C20 (30C60)},
      MRNUMBER = {1024425},
      MRREVIEWER = {Gudlaugur Thorbergsson},
      ZBLNUMBER = {0722.53028},
      }
  • [PPS] Go to document Y. Peres, P. Gábor, and A. Scolnicov, "Critical percolation on certain nonunimodular graphs," New York J. Math., vol. 12, pp. 1-18 (electronic), 2006.
    @article{PPS,
      author={Peres, Y. and G{á}bor, P. and Scolnicov, A.},
      TITLE = {Critical percolation on certain nonunimodular graphs},
      JOURNAL = {New York J. Math.},
      FJOURNAL = {New York Journal of Mathematics},
      VOLUME = {12},
      YEAR = {2006},
      PAGES = {1--18 (electronic)},
      ISSN = {1076-9803},
      MRCLASS = {60K35 (20F65 82B43)},
      MRNUMBER = {2217160},
      MRREVIEWER = {Zhong Gen Su},
      ZBLNUMBER = {1103.60079},
      URL = {http://nyjm.albany.edu:8000/j/2006/12_1.html},
     }
  • [Pr] Go to document D. Preiss, "Differentiability of Lipschitz functions on Banach spaces," J. Funct. Anal., vol. 91, iss. 2, pp. 312-345, 1990.
    @article {Pr, MRKEY = {1058975},
      AUTHOR = {Preiss, D.},
      TITLE = {Differentiability of {L}ipschitz functions on {B}anach spaces},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {91},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {312--345},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {46G05 (49J50 58B10)},
      MRNUMBER = {1058975},
      MRREVIEWER = {Simon Fitzpatrick},
      DOI = {10.1016/0022-1236(90)90147-D},
      ZBLNUMBER = {0711.46036},
      }
  • [S1] Go to document R. E. Schwartz, "The quasi-isometry classification of rank one lattices," Inst. Hautes Études Sci. Publ. Math., vol. 82, pp. 133-168, 1995.
    @article {S1, MRKEY = {1383215},
      AUTHOR = {Schwartz, Richard Evan},
      TITLE = {The quasi-isometry classification of rank one lattices},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {82},
      YEAR = {1995},
      PAGES = {133--168},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {22E40 (22E46)},
      MRNUMBER = {1383215},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.1007/BF02698639},
      ZBLNUMBER = {0852.22010},
      }
  • [S2] Go to document R. E. Schwartz, "Quasi-isometric rigidity and Diophantine approximation," Acta Math., vol. 177, iss. 1, pp. 75-112, 1996.
    @article {S2, MRKEY = {1417087},
      AUTHOR = {Schwartz, Richard Evan},
      TITLE = {Quasi-isometric rigidity and {D}iophantine approximation},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {177},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {75--112},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {53C35 (11J99 22E40 53C23)},
      MRNUMBER = {1417087},
      MRREVIEWER = {Alexander Starkov},
      DOI = {10.1007/BF02392599},
      ZBLNUMBER = {1029.11038},
      }
  • [Sh] Go to document Y. Shalom, "Harmonic analysis, cohomology, and the large-scale geometry of amenable groups," Acta Math., vol. 192, iss. 2, pp. 119-185, 2004.
    @article {Sh, MRKEY = {2096453},
      AUTHOR = {Shalom, Yehuda},
      TITLE = {Harmonic analysis, cohomology, and the large-scale geometry of amenable groups},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {192},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {119--185},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {20F65 (43A07)},
      MRNUMBER = {2096453},
      MRREVIEWER = {Peter A. Linnell},
      DOI = {10.1007/BF02392739},
      ZBLNUMBER = {1064.43004},
      }
  • [SW] Go to document P. M. Soardi and W. Woess, "Amenability, unimodularity, and the spectral radius of random walks on infinite graphs," Math. Z., vol. 205, iss. 3, pp. 471-486, 1990.
    @article {SW, MRKEY = {1082868},
      AUTHOR = {Soardi, Paolo M. and Woess, Wolfgang},
      TITLE = {Amenability, unimodularity, and the spectral radius of random walks on infinite graphs},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {205},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {471--486},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {43A07 (60B15 60J15)},
      MRNUMBER = {1082868},
      MRREVIEWER = {Donald I. Cartwright},
      DOI = {10.1007/BF02571256},
      ZBLNUMBER = {0693.43001},
      }
  • [Wo] Go to document W. Woess, "Topological groups and infinite graphs," Discrete Math., vol. 95, iss. 1-3, pp. 373-384, 1991.
    @article {Wo, MRKEY = {1141949},
      AUTHOR = {Woess, Wolfgang},
      TITLE = {Topological groups and infinite graphs},
      NOTE = {Directions in infinite graph theory and combinatorics (Cambridge, 1989)},
      JOURNAL = {Discrete Math.},
      FJOURNAL = {Discrete Mathematics},
      VOLUME = {95},
      YEAR = {1991},
      NUMBER = {1-3},
      PAGES = {373--384},
      ISSN = {0012-365X},
      CODEN = {DSMHA4},
      MRCLASS = {22D99 (05C25)},
      MRNUMBER = {1141949},
      MRREVIEWER = {P. Gerl},
      DOI = {10.1016/0012-365X(91)90348-6},
      ZBLNUMBER = {0757.05060},
      }
  • [Wo2] Go to document W. Woess, "Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions," Combin. Probab. Comput., vol. 14, iss. 3, pp. 415-433, 2005.
    @article {Wo2, MRKEY = {2138121},
      AUTHOR = {Woess, Wolfgang},
      TITLE = {Lamplighters, {D}iestel-{L}eader graphs, random walks, and harmonic functions},
      JOURNAL = {Combin. Probab. Comput.},
      FJOURNAL = {Combinatorics, Probability and Computing},
      VOLUME = {14},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {415--433},
      ISSN = {0963-5483},
      MRCLASS = {60C05 (05C25 60G50 60J45)},
      MRNUMBER = {2138121},
      MRREVIEWER = {Donald I. Cartwright},
      DOI = {10.1017/S0963548304006443},
      ZBLNUMBER = {1066.05075},
      }
  • [W] Go to document K. Wortman, "A finitely presented solvable group with a small quasi-isometry group," Michigan Math. J., vol. 55, iss. 1, pp. 3-24, 2007.
    @article {W,
      author = {Wortman, Kevin},
      TITLE = {A finitely presented solvable group with a small quasi-isometry group},
      JOURNAL = {Michigan Math. J.},
      FJOURNAL = {Michigan Mathematical Journal},
      VOLUME = {55},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {3--24},
      ISSN = {0026-2285},
      MRCLASS = {20F65 (20F16)},
      MRNUMBER = {2320169},
      MRREVIEWER = {Laurent Bartholdi},
      ZBLNUMBER = {1137.20034},
      DOI = {10.1307/mmj/1177681982},
      }

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