Groups of oscillating intermediate growth

Abstract

We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of functions. In particular, we can have growth oscillating between $e^{n^\alpha}$ and any prescribed function, growing as rapidly as desired. Our construction is built on top of any of the Grigorchuk groups of intermediate growth and is a variation on the limit of permutational wreath product.

  • [Bar1] Go to document L. Bartholdi, "The growth of Grigorchuk’s torsion group," Internat. Math. Res. Notices, iss. 20, pp. 1049-1054, 1998.
    @article {Bar1, MRKEY = {1656258},
      AUTHOR = {Bartholdi, Laurent},
      TITLE = {The growth of {G}rigorchuk's torsion group},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1998},
      NUMBER = {20},
      PAGES = {1049--1054},
      ISSN = {1073-7928},
      MRCLASS = {20F32},
      MRNUMBER = {1656258},
      MRREVIEWER = {Victor M. Petrogradsky},
      DOI = {10.1155/S1073792898000622},
      ZBLNUMBER = {0942.20027},
      }
  • [Bar2] Go to document L. Bartholdi, "Lower bounds on the growth of a group acting on the binary rooted tree," Internat. J. Algebra Comput., vol. 11, iss. 1, pp. 73-88, 2001.
    @article {Bar2, MRKEY = {1818662},
      AUTHOR = {Bartholdi, Laurent},
      TITLE = {Lower bounds on the growth of a group acting on the binary rooted tree},
      JOURNAL = {Internat. J. Algebra Comput.},
      FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {11},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {73--88},
      ISSN = {0218-1967},
      MRCLASS = {20F05 (20E08)},
      MRNUMBER = {1818662},
      DOI = {10.1142/S0218196701000395},
      ZBLNUMBER = {1028.20025},
      }
  • [BE] L. Bartholdi and A. G. Erschler, Growth of permutational extensions.
    @misc{BE,
      author={Bartholdi, Laurent and Erschler, A. G.},
      TITLE={Growth of permutational extensions},
      ARXIV={1011.5266},
     }
  • [BE2] L. Bartholdi and A. G. Erschler, Groups of given intermediate word growth.
    @misc{BE2,
      author={Bartholdi, Laurent and Erschler, A. G.},
      TITLE={Groups of given intermediate word growth},
      ARXIV={1108.4511},
     }
  • [BGS] Go to document L. Bartholdi, R. I. Grigorchuk, and Z. vSuni’k, "Branch groups," in Handbook of Algebra, Vol. 3, Amsterdam: North-Holland, 2003, pp. 989-1112.
    @incollection {BGS, MRKEY = {2035113},
      AUTHOR = {Bartholdi, Laurent and Grigorchuk, Rostislav I. and {Š}uni{\'k},
      Zoran},
      TITLE = {Branch groups},
      BOOKTITLE = {Handbook of Algebra, {V}ol. 3},
      PAGES = {989--1112},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {2003},
      MRCLASS = {20E08 (05C25 20F10 20F65 22D10 37B05 43A07 68Q70)},
      MRNUMBER = {2035113},
      MRREVIEWER = {Tullio G. Ceccherini-Silberstein},
      DOI = {10.1016/S1570-7954(03)80078-5},
      ZBLNUMBER = {1140.20306},
      }
  • [BS] Go to document I. Benjamini and O. Schramm, "Recurrence of distributional limits of finite planar graphs," Electron. J. Probab., vol. 6, p. 23, 2001.
    @article {BS, MRKEY = {1873300},
      AUTHOR = {Benjamini, Itai and Schramm, Oded},
      TITLE = {Recurrence of distributional limits of finite planar graphs},
      JOURNAL = {Electron. J. Probab.},
      FJOURNAL = {Electronic Journal of Probability},
      VOLUME = {6},
      YEAR = {2001},
      PAGES = {no. 23, 13 pp.},
      ISSN = {1083-6489},
      MRCLASS = {82B41 (05C80 52C26 60G50)},
      MRNUMBER = {1873300},
      MRREVIEWER = {Olle H{ä}ggstr{ö}m},
      DOI = {10.1214/EJP.v6-96},
      ZBLNUMBER = {1010.82021},
     }
  • [BG1] Go to document J. Bourgain and A. Gamburd, "Uniform expansion bounds for Cayley graphs of ${ SL}_2(\Bbb F_p)$," Ann. of Math., vol. 167, iss. 2, pp. 625-642, 2008.
    @article {BG1, MRKEY = {2415383},
      AUTHOR = {Bourgain, Jean and Gamburd, Alex},
      TITLE = {Uniform expansion bounds for {C}ayley graphs of {${\rm SL}\sb 2(\Bbb F\sb p)$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {167},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {625--642},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20F65 (05C25 05E15 11B30 20G40)},
      MRNUMBER = {2415383},
      MRREVIEWER = {Ben Joseph Green},
      DOI = {10.4007/annals.2008.167.625},
      ZBLNUMBER = {1216.20042},
      }
  • [BG] Go to document E. Breuillard and A. Gamburd, "Strong uniform expansion in ${ SL}(2,p)$," Geom. Funct. Anal., vol. 20, iss. 5, pp. 1201-1209, 2010.
    @article {BG, MRKEY = {2746951},
      AUTHOR = {Breuillard, Emmanuel and Gamburd, Alex},
      TITLE = {Strong uniform expansion in {${\rm SL}(2,p)$}},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {20},
      YEAR = {2010},
      NUMBER = {5},
      PAGES = {1201--1209},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {20G40 (05C25 60B15 60B99)},
      MRNUMBER = {2746951},
      MRREVIEWER = {Goulnara N. Arzhantseva},
      DOI = {10.1007/s00039-010-0094-3},
      ZBLNUMBER = {05822839},
      }
  • [Bri] Go to document J. Brieussel, Growth of certain groups of automorphisms of rooted trees, 2008.
    @misc{Bri,
      author={Brieussel, J{é}r{é}mie},
      TITLE = {Growth of certain groups of automorphisms of rooted trees},
      NOTE={(in French), Ph.D. dissertation, University of Paris~7},
      YEAR={2008},
      URL={http://www.institut.math.jussieu.fr/theses/2008/brieussel/},
     }
  • [Bri1] Go to document J. Brieussel, "Amenability and non-uniform growth of some directed automorphism groups of a rooted tree," Math. Z., vol. 263, iss. 2, pp. 265-293, 2009.
    @article {Bri1, MRKEY = {2534118},
      AUTHOR = {Brieussel, J{é}r{é}mie},
      TITLE = {Amenability and non-uniform growth of some directed automorphism groups of a rooted tree},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {263},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {265--293},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {43A07 (20E08)},
      MRNUMBER = {2534118},
      MRREVIEWER = {Laurent Bartholdi},
      DOI = {10.1007/s00209-008-0417-3},
      ZBLNUMBER = {1227.43001},
      }
  • [Bri2] J. Brieussel, Growth behaviors in the range $e^{r^\alpha}$.
    @misc{Bri2,
      author = {Brieussel, J{é}r{é}mie},
      TITLE = {Growth behaviors in the range $e^{r^\alpha}$},
      ARXIV={1107.1632},
      SORTYEAR={2013},
     }
  • [BKP] J. Brieussel, M. Kassabov, and I. Pak, The oscillating rate of escape of random walks on groups, 2013.
    @misc{BKP,
      author = {Brieussel, J{é}r{é}mie and Kassabov, M. and Pak, I.},
      TITLE = {The oscillating rate of escape of random walks on groups},
      YEAR={2013},
      NOTE={in preparation},
     }
  • [Cra] Go to document D. A. Craven, "Lower bounds for representation growth," J. Group Theory, vol. 13, iss. 6, pp. 873-890, 2010.
    @article {Cra, MRKEY = {2736162},
      AUTHOR = {Craven, David A.},
      TITLE = {Lower bounds for representation growth},
      JOURNAL = {J. Group Theory},
      FJOURNAL = {Journal of Group Theory},
      VOLUME = {13},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {873--890},
      ISSN = {1433-5883},
      CODEN = {JGTHFQ},
      MRCLASS = {20C15 (20C30 20E18 20E26 20F69)},
      MRNUMBER = {2736162},
      MRREVIEWER = {Benjamin Klopsch},
      DOI = {10.1515/JGT.2010.029},
      ZBLNUMBER = {1211.20008},
      }
  • [Ers1] Go to document A. Erschler, "Boundary behavior for groups of subexponential growth," Ann. of Math., vol. 160, iss. 3, pp. 1183-1210, 2004.
    @article {Ers1, MRKEY = {2144977},
      AUTHOR = {Erschler, Anna},
      TITLE = {Boundary behavior for groups of subexponential growth},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {160},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {1183--1210},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20F65 (20F05 20P05 60G50)},
      MRNUMBER = {2144977},
      MRREVIEWER = {Anders Karlsson},
      DOI = {10.4007/annals.2004.160.1183},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1089.20025},
      }
  • [Ers2] Go to document A. Erschler, "On the degrees of growth of finitely generated groups," Funktsional. Anal. i Prilozhen., vol. 39, iss. 4, pp. 86-89, 2005.
    @article {Ers2, MRKEY = {2197519},
      AUTHOR = {Erschler, Anna},
      TITLE = {On the degrees of growth of finitely generated groups},
      JOURNAL = {Funktsional. Anal. i Prilozhen.},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Funktsional\cprime nyĭAnaliz i ego Prilozheniya},
      VOLUME = {39},
      YEAR = {2005},
      NUMBER = {4},
      PAGES = {86--89},
      ISSN = {0374-1990},
      MRCLASS = {20F05},
      MRNUMBER = {2197519},
      MRREVIEWER = {Waldemar M. Ho{\l}ubowski},
      DOI = {10.1007/s10688-005-0055-z},
      ZBLNUMBER = {1122.20016},
     }
  • [Gri1] R. I. Grigorchuk, "On Burnside’s problem on periodic groups," Funktsional. Anal. i Prilozhen., vol. 14, iss. 1, pp. 53-54, 1980.
    @article {Gri1, MRKEY = {0565099},
      AUTHOR = {Grigorchuk, R. I.},
      TITLE = {On {B}urnside's problem on periodic groups},
      JOURNAL = {Funktsional. Anal. i Prilozhen.},
      FJOURNAL = {Akademiya Nauk SSSR. Funktsional\cprime nyĭAnaliz i ego Prilozheniya},
      VOLUME = {14},
      YEAR = {1980},
      NUMBER = {1},
      PAGES = {53--54},
      ISSN = {0374-1990},
      MRCLASS = {20F05},
      MRNUMBER = {0565099},
      MRREVIEWER = {N. S. Chernikov},
      ZBLNUMBER = {0595.20029},
     }
  • [Gri2] R. I. Grigorchuk, "On the Milnor problem of group growth," Soviet Math. Dokl., vol. 28, pp. 23-26, 1983.
    @article {Gri2, MRKEY={0712546},
      AUTHOR = {Grigorchuk, R. I.},
      TITLE = {On the {M}ilnor problem of group growth},
      JOURNAL = {Soviet Math. Dokl.},
      FJOURNAL = {Doklady Akademii Nauk SSSR},
      VOLUME = {28},
      YEAR = {1983},
      PAGES = {23--26},
      MRNUMBER = {0712546},
      ZBLNUMBER = {0547.20025},
     }
  • [Gri3] Go to document R. I. Grigorchuk, "Degrees of growth of finitely generated groups, and the theory of invariant means," Math. USSR-Izv., vol. 25, pp. 259-300, 1985.
    @article {Gri3, MRKEY = {0764305},
      AUTHOR = {Grigorchuk, R. I.},
      TITLE = {Degrees of growth of finitely generated groups, and the theory of invariant means},
      JOURNAL = {Math. USSR-Izv.},
      VOLUME = {25},
      YEAR = {1985},
      PAGES = {259--300},
      ZBLNUMBER = {0583.20023},
      MRNUMBER = {0764305},
      DOI = {10.1070/IM1985v025n02ABEH001281},
     }
  • [Gri4] Go to document R. I. Grigorchuk, "Degrees of growth of $p$-groups and torsion-free groups," Math. USSR-Sb., vol. 54, pp. 185-205, 1986.
    @article {Gri4, MRKEY = {0784354},
      AUTHOR = {Grigorchuk, R. I.},
      TITLE = {Degrees of growth of {$p$}-groups and torsion-free groups},
      JOURNAL = {Math. USSR-Sb.},
      VOLUME = {54},
      YEAR = {1986},
      PAGES = {185--205},
      MRNUMBER = {0784354},
      ZBLNUMBER = {0583.20024},
      DOI = {10.1070/SM1986v054n01ABEH002967},
     }
  • [Gri-one] Go to document R. I. Grigorchuk, "Solved and unsolved problems around one group," in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Basel: Birkhäuser, 2005, vol. 248, pp. 117-218.
    @incollection {Gri-one, MRKEY = {2195454},
      AUTHOR = {Grigorchuk, R. I.},
      TITLE = {Solved and unsolved problems around one group},
      BOOKTITLE = {Infinite Groups: Geometric, Combinatorial and Dynamical Aspects},
      SERIES = {Progr. Math.},
      VOLUME = {248},
      PAGES = {117--218},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2005},
      MRCLASS = {20-02 (20E08 20E18 20F10 20F38 20F50 20F65 20F69)},
      MRNUMBER = {2195454},
      MRREVIEWER = {Zoran {Š}uni{ć}},
      DOI = {10.1007/3-7643-7447-0_5},
      ZBLNUMBER = {1165.20021},
      }
  • [Gri5] Go to document R. I. Grigorchuk, "Some topics of dynamics of group actions on rooted trees," Proc. Steklov Inst. Math., vol. 273, pp. 1-118, 2011.
    @article{Gri5,
      author={Grigorchuk, R. I.},
      TITLE={Some topics of dynamics of group actions on rooted trees},
      NOTE={in Russian},
      JOURNAL={Proc. Steklov Inst. Math.},
      VOLUME={273},
      YEAR={2011},
      PAGES={1--118},
      ZBLNUMBER = {05963531},
      DOI = {10.1134/S0081543811040067},
     }
  • [Gri6] R. I. Grigorchuk, Milnor’s problem on the growth of groups and its consequences, 2011.
    @misc{Gri6,
      author={Grigorchuk, R. I.},
      TITLE={Milnor's problem on the growth of groups and its consequences},
      YEAR={2011},
      SORTYEAR={2012},
      ARXIV={1111.0512},
     }
  • [GH] Go to document R. I. Grigorchuk and P. de la Harpe, "On problems related to growth, entropy, and spectrum in group theory," J. Dynam. Control Systems, vol. 3, iss. 1, pp. 51-89, 1997.
    @article {GH, MRKEY = {1436550},
      AUTHOR = {Grigorchuk, R. I. and de la Harpe, P.},
      TITLE = {On problems related to growth, entropy, and spectrum in group theory},
      JOURNAL = {J. Dynam. Control Systems},
      FJOURNAL = {Journal of Dynamical and Control Systems},
      VOLUME = {3},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {51--89},
      ISSN = {1079-2724},
      CODEN = {JDCSEK},
      MRCLASS = {20F32 (20-02)},
      MRNUMBER = {1436550},
      MRREVIEWER = {Susan Hermiller},
      DOI = {10.1007/BF02471762},
      ZBLNUMBER = {0949.20033},
      }
  • [GNS] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiui, "Automata, dynamical systems, and groups," Tr. Mat. Inst. Steklova, vol. 231, iss. Din. Sist., Avtom. i Beskon. Gruppy, pp. 134-214, 2000.
    @article {GNS, MRKEY = {1841755},
      AUTHOR = {Grigorchuk, R. I. and Nekrashevich, V. V. and Sushchanski{\u\i},
      V. I.},
      TITLE = {Automata, dynamical systems, and groups},
      JOURNAL = {Tr. Mat. Inst. Steklova},
      FJOURNAL = {Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossiĭskaya Akademiya Nauk},
      VOLUME = {231},
      YEAR = {2000},
      NUMBER = {Din. Sist., Avtom. i Beskon. Gruppy},
      PAGES = {134--214},
      ISSN = {0371-9685},
      MRCLASS = {37B15 (20F10 68Q45 68Q70)},
      MRNUMBER = {1841755},
      ZBLNUMBER = {1155.37311},
      }
  • [GP] R. I. Grigorchuk and I. Pak, "Groups of intermediate growth: an introduction," Enseign. Math., vol. 54, iss. 3-4, pp. 251-272, 2008.
    @article {GP, MRKEY = {2478087},
      AUTHOR = {Grigorchuk, R. I. and Pak, Igor},
      TITLE = {Groups of intermediate growth: an introduction},
      JOURNAL = {Enseign. Math.},
      FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. 2e Série},
      VOLUME = {54},
      YEAR = {2008},
      NUMBER = {3-4},
      PAGES = {251--272},
      ISSN = {0013-8584},
      CODEN = {ENMAAR},
      MRCLASS = {20F69 (20F10 20F50)},
      MRNUMBER = {2478087},
      MRREVIEWER = {Goulnara N. Arzhantseva},
      ZBLNUMBER = {1204.20049},
      }
  • [Gro] Go to document M. Gromov, "Hyperbolic groups," in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263.
    @incollection {Gro, MRKEY = {0919829},
      AUTHOR = {Gromov, M.},
      TITLE = {Hyperbolic groups},
      BOOKTITLE = {Essays in Group Theory},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {8},
      PAGES = {75--263},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {20F32 (20F06 20F10 22E40 53C20 57R75 58F17)},
      MRNUMBER = {0919829},
      MRREVIEWER = {Christopher W. Stark},
      DOI = {10.1007/978-1-4613-9586-7_3},
      ZBLNUMBER = {0634.20015},
      }
  • [Har1] P. de la Harpe, Topics in Geometric Group Theory, Chicago, IL: University of Chicago Press, 2000.
    @book {Har1, MRKEY = {1786869},
      KEY={Har00},
      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},
      }
  • [Har2] Go to document P. de la Harpe, "Uniform growth in groups of exponential growth," in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II, 2002, pp. 1-17.
    @inproceedings {Har2, MRKEY = {1950882},
      KEY={Har02},
      AUTHOR = {de la {Harpe},
      Pierre},
      TITLE = {Uniform growth in groups of exponential growth},
      BOOKTITLE = {Proceedings of the {C}onference on {G}eometric and {C}ombinatorial {G}roup {T}heory, {P}art {II}},
      VENUE={{H}aifa, 2000},
      SERIES = {Geom. Dedicata},
      FJOURNAL = {Geometriae Dedicata},
      VOLUME = {95},
      YEAR = {2002},
      PAGES = {1--17},
      ISSN = {0046-5755},
      CODEN = {GEMDAT},
      MRCLASS = {20E07 (20F65)},
      MRNUMBER = {1950882},
      MRREVIEWER = {Victor M. Petrogradsky},
      DOI = {10.1023/A:1021273024728},
      ZBLNUMBER = {1025.20027},
      }
  • [HLW] Go to document S. Hoory, N. Linial, and A. Wigderson, "Expander graphs and their applications," Bull. Amer. Math. Soc., vol. 43, iss. 4, pp. 439-561, 2006.
    @article {HLW, MRKEY = {2247919},
      AUTHOR = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi},
      TITLE = {Expander graphs and their applications},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {43},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {439--561},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {68Q15 (00-02 05C25 05C80 60G50 68Q17 68R10)},
      MRNUMBER = {2247919},
      MRREVIEWER = {Mark R. Jerrum},
      DOI = {10.1090/S0273-0979-06-01126-8},
      ZBLNUMBER = {1147.68608},
     }
  • [Jai] Go to document A. Jaikin-Zapirain, "Zeta function of representations of compact $p$-adic analytic groups," J. Amer. Math. Soc., vol. 19, iss. 1, pp. 91-118, 2006.
    @article {Jai, MRKEY = {2169043},
      AUTHOR = {Jaikin-Zapirain, A.},
      TITLE = {Zeta function of representations of compact {$p$}-adic analytic groups},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {19},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {91--118},
      ISSN = {0894-0347},
      MRCLASS = {20E18 (11M41 20C15 22E35)},
      MRNUMBER = {2169043},
      MRREVIEWER = {Alexander Moret{ó}},
      DOI = {10.1090/S0894-0347-05-00501-1},
      ZBLNUMBER = {1092.20023},
      }
  • [KN] Go to document M. Kassabov and N. Nikolov, "Cartesian products as profinite completions," Int. Math. Res. Not., vol. 2006, p. I, 2006.
    @article {KN, MRKEY = {2264720},
      AUTHOR = {Kassabov, Martin and Nikolov, Nikolay},
      TITLE = {Cartesian products as profinite completions},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
      PAGES = {Art. ID 72947, 17},
      ISSN = {1073-7928},
      MRCLASS = {20B25},
      MRNUMBER = {2264720},
      DOI = {10.1155/IMRN/2006/72947},
      VOLUME = {2006},
      ZBLNUMBER = {1129.20019},
     }
  • [LL] Go to document M. Larsen and A. Lubotzky, "Representation growth of linear groups," J. Eur. Math. Soc. $($JEMS$)$, vol. 10, iss. 2, pp. 351-390, 2008.
    @article {LL, MRKEY = {2390327},
      AUTHOR = {Larsen, Michael and Lubotzky, Alexander},
      TITLE = {Representation growth of linear groups},
      JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
      FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {10},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {351--390},
      ISSN = {1435-9855},
      MRCLASS = {20F69 (11M41 20E07)},
      MRNUMBER = {2390327},
      MRREVIEWER = {Nikolay V. Nikolov},
      DOI = {10.4171/JEMS/113},
      ZBLNUMBER = {1142.22006},
      }
  • [Leo] Go to document Y. G. Leonov, "On a lower bound for the growth function of the Grigorchuk group," Mat. Zametki, vol. 67, iss. 3, pp. 475-477, 2000.
    @article {Leo, MRKEY = {1779480},
      AUTHOR = {Leonov, Yu. G.},
      TITLE = {On a lower bound for the growth function of the {G}rigorchuk group},
      JOURNAL = {Mat. Zametki},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Matematicheskie Zametki},
      VOLUME = {67},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {475--477},
      ISSN = {0025-567X},
      MRCLASS = {20F05 (20F50)},
      MRNUMBER = {1779480},
      DOI = {10.1007/BF02676677},
      ZBLNUMBER = {0984.20020},
      }
  • [Lub-book] A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Basel: Birkhäuser, 1994, vol. 125.
    @book {Lub-book, MRKEY = {1308046},
      AUTHOR = {Lubotzky, Alexander},
      TITLE = {Discrete Groups, Expanding Graphs and Invariant Measures},
      SERIES = {Progr. Math.},
      VOLUME = {125},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1994},
      PAGES = {xii+195},
      ISBN = {3-7643-5075-X},
      MRCLASS = {22E40 (05C25 11F70 28C10 43A07)},
      MRNUMBER = {1308046},
      MRREVIEWER = {Wolfgang Woess},
      ZBLNUMBER = {0826.22012},
      }
  • [Lub] A. Lubotzky, "Subgroup growth," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 309-317.
    @inproceedings {Lub, MRKEY = {1403931},
      AUTHOR = {Lubotzky, Alexander},
      TITLE = {Subgroup growth},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      VENUE={{Z}ürich, 1994},
      PAGES = {309--317},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1995},
      MRCLASS = {20E15 (20E18 20E26 20F18 20H05)},
      MRNUMBER = {1403931},
      MRREVIEWER = {John S. Wilson},
      ZBLNUMBER = {0857.20013},
      }
  • [LS] Go to document A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212.
    @book {LS, MRKEY = {1978431},
      AUTHOR = {Lubotzky, Alexander and Segal, Dan},
      TITLE = {Subgroup Growth},
      SERIES = {Progr. Math.},
      VOLUME = {212},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2003},
      PAGES = {xxii+453},
      ISBN = {3-7643-6989-2},
      MRCLASS = {20E07 (20E18 20E26 20F69)},
      MRNUMBER = {1978431},
      MRREVIEWER = {Avinoam Mann},
      DOI = {10.1007/978-3-0348-8965-0},
      ZBLNUMBER = {1071.20033},
      }
  • [MP] Go to document R. Muchnik and I. Pak, "On growth of Grigorchuk groups," Internat. J. Algebra Comput., vol. 11, iss. 1, pp. 1-17, 2001.
    @article {MP, MRKEY = {1818659},
      AUTHOR = {Muchnik, Roman and Pak, Igor},
      TITLE = {On growth of {G}rigorchuk groups},
      JOURNAL = {Internat. J. Algebra Comput.},
      FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {11},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {1--17},
      ISSN = {0218-1967},
      MRCLASS = {20F05 (20E08 68R15)},
      MRNUMBER = {1818659},
      MRREVIEWER = {Muhammad Zafrullah},
      DOI = {10.1142/S0218196701000450},
      ZBLNUMBER = {1024.20031},
      }
  • [Nek] V. Nekrashevych, Self-Similar Groups, Providence, RI: Amer. Math. Soc., 2005, vol. 117.
    @book {Nek, MRKEY = {2162164},
      AUTHOR = {Nekrashevych, Volodymyr},
      TITLE = {Self-Similar Groups},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {117},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {xii+231},
      ISBN = {0-8218-3831-8},
      MRCLASS = {20E08 (20F65 37B15 37F10)},
      MRNUMBER = {2162164},
      MRREVIEWER = {Laurent Bartholdi},
      ZBLNUMBER = {1087.20032},
      }
  • [Neu] Go to document B. H. Neumann, "Some remarks on infinite groups," J. London Math. Soc., vol. 12, pp. 120-127, 1937.
    @article{Neu,
      author={Neumann, B. H.},
      TITLE={Some remarks on infinite groups},
      JOURNAL={J. London Math. Soc.},
      VOLUME={12},
      YEAR={1937},
      PAGES={120--127},
      ZBLNUMBER = {0016.29501},
      DOI = {10.1112/jlms/s1-12.46.120},
     }
  • [Neumann] Go to document P. M. Neumann, "Some questions of Edjvet and Pride about infinite groups," Illinois J. Math., vol. 30, iss. 2, pp. 301-316, 1986.
    @article {Neumann, MRKEY = {0840129},
      AUTHOR = {Neumann, Peter M.},
      TITLE = {Some questions of {E}djvet and {P}ride about infinite groups},
      JOURNAL = {Illinois J. Math.},
      FJOURNAL = {Illinois Journal of Mathematics},
      VOLUME = {30},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {301--316},
      ISSN = {0019-2082},
      CODEN = {IJMTAW},
      MRCLASS = {20E15 (20E22)},
      MRNUMBER = {0840129},
      MRREVIEWER = {John S. Wilson},
      URL = {http://projecteuclid.org/euclid.ijm/1256044639},
      ZBLNUMBER = {0598.20029},
      }
  • [Pes] Go to document V. G. Pestov, "Hyperlinear and sofic groups: a brief guide," Bull. Symbolic Logic, vol. 14, iss. 4, pp. 449-480, 2008.
    @article {Pes, MRKEY = {2460675},
      AUTHOR = {Pestov, Vladimir G.},
      TITLE = {Hyperlinear and sofic groups: a brief guide},
      JOURNAL = {Bull. Symbolic Logic},
      FJOURNAL = {Bulletin of Symbolic Logic},
      VOLUME = {14},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {449--480},
      ISSN = {1079-8986},
      MRCLASS = {20F69 (03C20 37B10 46L05 46L10)},
      MRNUMBER = {2460675},
      MRREVIEWER = {Alain Valette},
      DOI = {10.2178/bsl/1231081461},
      ZBLNUMBER = {1206.20048},
      }
  • [Pyb1] Go to document L. Pyber, "Old groups can learn new tricks," in Groups, Combinatorics & Geometry, World Sci. Publ., River Edge, NJ, 2003, pp. 243-255.
    @incollection {Pyb1, MRKEY = {1994970},
      AUTHOR = {Pyber, L{á}szl{ó}},
      TITLE = {Old groups can learn new tricks},
      BOOKTITLE = {Groups, Combinatorics \& Geometry},
      VENUE={{D}urham, 2001},
      PAGES = {243--255},
      PUBLISHER = {World Sci. Publ., River Edge, NJ},
      YEAR = {2003},
      MRCLASS = {20E07 (20-02 20E18)},
      MRNUMBER = {1994970},
      DOI = {10.1142/9789812564481_0014},
      ZBLNUMBER = {1046.20024},
      }
  • [Pyb2] Go to document L. Pyber, "Groups of intermediate subgroup growth and a problem of Grothendieck," Duke Math. J., vol. 121, iss. 1, pp. 169-188, 2004.
    @article {Pyb2, MRKEY = {2031168},
      AUTHOR = {Pyber, L{á}szl{ó}},
      TITLE = {Groups of intermediate subgroup growth and a problem of {G}rothendieck},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {121},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {169--188},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {20E07 (20E18)},
      MRNUMBER = {2031168},
      MRREVIEWER = {Jan-Christoph Schlage-Puchta},
      DOI = {10.1215/S0012-7094-04-12115-3},
      ZBLNUMBER = {1057.20019},
      }
  • [Ruiz] Go to document S. M. Ruiz, "A result on prime numbers," Math. Gazette, vol. 81, pp. 269-270, 1997.
    @article{Ruiz,
      author={Ruiz, S. M.},
      TITLE={A result on prime numbers},
      JOURNAL={Math. Gazette},
      VOLUME={81},
      YEAR={1997},
      PAGES={269--270},
      DOI = {10.2307/3619207},
      }
  • [Seg] Go to document D. Segal, "The finite images of finitely generated groups," Proc. London Math. Soc., vol. 82, iss. 3, pp. 597-613, 2001.
    @article {Seg, MRKEY = {1816690},
      AUTHOR = {Segal, Dan},
      TITLE = {The finite images of finitely generated groups},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {82},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {597--613},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {20E07 (20E18)},
      MRNUMBER = {1816690},
      MRREVIEWER = {Gary L. Walls},
      DOI = {10.1112/plms/82.3.597},
      ZBLNUMBER = {1022.20011},
      }
  • [ST] Go to document Y. Shalom and T. Tao, "A finitary version of Gromov’s polynomial growth theorem," Geom. Funct. Anal., vol. 20, iss. 6, pp. 1502-1547, 2010.
    @article {ST, MRKEY = {2739001},
      AUTHOR = {Shalom, Yehuda and Tao, Terence},
      TITLE = {A finitary version of {G}romov's polynomial growth theorem},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {20},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {1502--1547},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {20F65 (31C05)},
      MRNUMBER = {2739001},
      MRREVIEWER = {Alexander Lubotzky},
      DOI = {10.1007/s00039-010-0096-1},
      ZBLNUMBER = {05833801},
      }
  • [Ufn] V. A. Ufnarovskij, "Combinatorial and asymptotic methods in algebra," in Algebra, VI, New York: Springer-Verlag, 1995, vol. 57, pp. 1-196.
    @incollection {Ufn, MRKEY = {1360005},
      AUTHOR = {Ufnarovskij, V. A.},
      TITLE = {Combinatorial and asymptotic methods in algebra},
      BOOKTITLE = {Algebra, {VI}},
      SERIES = {Encyclopaedia Math. Sci.},
      VOLUME = {57},
      PAGES = {1--196},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      MRCLASS = {16S15 (13D40 16P90 16W50)},
      MRNUMBER = {1360005},
      ZBLNUMBER = {0826.16001},
     }
  • [Voll] Go to document C. Voll, "Functional equations for zeta functions of groups and rings," Ann. of Math., vol. 172, iss. 2, pp. 1181-1218, 2010.
    @article {Voll, MRKEY = {2680489},
      AUTHOR = {Voll, Christopher},
      TITLE = {Functional equations for zeta functions of groups and rings},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {172},
      YEAR = {2010},
      NUMBER = {2},
      PAGES = {1181--1218},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {20E07 (11S40 16P90 39B52)},
      MRNUMBER = {2680489},
      MRREVIEWER = {Alexander Fel{\cprime}shtyn},
      DOI = {10.4007/annals.2010.172.1185},
      ZBLNUMBER = {05808584},
      }
  • [Wil] Go to document J. S. Wilson, "On exponential growth and uniformly exponential growth for groups," Invent. Math., vol. 155, iss. 2, pp. 287-303, 2004.
    @article {Wil, MRKEY = {2031429},
      AUTHOR = {Wilson, John S.},
      TITLE = {On exponential growth and uniformly exponential growth for groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {155},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {287--303},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20F69 (20E22 20F05 20F65)},
      MRNUMBER = {2031429},
      MRREVIEWER = {Victor M. Petrogradsky},
      DOI = {10.1007/s00222-003-0321-8},
      ZBLNUMBER = {1065.20054},
      }
  • [Woe] Go to document W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge: Cambridge Univ. Press, 2000, vol. 138.
    @book {Woe, MRKEY = {1743100},
      AUTHOR = {Woess, Wolfgang},
      TITLE = {Random Walks on Infinite Graphs and Groups},
      SERIES = {Cambridge Tracts Math.},
      VOLUME = {138},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2000},
      PAGES = {xii+334},
      ISBN = {0-521-55292-3},
      MRCLASS = {60B15 (60G50 60J10)},
      MRNUMBER = {1743100},
      MRREVIEWER = {Donald I. Cartwright},
      DOI = {10.1017/CBO9780511470967},
      ZBLNUMBER = {0951.60002},
      }

Authors

Martin Kassabov

Mathematics
University of Southampton
Southampton SO17 1BG
United Kingdom
and
Department of Mathematics
Cornell University
Ithaca, NY 14853-4201

Igor Pak

Department of Mathematics
University of California, Los Angeles
Box 951555
Los Angeles, CA 90095-1555