• Mark V. Sapir
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we introduce a new tool – van Kampen diagrams (although these are sibling of diagrams introduced in Section  1.7.4). We start with defining van Kampen diagrams and explaining basic methods of using them: the bands, the Swiss cheese method and the small cancelation theory. Then we explain Golod’s solution of the unbounded Burnside problem and a road map of Olshanskii’s proof of the Novikov–Adian theorem solving the bounded Burnside problem: there exists an infinite finitely generated group satisfying the identity x n  = 1 for all odd n ≥ 665. The theorem was one of the main achievements in group theory of the 20th century. Its initial proof occupied more than 300 pages and was extremely complicated. Olshanskii managed to find a much simpler proof (for much bigger n). Our road map explains the main ideas and “points of interest” of Olshanskii’s proof.


  1. 1.
    M. Abért, Group laws and free subgroups in topological groups. Bull. Lond. Math. Soc. 37(4), 525–534 (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    S.I. Adian, Infinite irreducible systems of group identities. Izv. Akad. Nauk SSSR Ser. Mat. 34, 715–734 (1970)MathSciNetGoogle Scholar
  3. 3.
    S.I. Adian, The Burnside Problem and Identities in Groups (Springer, Berlin/ New York, 1979)Google Scholar
  4. 4.
    S.I. Adian, Random walks on free periodic groups. Izv. Akad. Nauk SSSR Ser. Mat. 46(6), 1139–1149 (1982)MathSciNetGoogle Scholar
  5. 9.
    R. Aharoni, P. Haxell, Hall’s theorem for hypergraphs. J. Graph Theory 35(2), 83–88 (2000)Google Scholar
  6. 10.
    I. Agol (with an appendix by Ian Agol, Daniel Groves and Jason Manning). The virtual Haken conjecture, arXiv:1204.2810Google Scholar
  7. 12.
    J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, Notes on hyperbolic groups, group theory from a geometric viewpoint, in Proceedings of ICTP, Trieste (World Scientific, Singapore, 1991), pp. 3–63Google Scholar
  8. 21.
    S. Banach, A. Tarski, Sur la decomposition des ensembles de points en porties respectivement congruents. Fund. Math. 6, 244–277 (l924)Google Scholar
  9. 22.
    L. Bartholdi, The growth of Grigorchuk’s torsion group. Int. Math. Res. Not. 20, 1049–1054 (1998)CrossRefMathSciNetGoogle Scholar
  10. 23.
    L. Bartholdi, Lower bounds on the growth of a group acting on the binary rooted tree. Int. J. Algebra Comput. 11(1), 73–88 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 25.
    H. Bass, The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. 25, 603–614 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 47.
    Collected by M. Bestvina, Questions in geometric group theory (preprint), available at
  13. 48.
    O. Bogopolski, Introduction to Group Theory (trans., revised and expanded from the 2002 Russian original). EMS Textbooks in Mathematics (European Mathematical Society (EMS), Zürich, 2008)Google Scholar
  14. 51.
    M.R. Bridson, The geometry of the word problem, in Invitations to Geometry and Topology. Oxford Graduate Texts in Mathematics, vol. 7 (Oxford University Press, Oxford, 2002), pp. 29–91Google Scholar
  15. 53.
    M.G. Brin, C.C. Squier, Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79(3), 485–498 (1985)Google Scholar
  16. 54.
    K. Brown, Finiteness properties of groups, in Proceedings of the Northwestern Conference on Cohomology of Groups, Evanston, 1985. J. Pure Appl. Algebra 44(1–3), 45–75 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 56.
    R.M. Bryant, Some infinitely based varieties of groups. Aust. J. Math. 16(1), 29–33 (1973)CrossRefzbMATHGoogle Scholar
  18. 59.
    J. Burillo, Growth of positive words in Thompson’s group F. Commun. Algebra 32(8), 3087–3094 (2004)Google Scholar
  19. 61.
    J.W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata 16(2), 123–148 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 62.
    J.W. Cannon, W.J. Floyd, W.R. Parry, Introductorary notes on Richard Thompson’s groups. L’Enseignement Mathématique 42(2), 215–256 (1996)zbMATHMathSciNetGoogle Scholar
  21. 69.
    J.M. Cohen, Cogrowth and amenability of discrete groups. J. Funct. Anal. 48(3), 301–309 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 73.
    M. Coornaert, A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups. Lecture Notes in Mathematics, vol. 1539 (Springer, Berlin, 1993)Google Scholar
  23. 79.
    M.M. Day, Amenable semigroups. Ill. J. Math. 1, 509–544 (1957)zbMATHGoogle Scholar
  24. 80.
    P. Dehornoy, Geometric presentations for Thompson’s groups. J. Pure Appl. Algebra 203(1–3), 1–44 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 81.
    P. de la Harpe, Topics in Geometric Group Theory. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, 2000)Google Scholar
  26. 82.
    P. de la Harp, R.I. Grigorchuk, T. Ceccherini-Silberstein, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova 224 (1999), Algebra. Topol. Differ. Uravn. i ikh Prilozh. 68–111; trans. in Proc. Steklov Inst. Math. 224(1), 57–97 (1999)Google Scholar
  27. 85.
    W. Dicks, Joel Friedman’s proof of the strengthened Hanna Neumann conjecture (preprint),
  28. 86.
    W. Dicks, Simplified Mineyev (preprint),
  29. 91.
    C. Druţu, M. Kapovich, Lectures on geometric group theory (2012, preprint)Google Scholar
  30. 94.
    M. Elder, É. Fusy, A. Rechnitzer, Counting elements and geodesics in Thompson’s group F. J. Algebra 324(1), 102–121 (2010)Google Scholar
  31. 95.
    G. Endimioni, Conditions de finitude pour un groupe d’exposant fini (Finiteness conditions for a group with finite exponent). J. Algebra 155, 290–297 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 96.
    D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, Word Processing in Groups (Jones and Bartlett, Boston, 1992)zbMATHGoogle Scholar
  33. 97.
    M. Ershov, Golod–Shafarevich groups: a survey. Int. J. Algebra Comput. 22(5), 1230001, 68 (2012)Google Scholar
  34. 98.
    M. Ershov, G. Golan, M. Sapir, The Tarski numbers of groups (2014, preprint)Google Scholar
  35. 99.
    D.S. Farley, Proper isometric actions of Thompson’s groups on Hilbert space. Int. Math. Res. Not. 45, 2409–2414 (2003)CrossRefMathSciNetGoogle Scholar
  36. 100.
    D.S. Farley, Finiteness and CAT(0) properties of diagram groups. Topology 42(5), 1065–1082 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 101.
    A.F. Filippov, An elementary proof of Jordan’s theorem. Uspehi Matem. Nauk (N.S.) 5(39), 173–176 (1950)Google Scholar
  38. 104.
    J. Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture (submitted). arXiv:1105.0129Google Scholar
  39. 106.
    F. Galvin, R. Thompson, Unpublished notesGoogle Scholar
  40. 108.
    R. Geoghegan (ed.), Open Problems in Infinite-Dimensional Topology: The Proceedings of the 1979 Topology Conference, Ohio University, Athens, 1979. Topology Proceedings, vol. 4 (1979) no. 1) (1980), pp. 287–338Google Scholar
  41. 109.
    R. Geoghegan, F. Guzmán, Associativity and Thompson’s group, in Topological and Asymptotic Aspects of Group Theory. Contemporary Mathematics, vol. 394 (American Mathematical Society, Providence, 2006), pp. 113–135Google Scholar
  42. 110.
    E. Ghys, P. de la Harpe (eds.), Sur les groupes hyperboliques d’aprés Mikhael Gromov. Progress in Mathematics, vol. 83 (Birkhäuser, Boston, 1990)Google Scholar
  43. 112.
    E.S. Golod, On nil-algebras and finitely approximable p-groups. Izv. Akad. Nauk SSSR. Ser. Mat. 28, 273–276 (1964)MathSciNetGoogle Scholar
  44. 113.
    E.S. Golod, Some problems of Burnside type, in Proceedings of International Congress of Mathematicians, Moscow, 1966 (Izdat. “Mir”, Moscow, 1968), pp. 284–289Google Scholar
  45. 117.
    F.P. Greenleaf, Invariant Means on Topological Groups and Their Applications (Van Nostrand Reinhold, New York, 1969)zbMATHGoogle Scholar
  46. 118.
    M. Greendlinger, Dehn’s algorithm for the word problem. Commun. Pure Appl. Math. 13, 67–83 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 119.
    R.I. Grigorcuk, On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)MathSciNetGoogle Scholar
  48. 120.
    R.I. Grigorchuk, Symmetrical random walks on discrete groups, in Multicomponent Random Systems. Advances in Probability and Related Topics, vol. 6 (Dekker, New York, 1980), pp. 285–3251Google Scholar
  49. 121.
    R.I. Grigorchuk, Symmetric random walks on discrete groups. Uspehi Mat. Nauk 32(6)(198), 217–218 (1977)Google Scholar
  50. 122.
    R.I. Grigorchuk, On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983)MathSciNetGoogle Scholar
  51. 123.
    R.I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48(5), 939–985 (1984)MathSciNetGoogle Scholar
  52. 126.
    M. Gromov, Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Publ. Math. Inst. Hautes Études Sci. 53, 53–73 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 127.
    M. Gromov, Hyperbolic groups, in Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8 (Springer, New York, 1987), pp. 75–263Google Scholar
  54. 130.
    V.S. Guba, The Dehn function of Richard Thompson’s group F is quadratic. Invent. Math. 163(2), 313–342 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 131.
    V.S. Guba, Notes on the Burnside problem for odd exponents, private communication, Oct 2012Google Scholar
  56. 133.
    V.S. Guba, M. Sapir, Diagram Groups. Memoirs of the AMS, vol. 130 (American Mathematical Society, Providence, 1997)Google Scholar
  57. 134.
    V.S. Guba, M. Sapir, The Dehn function and a regular set of normal forms for R. Thompson’s group F. J. Aust. Math. Soc. Ser. A 62(3), 315–328 (1997)Google Scholar
  58. 135.
    V.S. Guba, M. Sapir, On subgroups of R. Thompson’s group F and other diagram groups. Mat. Sb. 190(8), 3–60 (1999)Google Scholar
  59. 136.
    V.S. Guba, M. Sapir, Diagram groups and directed 2-complexes: homotopy and homology. J. Pure Appl. Algebra 205(1), 1–47 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  60. 137.
    V.S. Guba, M. Sapir, Diagram groups are totally orderable. J. Pure Appl. Algebra 205(1), 48–73 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 138.
    Y. Guivarc’h, Groupes de Lie à croissance polynomiale. C. R. Acad. Sci. Paris, Sér. A–B 271, A237–A239 (1970)Google Scholar
  62. 139.
    Y. Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. Fr. 101, 333–379 (1973)Google Scholar
  63. 141.
    C.K. Gupta, A.N. Krasilnikov, The finite basis question for varieties of groups – some recent results. Special issue in honor of Reinhold Baer (1902–1979). Ill. J. Math. 47(1–2), 273–283 (2003)Google Scholar
  64. 142.
    M. Hall Jr., The Theory of Groups (Macmillan, New York, 1959)zbMATHGoogle Scholar
  65. 148.
    P.J. Heawood, Map-colour theorems. Q. J. Math. Oxf. 24, 332–338 (1890)zbMATHGoogle Scholar
  66. 150.
    S.M. Hermiller, Rewriting systems for Coxeter groups. J. Pure Appl. Algebra 92(2), 137–148 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  67. 153.
    G. Higman, Finitely Presented Infinite Simple Groups. Notes on Pure Mathematics, vol. 8 (Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974)Google Scholar
  68. 154.
    G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups. J. Lond. Math. Soc. 24, 247–254 (1949)Google Scholar
  69. 155.
    W. Imrich, On finitely generated subgroups of free groups. Arch. Math. 28, 21–24 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  70. 156.
    S.V. Ivanov, The Burnside problem for all sufficiently large exponents. Int. J. Algebra Comput. 4(1, 2), 1–300 (1994)Google Scholar
  71. 157.
    S.V. Ivanov, Embedding free Burnside groups in finitely presented groups. Geom. Dedicata 111, 87–105 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 158.
    S.V. Ivanov, A.M. Storozhev, On varieties of groups in which all periodic groups are abelian, in Group Theory, Statistics, and Cryptography. Contemporary Mathematics, vol. 360 (American Mathematical Society, Providence, 2004), pp. 55–62Google Scholar
  73. 168.
    I. Kapovich, A. Miasnikov, P. Schupp, V. Shpilrain, Average-case complexity and decision problems in group theory. Adv. Math. 190(2), 343–359 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  74. 169.
    M. Kapovich, Representations of polygons of finite groups. Geom. Topol. 9, 1915–1951 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  75. 170.
    M.I. Kargapolov, Yu.I. Merzlyakov, Fundamentals of Group Theory, 4th edn. (Fizmatlit “Nauka”, Moscow, 1996)Google Scholar
  76. 175.
    H. Kesten, Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336–354 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  77. 180.
    Yu.G. Kleiman, On the basis of products of varieties of groups. Izv. AN SSSR, Ser. Mat. 37, 95–97 (1973)Google Scholar
  78. 181.
    B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth. J. Am. Math. Soc. 23(3), 815–829 (2010)Google Scholar
  79. 186.
    A.N. Krasil’nikov, On the finiteness of the basis of identities of groups with a nilpotent commutator group. Izv. Akad. Nauk SSSR Ser. Mat. 54, 1181–1195 (1990); trans. in Math. USSR Izv. 37, 539–553 (1991)Google Scholar
  80. 199.
    J.M. Lever, The Elizabethan Love Sonnet (Barnes & Noble, London, 1968)Google Scholar
  81. 204.
    Y. Lodha, J. Moore, A geometric solution to the von Neumann-Day problem for finitely presented groups (2013), arXiv:1308.4250Google Scholar
  82. 210.
    R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory (Springer, Berlin/ New York, 1977)Google Scholar
  83. 212.
    I.G. Lysenok, A system of defining relations for the Grigorchuk group. Mat. Zametki 38, 503–511 (1985)MathSciNetGoogle Scholar
  84. 213.
    I.G. Lysenok, Infinite Burnside groups of even exponent. Izvestiya: Math. 60(3), 453–654 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  85. 219.
    A. Mann, How Groups Grow. London Mathematical Society Lecture Note Series, vol. 395 (Cambridge University Press, Cambridge, 2012)Google Scholar
  86. 221.
    S. Margolis, J. Meakin, M. Sapir, Algorithmic problems in groups, semigroups and inverse semigroups, in Semigroups, Formal Languages and Groups, York, 1993, pp. 147–214Google Scholar
  87. 223.
    A discussion # 60897 of invariant means on the group of integers,
  88. 230.
    Yu.I. Merzlyakov, Infinite finitely generated periodic groups. Dokl. Akad. Nauk SSSR 268(4), 803–805 (1983)MathSciNetGoogle Scholar
  89. 232.
    K.V. Mikhajlovskii, A.Yu. Olshanskii, Some constructions relating to hyperbolic groups, in Geometry and Cohomology in Group Theory, Durham, 1994. London Mathematical Society Lecture Note Series, vol. 252 (Cambridge University Press, Cambridge, 1998), pp. 263–290Google Scholar
  90. 233.
    J. Milnor, Problem 5603. Amer. Math. Mon. 75, 685–686 (1968)CrossRefMathSciNetGoogle Scholar
  91. 234.
    I. Mineyev, submultiplicativity and the Hanna Neumann conjecture. Ann. Math. (2) 175(1), 393–414 (2012)Google Scholar
  92. 235.
    N. Monod, Groups of piecewise projective homeomorphisms. Proc. Natl. Acad. Sci. U.S.A 110(12), 4524–4527 (2013)Google Scholar
  93. 237.
    J. Moore, Fast growth in the Følner function for Thompson’s group F. Groups Geom. Dyn. 7(3), 633–651 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  94. 246.
    H. Neumann, On the intersection of finitely generated free groups. Publ. Math. Debr. 4, 186–189 (1956)Google Scholar
  95. 247.
    H. Neumann, On the intersection of finitely generated free groups. Addendum. Publ. Math. Debr. 5, 128 (1957)Google Scholar
  96. 248.
    H. Neumann, Varieties of Groups (Springer, New York, 1967)CrossRefzbMATHGoogle Scholar
  97. 249.
    W.D. Neumann, On intersections of finitely generated subgroups of free groups, in Groups-Canberra 1989. Lecture Notes in Mathematics, vol. 1456 (Springer, Berlin/New York, 1990), pp. 161–170Google Scholar
  98. 251.
    P.S. Novikov, S.I. Adian, On infinite periodic groups. I, II, III. Izv. Akad. Nauk SSSR. Ser. Mat. 32, 212–244; 251–524; 709–731 (1968)Google Scholar
  99. 256.
    A.Yu. Olshanskii, On the problem of a finite basis of identities in groups. Izv. Akad. Nauk SSSR, Ser. Mat. 34, 376–384 (1970)Google Scholar
  100. 257.
    A.Yu. Olshanskii, On the question of the existence of an invariant mean on a group. Uspekhi Mat. Nauk 35 no. 4(214), 199–200 (1980)Google Scholar
  101. 258.
    A.Yu. Olshanskii, The Novikov-Adyan theorem. Mat. Sb. (N.S.) 118(160)(2), 203–235 (1982)Google Scholar
  102. 259.
    A.Yu. Olshanskii, Varieties in which all finite groups are abelian. Mat. Sb. (N.S.) 126(168)(1), 59–82 (1985)Google Scholar
  103. 260.
    A.Yu. Olshanskii, The Geometry of Defining Relations in Groups. Nauka, Moscow, 1989 (trans. from the 1989 Russian original by Yu. A. Bakhturin). Mathematics and Its Applications (Soviet Series), vol. 70 (Kluwer, Dordrecht, 1991)Google Scholar
  104. 261.
    A.Yu. Olshanskii, Hyperbolicity of groups with subquadratic isoperimetric inequalities. Int. J. Algebra Comput. 1, 282–290 (1991)MathSciNetGoogle Scholar
  105. 262.
    A.Yu. Olshanskii, D.V. Osin, Large groups and their periodic quotients. Proc. Am. Math. Soc. 136(3), 753–759 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  106. 263.
    A. Olshanskii, D. Osin, M. Sapir, Lacunary hyperbolic groups (with appendix by M. Kapovich and B. Kleiner). Geom. Topol. 13, 2051–2140 (2009)Google Scholar
  107. 264.
    A. Olshanskii, M. Sapir, Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci. 96(2002), 43–169 (2003)CrossRefGoogle Scholar
  108. 265.
    D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems. Ann. Math. (2) 172(1), 1–39 (2010)Google Scholar
  109. 273.
    P.F. Reichmeider, The Equivalence of Some Combinatorial Matching Theorems (Polygonal Publishing House, Washington, 1984)zbMATHGoogle Scholar
  110. 276.
    I.N. Sanov, Solution of Burnside’s problem for exponent 4. Leningr. State Univ. Ann. [Uchenye Zapiski] Math. Ser. 10, 166–170 (1940)MathSciNetGoogle Scholar
  111. 281.
    M. Sapir, Asymptotic invariants, complexity of groups and related problems. Bull. Math. Sci. 1(2), 277–364 (1993)CrossRefMathSciNetGoogle Scholar
  112. 282.
    M. Sapir, Some group theory problems. Int. J. Algebra Comput. 17(5–6), 1189–1214 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  113. 287.
    A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency. Vol. A. Paths, Flows, Matchings. Chapters 1 38. Algorithms and Combinatorics, vol. 24A (Springer, Berlin, 2003)Google Scholar
  114. 288.
    P. Schupp, On Dehn’s algorithm and the conjugacy problem. Math. Ann. 178, 119–130 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  115. 289.
    J.-P. Serre, Trees (Springer, Berlin/New York, 1980)CrossRefzbMATHGoogle Scholar
  116. 307.
    C.C. Squier, Word problems and a homological finiteness conditions for monoids. J. Pure Appl. Algebra 49, 201–217 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  117. 308.
    R.P. Stanley, Enumerative Combinatorics (Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin). Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999)Google Scholar
  118. 309.
    V.I. Sushchansky, Periodic p-groups of permutations and the unrestricted Burnside problem. Dokl. Akad. Nauk SSSR 247(3), 557–561 (1979)MathSciNetGoogle Scholar
  119. 311.
    S. Świerczkowski, On a free group of rotations of the Euclidean space. Indag. Math. 20, 376–378 (1958)Google Scholar
  120. 312.
    R. Szwarc, A short proof of the Grigorchuk-Cohen cogrowth theorem. Proc. Am. Math. Soc. 106(3), 663–665 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  121. 313.
    J. Tits, Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  122. 315.
    W.P. Thurston, Conway’s tiling groups. Am. Math. Mon. 97(8), 757–773 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  123. 322.
    L. van den Dries, A.J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89(2), 349–374 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  124. 323.
    M.R. Vaughan-Lee, Uncountably many varieties of groups. Bull. Lond. Math. Soc. 2, 280–286 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  125. 324.
    M.R. Vaughan-Lee, The Restricted Burnside Problem, 2nd edn. Volume 8 of London Mathematical Society Monographs. New Series. (The Clarendon Press/Oxford University Press, Oxford/New York, 1993)Google Scholar
  126. 328.
    J. von Neumann, Zur allgemeinen Theorie des Masses. Fund. Math. 13, 73–116 (1929); Collected Works, vol. I (Pergamon Press, New York/Oxford/London/Paris, 1961), pp. 599–643Google Scholar
  127. 329.
    S. Wagon, The Banach–Tarski Paradox. Corrected reprint of the 1985 original (Cambridge University Press, Cambridge, 1993)Google Scholar
  128. 330.
    D.T. Wise, Cubulating small cancellation groups. Geom. Funct. Anal. 14(1), 150–214 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  129. 331.
    W. Woess, Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41(4), 363–370 (1983)Google Scholar
  130. 332.
    J.A. Wolf, Growth of finitely generated solvable groups and curvature of riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968)zbMATHGoogle Scholar
  131. 333.
    E.I. Zel’manov, The solution of the restricted Burnside problem for groups of odd exponent. Izv. Akad. Nauk. SSSR. Ser. Mat. 54(1), 42–59 (1990); trans. in Math. USSR-Izv. 36(1), 41–60 (1991)Google Scholar
  132. 334.
    E.I. Zel’manov, The solution of the restricted Burnside problem for 2-groups. Mat. Sb. (N.S.), 182(4), 568–592 (1991)Google Scholar
  133. 335.
    E.I. Zel’manov, On additional laws in the Burnside problem for periodic groups. Int. J. Algebra Comput. 3(4), 583–600 (1993)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mark V. Sapir
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations