Skip to main content
SpringerLink
Log in

Solved and Unsolved Problems Around One Group

  • Chapter
Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Part of the book series: Progress in Mathematics ((PM,volume 248))

Abstract

This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ.

Among the topics touched in the paper are: actions on rooted trees and groups generated by finite automata, self-similarity, contracting properties, various notions of length, topology in the space of finitely generated groups, Kolmogorov complexity in relation to algorithmic problems, algorithmic problems such as word, conjugacy, and isomorphism problem, L-presentations of self-similar groups by generators and relators, branch groups and their subgroup structure, maximal and weakly maximal subgroups and the congruence subgroup property for groups acting on rooted trees, groups of finite type, intermediate growth and growth functions, amenability, Schreier graphs associated to groups acting on rooted trees and their asymptotic characteristics, C*-algebras associated to self-similar groups and spectral properties of Markov and Hecke type operators, multidimensional rational maps which arise in the study of the spectral problem, etc.

The paper contains many open problems, some of which have long history but some are completely new.

The author was supported by NSF grants DMS-0308985 and DMS-0456185 and by Swiss National Foundation for Scientific Research.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. I. Adian. The Burnside problem and identities in groups, volume 95 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  2. S. I. Adian. Random walks on free periodic groups. Izv. Akad. Nauk SSSR Ser. Mat., 46(6):1139–1149, 1343, 1982.

    MathSciNet  Google Scholar 

  3. W. Aitken, F. Hajir, and Chr. Maire. Finitely ramified iterated extensions.

    Google Scholar 

  4. S. V. Alešin. Finite automata and the Burnside problem for periodic groups. Mat. Zametki, 11:319–328, 1972.

    MathSciNet  Google Scholar 

  5. M. F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), pages 43–72. Astérisque, No. 32–33. Soc. Math. France, Paris, 1976.

    MathSciNet  Google Scholar 

  6. M. Abert and B. Virag. Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc., 18(1):157–192, 2005. (available at http://www.arxiv.org/abs/math.GR/0212191).

    MathSciNet  Google Scholar 

  7. G. M. Adel’son-Vel’skii and Yu. A. Šreider. The Banach mean on groups. Uspehi Mat. Nauk (N.S.), 12(6 (78)):131–136, 1957.

    MathSciNet  Google Scholar 

  8. L. Bartholdi. The growth of Grigorchuk’s torsion group. Internat. Math. Res. Notices, (20):1049–1054, 1998.

    MATH  MathSciNet  Google Scholar 

  9. L. Bartholdi. Lower bounds on the growth of a group acting on the binary rooted tree. Internat. J. Algebra Comput., 11(1):73–88, 2001.

    MATH  MathSciNet  Google Scholar 

  10. L. Bartholdi. Endomorphic presentations of branch groups. J. Algebra, 268(2):419–443, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Bartholdi. Lie algebras and growth in branch groups. Pacific J. Math., 218(2):241–282, 2005.

    MathSciNet  Google Scholar 

  12. G. Baumslag. Topics in combinatorial group theory. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

    Google Scholar 

  13. M. Bekka, M. Cowling, and P. de la Harpe. Some groups whose reduced C*-algebra is simple. Inst. Hautes Études Sci. Publ. Math., (80):117–134 (1995), 1994.

    Google Scholar 

  14. I. Bondarenko, T. Ceccherini-Silberstein, and V. Nekrashevych. Amenable graphs with dense non-cocompact pseudogroup of holonomy. In preparation.

    Google Scholar 

  15. M. Bachir Bekka and P. de la Harpe. Irreducibility of unitary group representations and reproducing kernels Hilbert spaces. Expo. Math., 21(2):115–149, 2003. Appendix by the authors in collaboration with Rostislav Grigorchuk.

    MathSciNet  Google Scholar 

  16. L. Bartholdi and R. I. Grigorchuk. On the spectrum of Hecke type operators related to some fractal groups. Tr. Mat. Inst. Steklova, 231 (Din. Sist., Avtom. i Beskon. Gruppy):5–45, 2000.

    MathSciNet  Google Scholar 

  17. L. Bartholdi and R. I. Grigorchuk. Spectra of non-commutative dynamical systems and graphs related to fractal groups. C. R. Acad. Sci. Paris Sér. I Math., 331(6):429–434, 2000.

    MathSciNet  Google Scholar 

  18. L. Bartholdi and R. I. Grigorchuk. On parabolic subgroups and Hecke algebras of some fractal groups. Serdica Math. J., 28(1):47–90, 2002.

    MathSciNet  Google Scholar 

  19. M. R. Bridson and R. H. Gilman. Context-free languages of sub-exponential growth. J. Comput. System Sci., 64(2):308–310, 2002.

    Article  MathSciNet  Google Scholar 

  20. L. Bartholdi, R. Grigorchuk, and V. Nekrashevych. From fractal groups to fractal sets. In Fractals in Graz 2001, Trends Math., pages 25–118. Birkhäuser, Basel, 2003.

    Google Scholar 

  21. L. Bartholdi, R. I. Grigorchuk, and Z. Šunik. Branch groups. In Handbook of algebra, Vol. 3, pages 989–1112. North-Holland, Amsterdam, 2003.

    Google Scholar 

  22. L. Bartholdi, V. Kaimanovich, V. Nekrashevych, and B Virag. Amenability of groups generated by bounded automata. In preparation.

    Google Scholar 

  23. M. B. Bekka and N. Louvet. Some properties of C*-algebras associated to discrete linear groups. In C*-algebras (Münster, 1999), pages 1–22. Springer, Berlin, 2000.

    Google Scholar 

  24. H. Bass, J. Milnor, and J.-P. Serre. Solution of the congruence subgroup problem for SLn (n≥3) and Sp2n (n≥2). Inst. Hautes Études Sci. Publ. Math., (33):59–137, 1967.

    MathSciNet  Google Scholar 

  25. H. Bass, M. V. Otero-Espinar, D. Rockmore, and C. Tresser. Cyclic renormalization and automorphism groups of rooted trees, volume 1621 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996.

    Google Scholar 

  26. N. Boston. p-adic Galois representations and pro-p Galois groups. In New horizons in pro-p groups, volume 184 of Progr. Math., pages 329–348. Birkhäuser Boston, Boston, MA, 2000.

    Google Scholar 

  27. G. Baumslag and J. E. Roseblade. Subgroups of direct products of free groups. J. London Math. Soc. (2), 30(1):44–52, 1984.

    MathSciNet  Google Scholar 

  28. L. Bartholdi and Z. Šunik. On the word and period growth of some groups of tree automorphisms. Comm. Algebra, 29(11):4923–4964, 2001.

    Article  MathSciNet  Google Scholar 

  29. L. Bartholdi and B. Virag. Amenability via random walks. (available at http://arxiv.org/abs/math.GR/0305262), 2003.

    Google Scholar 

  30. M. R. Bridson and D. T. Wise. VH complexes, towers and subgroups of F × F. Math. Proc. Cambridge Philos. Soc., 126(3):481–497, 1999.

    Article  MathSciNet  Google Scholar 

  31. C. Champetier and V. Guirardel. Monoïdes libres dans les groupes hyperboliques. In Séminaire de Théorie Spectrale et Géométrie, Vol. 18, Année 1999–2000, volume 18 of Sémin. Théor. Spectr. Géom., pages 157–170. Univ. Grenoble I, Saint, 2000.

    MathSciNet  Google Scholar 

  32. C. Champetier. Properiétés génériques des groupes de type fini. These de doctorat, Université de Lyon I, 1991.

    Google Scholar 

  33. C. Champetier. L’espace des groupes de type fini. Topology, 39(4):657–680, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  34. Ching Chou. Elementary amenable groups. Illinois J. Math., 24(3):396–407, 1980.

    MATH  MathSciNet  Google Scholar 

  35. Fan R. K. Chung. Spectral graph theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.

    Google Scholar 

  36. B. Chandler and W. Magnus. The history of combinatorial group theory, volume 9 of Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 1982.

    Google Scholar 

  37. T. Ceccherini-Silberstein, R. I. Grigorchuk, and P. de la Harpe. Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova, 224 (Algebra. Topol. Differ. Uravn. i ikh Prilozh.):68–111, 1999.

    Google Scholar 

  38. T. Ceccherini-Silberstein, Yu. Leonov, F. Scarabotti, and F. Tolli. Generalized Kaloujnine groups, universality and height of automorphisms. Internat. J. Algebra Comput., 15(3):503–527, 2004.

    MathSciNet  Google Scholar 

  39. T. Ceccherini-Silberstein, A. Machì, and F. Scarabotti. The Grigorchuk group of intermediate growth. Rend. Circ. Mat. Palermo (2), 50(1):67–102, 2001.

    MathSciNet  Google Scholar 

  40. T. Ceccherini-Silberstein, F. Scarabotti, and Tolli F. Finite Gelfand pairs and their applications to probability and statistics. To appear.

    Google Scholar 

  41. T. Ceccherini-Silberstein, F. Scarabotti, and Tolli F. Trees, wreath products and finite Gelfand pairs. To appear.

    Google Scholar 

  42. M. M. Day. Amenable semigroups. Illinois J. Math., 1:509–544, 1957.

    MATH  MathSciNet  Google Scholar 

  43. P. Diaconis. Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA, 1988.

    Google Scholar 

  44. P. de la Harpe. Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000.

    Google Scholar 

  45. P. de la Harpe. Mesures finiment additives et paradoxes. Panoramas et Synthesis, 18:39–61, 2004.

    MATH  Google Scholar 

  46. S. Eilenberg. Automata, languages, and machines. Vol. A. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1974.

    Google Scholar 

  47. A. Erschler. Piecewise automatic groups. to appear.

    Google Scholar 

  48. A. Erschler. On isoperimetric profiles of finitely generated groups. Geom. Dedicata, 100:157–171, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  49. A. Erschler. Boundary behavior for groups of subexponential growth. Annals of Math., 160(3):1183–1210, 2004.

    MathSciNet  Google Scholar 

  50. A. Erschler. Not residually finite groups of intermediate growth, commensurability and non-geometricity. J. Algebra, 272(1):154–172, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  51. E. Følner. Note on groups with and without full Banach mean value. Math. Scand., 5:5–11, 1957.

    MATH  MathSciNet  Google Scholar 

  52. R. Grigorchuk and P. de la Harpe. On problems related to growth, entropy, and spectrum in group theory. J. Dynam. Control Systems, 3(1):51–89, 1997.

    MathSciNet  Google Scholar 

  53. R. I. Grigorchuk and P. F. Kurchanov. Some questions of group theory related to geometry. In Algebra, VII, volume 58 of Encyclopaedia Math. Sci., pages 167–232, 233–240. Springer, Berlin, 1993.

    Google Scholar 

  54. R. I. Grigorchuk and I. Lysenok. Burnside problem. In The Consice handbook of algebra, A.V. Mikhalev and Gunter F. Pilz (eds.), pages 111–115. Kluwer Academic Publishers, 2002.

    Google Scholar 

  55. R. I. Grigorchuk, P. Linnell, T. Schick, and A. Żuk. On a question of Atiyah. C. R. Acad. Sci. Paris Sér. I Math., 331(9):663–668, 2000.

    Google Scholar 

  56. R. I. Grigorchuk and A. Machì. An example of an indexed language of intermediate growth. Theoret. Comput. Sci., 215(1–2):325–327, 1999.

    MathSciNet  Google Scholar 

  57. Y. Glasner and S. Mozes. Automata and square complexes. (available at http://arxiv.org/abs/math.GR/0306259), 2003.

    Google Scholar 

  58. R. I. Grigorchuk and T. Nagnibeda. Complete growth functions of hyperbolic groups. Invent. Math., 130(1):159–188, 1997.

    Article  MathSciNet  Google Scholar 

  59. R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii. Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):134–214, 2000.

    Google Scholar 

  60. F. Gecseg and I. Peák. Algebraic theory of automata. Akadémiai Kiadó, Budapest, 1972.

    Google Scholar 

  61. F. P. Greenleaf. Invariant means on topological groups and their applications. Van Nostrand Mathematical Studies, No. 16. Van Nostrand Reinhold Co., New York, 1969.

    Google Scholar 

  62. R. I. Grigorchuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53–54, 1980.

    MATH  MathSciNet  Google Scholar 

  63. R. I. Grigorchuk. On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR, 271(1):30–33, 1983.

    MATH  MathSciNet  Google Scholar 

  64. 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.

    MathSciNet  Google Scholar 

  65. R.I. Grigorchuk. The construction of p-groups of intermediate growth having continuum factor groups. Algebra i Logika, 23(4):383–394, 1984.

    MATH  MathSciNet  Google Scholar 

  66. R. I. Grigorchuk. A relationship between algorithmic problems and entropy characteristics of groups. Dokl. Akad. Nauk SSSR, 284(1):24–29, 1985.

    MATH  MathSciNet  Google Scholar 

  67. R. I. Grigorchuk. Semigroups with cancellations of degree growth. Mat. Zametki, 43(3):305–319, 428, 1988.

    MATH  MathSciNet  Google Scholar 

  68. R. I. Grigorchuk. Growth and amenability of a semigroup and its group of quotients. In Proceedings of the International Symposium on the Semigroup Theory and its Related Fields (Kyoto, 1990), pages 103–108, Matsue, 1990. Shimane Univ.

    Google Scholar 

  69. R. I. Grigorchuk. Weight functions on groups, and criteria for the amenability of Beurling algebras. Mat. Zametki, 60(3):370–382, 479, 1996.

    MATH  MathSciNet  Google Scholar 

  70. R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb., 189(1):79–100, 1998.

    MATH  MathSciNet  Google Scholar 

  71. R. I. Grigorchuk. On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata. In Groups St. Andrews 1997 in Bath, I, volume 260 of London Math. Soc. Lecture Note Ser., pages 290–317. Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  72. R. I. Grigorchuk. Branch groups. Mat. Zametki, 67(6):852–858, 2000.

    MATH  MathSciNet  Google Scholar 

  73. R. I. Grigorchuk. Just infinite branch groups. In New horizons in pro-p groups, volume 184 of Progr. Math., pages 121–179. Birkhäuser Boston, Boston, MA, 2000.

    Google Scholar 

  74. M. Gromov. Groups of polynomial growth and expanding maps. Publ. Math. IHES, 53:53–78, 1981.

    MATH  MathSciNet  Google Scholar 

  75. M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75–263. Springer, New York, 1987.

    Google Scholar 

  76. N. Gupta and S. Sidki. Some infinite p-groups. Algebra i Logika, 22(5):584–589, 1983.

    MathSciNet  Google Scholar 

  77. R. Grigorchuk and Z. Sunik. Featured review on the paper “Non-amenable finitely presented torsion-by-cyclic groups” by Olshanskii, A. and Sapir, M. Publ. Math. Inst. Hautes E’tudes Sci. no. 96 (2002), 43–169 (2003)., 2002.

    Google Scholar 

  78. R. I. Grigorchuk and S. N. Sidki. The group of automorphisms of a 3-generated 2-group of intermediate growth. Internat. J. Algebra Comput., 14(5–6):667–676, 2004.

    MathSciNet  Google Scholar 

  79. N. Gupta. On groups in which every element has finite order. Amer. Math. Monthly, 96(4):297–308, 1989.

    MATH  MathSciNet  Google Scholar 

  80. R. I. Grigorchuk and J. S. Wilson. A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. (2), 68(3):671–682, 2003.

    MathSciNet  Google Scholar 

  81. R. I. Grigorchuk and J. S. Wilson. The uniqueness of the actions of certain branch groups on rooted trees. Geom. Dedicata, 100:103–116, 2003.

    Article  MathSciNet  Google Scholar 

  82. Max Garzon and Yechezkel Zalcstein. The complexity of Grigorchuk groups with application to cryptography. Theoret. Comput. Sci., 88(1):83–98, 1991.

    Article  MathSciNet  Google Scholar 

  83. R. I. Grigorchuk and A. Żuk. On the asymptotic spectrum of random walks on infinite families of graphs. In Random walks and discrete potential theory (Cortona, 1997), Sympos. Math., XXXIX, pages 188–204. Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  84. R. I. Grigorchuk and A. Żuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata, 87(1–3):209–244, 2001.

    MathSciNet  Google Scholar 

  85. R. I. Grigorchuk and A. Żuk. On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput., 12(1–2):223–246, 2002.

    MathSciNet  Google Scholar 

  86. R. I. Grigorchuk and A. Żuk. Spectral properties of a torsion-free weakly branch group defined by a three state automaton. In Computational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001), volume 298 of Contemp. Math., pages 57–82. Amer. Math. Soc., Providence, RI, 2002.

    Google Scholar 

  87. R. I. Grigorchuk and A. Zuk. Elementary classes and amenability. In Proceedings of The Congress of Ukrainian Mathematicians, Functional Analysis, pages 59–84, 2003.

    Google Scholar 

  88. R. I. Grigorchuk and A. Żuk. The Ihara zeta function of infinite graphs, the KNS spectral measure and integrablemaps. In Random walks and geometry, pages 141–180. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.

    Google Scholar 

  89. J. Hořeiš. Transformations defined by finite automata (in Russian). Problemy Kibernetiki, 9:23–26, 1963.

    Google Scholar 

  90. D. F. Holt and C. E. Röver. On real-time word problems. J. London Math. Soc. (2), 67(2):289–301, 2003.

    MathSciNet  Google Scholar 

  91. J. E. Hopcroft and J. D. Ullman. Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading, Mass., 1979.

    Google Scholar 

  92. R. Incitti. The growth function of context-free languages. Theoret. Comput. Sci., 255(1–2):601–605, 2001.

    MATH  MathSciNet  Google Scholar 

  93. V. Kaimanovich. “Münchhausen trick” and amenability of self-similar groups. To appear in IJAC (2005), special issue “International Conference on Infinite Groups, Gaeta, 2003”.

    Google Scholar 

  94. L. Kaloujnine. Sur les p-groupes de Sylow du groupe symétrique du degré pm. C. R. Acad. Sci. Paris, 221:222–224, 1945.

    MATH  MathSciNet  Google Scholar 

  95. L. Kaloujnine. La structure des p-groupes de Sylow des groupes symétriques finis. Ann. Sci. École Norm. Sup. (3), 65:239–276, 1948.

    MATH  MathSciNet  Google Scholar 

  96. I. Kaplansky. “Problems in the theory of rings” revisited. Amer. Math. Monthly, 77:445–454, 1970.

    MATH  MathSciNet  Google Scholar 

  97. A. S. Kechris. Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

    Google Scholar 

  98. H. Kesten. Symmetric random walks on groups. Trans. Amer. Math. Soc., 92:336–354, 1959.

    MATH  MathSciNet  Google Scholar 

  99. B. P. Kitchens. Symbolic dynamics. Universitext. Springer-Verlag, Berlin, 1998.

    Google Scholar 

  100. O. Kharlampovich and A. Myasnikov. Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz. J. Algebra, 200(2):472–516, 1998.

    MathSciNet  Google Scholar 

  101. The Kourovka notebook. Rossiiskaya Akademiya Nauk Sibirskoe Otdelenie Institut Matematiki, Novosibirsk, 2002.

    Google Scholar 

  102. I. Kapovich and P. Schupp. Delzant’s t-invariant, one-relator groups and Kolmogorov complexity. (available at http://arxiv.org/abs/math.GR/0305353), 2005.

    Google Scholar 

  103. Yu. G. Leonov. The conjugacy problem in a class of 2-groups. Mat. Zametki, 64(4):573–583, 1998.

    MATH  MathSciNet  Google Scholar 

  104. Yu. G. Leonov. On a lower bound for the growth of a 3-generator 2-group. Mat. Sb., 192(11):77–92, 2001.

    MATH  MathSciNet  Google Scholar 

  105. A. I. Lihtman. On group rings of p-groups. Izv. Akad. Nauk SSSR Ser. Mat., 27:795–800, 1963.

    MATH  MathSciNet  Google Scholar 

  106. D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  107. Ya. Lavreniuk and V. Nekrashevych. Rigidity of branch groups acting on rooted trees. Geom. Dedicata, 89:159–179, 2002.

    Article  MathSciNet  Google Scholar 

  108. Yu. Leonov, V. Nekrashevych, and V. Sushchansky. Kaloujnine method, 2005. Preprint.

    Google Scholar 

  109. A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.

    Article  MathSciNet  Google Scholar 

  110. R. C. Lyndon and P. E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

    Google Scholar 

  111. A. Lubotzky and D. Segal. Subgroup growth, volume 212 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2003.

    Google Scholar 

  112. A. Lubotzky. Discrete groups, expanding graphs and invariant measures, volume 125 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1994.

    Google Scholar 

  113. W. Lück. L2-invariants: theory and applications to geometry and K-theory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2002.

    Google Scholar 

  114. Ming Li and P. Vitányi. An introduction to Kolmogorov complexity and its applications. Graduate Texts in Computer Science. Springer-Verlag, New York, second edition, 1997.

    Google Scholar 

  115. I. G. Lysënok. A set of defining relations for the Grigorchuk group. Mat. Zametki, 38(4):503–516, 634, 1985.

    MATH  MathSciNet  Google Scholar 

  116. J. L. Mennicke. Finite factor groups of the unimodular group. Ann. of Math. (2), 81:31–37, 1965.

    MATH  MathSciNet  Google Scholar 

  117. J. Milnor. Problem 5603. Amer. Math. Monthly, 75:685–686, 1968.

    MathSciNet  Google Scholar 

  118. Roman Muchnik and Igor Pak. On growth of Grigorchuk groups. Internat. J. Algebra Comput., 11(1):1–17, 2001.

    MathSciNet  Google Scholar 

  119. A. Nabutovsky. Fundamental group and contractible closed geodesics. Comm. Pure Appl. Math., 49(12):1257–1270, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  120. V. V. Nekrashevich. Iterated monodromy groups. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (4):18–20, 2003.

    MathSciNet  Google Scholar 

  121. V. V. Nekrashevych. Cuntz-Pimsner algebras of group actions. J. Operator Theory, 52(2):223–249, 2004.

    MathSciNet  Google Scholar 

  122. V. V. Nekrashevych. Self-similar groups. 2005.

    Google Scholar 

  123. A. Ju. Ol’šsanskii. On the question of the existence of an invariant mean on a group. Uspekhi Mat. Nauk, 35(4 (214)):199–200, 1980.

    Google Scholar 

  124. D. S. Passman. The semiprimitivity of group algebras. In Methods in ring theory (Levico Terme, 1997), volume 198 of Lecture Notes in Pure and Appl. Math., pages 199–211. Dekker, New York, 1998.

    Google Scholar 

  125. A. L. T. Paterson. Amenability, volume 29 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1988.

    Google Scholar 

  126. E. L. Pervova. Everywhere dense subgroups of a group of tree automorphisms. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):356–367, 2000.

    MATH  MathSciNet  Google Scholar 

  127. E. L. Pervova. The congruence property of AT-groups. Algebra Logika, 41(5):553–567, 634, 2002.

    MATH  MathSciNet  Google Scholar 

  128. E. L. Pervova. Profinite completions of some groups acting on trees. (available at http://arxiv.org/abs/math.GR/0411192), 2004.

    Google Scholar 

  129. E. L. Pervova. Maximal subgroups of non locally finite fractal groups., 2005. To appear in Inter. J. of Alg. and Comput.

    Google Scholar 

  130. K. M. Pilgrim. Dessins d’enfants and Hubbard trees. Ann. Sci. École Norm. Sup. (4), 33(5):671–693, 2000.

    MATH  MathSciNet  Google Scholar 

  131. R. T. Powers. Simplicity of the C*-algebra associated with the free group on two generators. Duke Math. J., 42:151–156, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  132. Ch. Pittet and L. Saloff-Coste. A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples. (available at http://www.math.cornell.edu/lsc/surv.ps.gz), 2001.

    Google Scholar 

  133. G. E. Revesz. Introduction to Formal Languages. Dover Publications, Inc., 1991.

    Google Scholar 

  134. A. V. Rozhkov. Centralizers of elements in a group of tree automorphisms. Izv. Ross. Akad. Nauk Ser. Mat., 57(6):82–105, 1993.

    MATH  Google Scholar 

  135. A. V. Rozhkov. The conjugacy problem in an automorphism group of an infinite tree. Mat. Zametki, 64(4):592–597, 1998.

    MATH  MathSciNet  Google Scholar 

  136. I. I. Reznikov and V. I. Sushchanskii. Growth functions of two-state automata over a two-element alphabet. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (2):76–81, 2002.

    Google Scholar 

  137. I. I. Reznikov and V. I. Sushchanskii. Two-state Mealy automata of intermediate growth over a two-letter alphabet. Mat. Zametki, 72(1):102–117, 2002.

    MathSciNet  Google Scholar 

  138. L. Ribes and P. A. Zalesskii. On the profinite topology on a free group. Bull. London Math. Soc., 25(1):37–43, 1993.

    MathSciNet  Google Scholar 

  139. Z. Sela. Diophantine geometry over groups. I. Makanin-Razborov diagrams. Publ. Math. Inst. Hautes Études Sci., (93):31–105, 2001.

    MATH  MathSciNet  Google Scholar 

  140. J.-P. Serre. Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp. J. Amer. Math. Soc., 10(1):75–102, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  141. Y. Shalom. Rigidity of commensurators and irreducible lattices. Invent. Math., 141(1):1–54, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  142. S. Sidki. On a 2-generated infinite 3-group: subgroups and automorphisms. J. Algebra, 110(1):24–55, 1987.

    MATH  MathSciNet  Google Scholar 

  143. S. Sidki. On a 2-generated infinite 3-group: the presentation problem. J. Algebra, 110(1):13–23, 1987.

    MATH  MathSciNet  Google Scholar 

  144. S. Sidki. A primitive ring associated to a Burnside 3-group. J. London Math. Soc. (2), 55(1):55–64, 1997.

    MATH  MathSciNet  Google Scholar 

  145. S. Sidki. Regular trees and their automorphisms, volume 56 of Monografias de Matemática [Mathematical Monographs]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1998.

    Google Scholar 

  146. S. Sidki. Finite automata of polynomial growth do not generate a free group. Geom. Dedicata, 108:193–204, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  147. A. Salomaa and M. Soittola. Automata-theoretic aspects of formal power series. Springer-Verlag, New York, 1978.

    Google Scholar 

  148. Z. Sunik. Hausdorff dimension of branched automaton groups, 2005. Preprint.

    Google Scholar 

  149. S. Sidki and J. S. Wilson. Free subgroups of branch groups. Arch. Math. (Basel), 80(5):458–463, 2003.

    MathSciNet  Google Scholar 

  150. A. Terras. Fourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  151. E. Thoma. Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math., 6:190–196, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  152. A. Vershik. Amenability and approximation of infinite groups. Selecta Math. Soviet., 2(4):311–330, 1982.

    MATH  MathSciNet  Google Scholar 

  153. J. von Neumann. Zur allgemeinen Theorie des Masses. Fund. Math., 13:73–116 and 333, 1929. = Collected works, vol. I, pages 599–643.

    Google Scholar 

  154. T. Vovkivsky. Infinite torsion groups arising as generalizations of the second Grigorchuk group. In Algebra (Moscow, 1998), pages 357–377. de Gruyter, Berlin, 2000.

    Google Scholar 

  155. M. Vorobets and Ya. Vorobets. On a free group of transformations defined by an automaton, 2005. preprint.

    Google Scholar 

  156. S. Wagon. The Banach-Tarski paradox. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  157. E.I. Zel’manov. Solution of the restricted burnside problem for groups of odd exponent. Izv. Akad. Nauk SSSR, Ser. Mat., 54(1):42–59, 1990.

    Google Scholar 

  158. E.I. Zel’manov. Solution of the restricted burnside problem for 2-groups. Mat. Sb., 182(4):568–592, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Texas A&M University, College Station, TX, 77843-3368, USA

    Rostislav Grigorchuk

Authors
  1. Rostislav Grigorchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut de Mathématiques, EPFL, 1015, Lausanne, Switzerland

    Laurent Bartholdi

  2. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 108, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  3. Section de mathématiques, Université de Genève, 1211, Geneva 4, Switzerland

    Tatiana Smirnova-Nagnibeda

  4. Institut de Mathématiques, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Andrzej Zuk

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Grigorchuk, R. (2005). Solved and Unsolved Problems Around One Group. In: Bartholdi, L., Ceccherini-Silberstein, T., Smirnova-Nagnibeda, T., Zuk, A. (eds) Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol 248. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7447-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation