Skip to main content
SpringerLink
Log in

Growth of periodic Grigorchuk groups

  • Published:
Inventiones mathematicae Aims and scope

Abstract

On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth \(v_{G,S}(n)\) satisfies \(\lim _{n\rightarrow \infty }\log \log v_{G,S}(n)/\log n=\alpha _{0}\), where \(\alpha _{0}=\frac{\log 2}{\log \lambda _{0}}\approx 0.7674\), \(\lambda _{0}\) is the positive root of the polynomial \(X^{3}-X^{2}-2X-4\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

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

    MathSciNet  Google Scholar 

  2. Azencott, R.: Espaces de Poisson des groupes localement compacts. Lecture Notes in Mathematics, vol. 148. Springer, Berlin (1970)

  3. Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)

    Article  MathSciNet  Google Scholar 

  4. Barlow, M.T., Grigor’yan, A., Kumagai, T.: Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626, 135–157 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bartholdi, L.: Growth of groups and wreath products. In: Ceccherini-Silberstein, T., Salvatori, M., Sava-Huss, E. (eds.) Groups, Graphs and Random Walks. London Mathematical Society Lecture Note Series, pp. 1–76. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781316576571.003

  8. Bartholdi, L., Erschler, A.: Growth of permutational extensions. Invent. Math. 189(2), 431–455 (2012)

    Article  MathSciNet  Google Scholar 

  9. Bartholdi, L., Erschler, A.: Groups of given intermediate word growth. Ann. Inst. Fourier (Grenoble) 64(5), 2003–2036 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bartholdi, L., Erschler, A.: Poisson–Furstenberg boundary and growth of groups. Probab. Theory Relat. Fields 168(1–2), 347–372 (2017)

    Article  MathSciNet  Google Scholar 

  11. Bartholdi, L., Grigorchuk, R.I., Šuniḱ Z.: Branch Groups. Handbook of Algebra, Vol. 3, pp. 989–1112 (2003)

  12. Bendikov, A., Saloff-Coste, L.: Random walks on groups and discrete subordination. Math. Nachr. 285(5–6), 580–605 (2012)

    Article  MathSciNet  Google Scholar 

  13. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  14. Brieussel, J.: Croissance et moyennabilité de certains groupes d’automorphismes d’un arbre enraciné. thése de doctorat, université Diderot Paris 7 (2008)

  15. Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Stat. 23(2 suppl.), 245–287 (1987)

    MathSciNet  MATH  Google Scholar 

  16. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140(1–2), 277–317 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Coulhon, T.: Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141(2), 510–539 (1996)

    Article  MathSciNet  Google Scholar 

  18. Coulhon, T., Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9(2), 293–314 (1993)

    Article  MathSciNet  Google Scholar 

  19. Davies, E.B.: Explicit constants for Gaussian upper bounds on heat kernels. Am. J. Math. 109(2), 319–333 (1987)

    Article  MathSciNet  Google Scholar 

  20. Davies, E.B.: Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  21. de la Harpe, P.: Topics in Geometric Group Theory, Chicago Lectures in Mathematics. University of Chicago Press, Chicago (2000)

    Google Scholar 

  22. Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15(1), 181–232 (1999)

    Article  MathSciNet  Google Scholar 

  23. Derriennic, Y.: Quelques applications du théorème ergodique sous-additif. In: Conference on Random Walks (Kleebach, 1979) (French), pp. 183–2014 (1980)

  24. Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk. SSSR 137, 1042–1045 (1961)

    MathSciNet  Google Scholar 

  25. Erschler, A.: Boundary behavior for groups of subexponential growth. Ann. of Math. (2) 160(3), 1183–1210 (2004)

    Article  MathSciNet  Google Scholar 

  26. Erschler, A.: Critical constants for recurrence of random walks on \(G\)-spaces. Ann. Inst. Fourier (Grenoble) 55(2), 493–509 (2005)

    Article  MathSciNet  Google Scholar 

  27. Erschler, A.: Piecewise automatic groups. Duke Math. J. 134(3), 591–613 (2006)

    Article  MathSciNet  Google Scholar 

  28. Frisch, J., Hartman, Y., Tamuz, O., Vahidi Ferdowsi, P.: Choquet-Deny groups and the infinite conjugacy class property. Ann. Math. (2) 190(1), 307–320 (2019)

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

    Article  MathSciNet  Google Scholar 

  30. Grigorchuk, R.I.: 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 

  31. Grigorchuk, R.I.: Some problems of the dynamics of group actions on rooted trees. Tr. Mat. Inst. Steklova 273(Sovremennye Problemy Matematiki), 72–191 (2011)

    MathSciNet  Google Scholar 

  32. Grigorchuk, R.I.: Milnor’s problem on the growth of groups and its consequences. In: Frontiers in Complex Dynamics, pp. 705–773 (2014)

  33. Grigor’yan, A.: Introduction to analysis on graphs. University Lecture Series, 71. vol. 8, pp. viii+150. American Mathematical Society, Providence, RI (2018). ISBN: 978-1-4704-4397-9

  34. Grigor’yan, A.: Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10(2), 395–452 (1994)

    Article  MathSciNet  Google Scholar 

  35. Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)

    Article  MathSciNet  Google Scholar 

  36. Kaĭmanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3), 457–490 (1983)

    Article  MathSciNet  Google Scholar 

  37. Lawler, G.F., Sokal, A.D.: Bounds on the \(L^2\) spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309(2), 557–580 (1988)

    MATH  Google Scholar 

  38. Leonov, Y.G.: A lower bound for the growth function of periods in Grigorchuk groups. Mat. Stud. 8(2), 192–197 (1997). 237

    MathSciNet  MATH  Google Scholar 

  39. Lysionok, I.G.: A system of defining relations for the Grigorchuk group. Mat. Zametki 38, 503–511 (1985)

    MathSciNet  Google Scholar 

  40. Mann, A.: How Groups Grow, London Mathematical Society Lecture Note Series, vol. 395. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  41. Merzlyakov, Y.I.: Infinite finitely generated periodic groups. Dokl. Akad. Nauk. SSSR 268(4), 803–805 (1983)

    MathSciNet  MATH  Google Scholar 

  42. Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447–449 (1968)

    Article  MathSciNet  Google Scholar 

  43. Muchnik, R., Pak, I.: On growth of Grigorchuk groups. Int. J. Algebra Comput. 11(1), 1–17 (2001)

    Article  MathSciNet  Google Scholar 

  44. Nekrashevych, V.: Self-similar groups, Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)

    Book  Google Scholar 

  45. Schilling, R.L., Song, R., Vondraek, Z.: Bernstein Functions: Theory and applications, Second, De Gruyter Studies in Mathematics, vol. 37. Walter de Gruyter & Co., Berlin (2012)

  46. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  Google Scholar 

  47. Woess, W.: Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  48. Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differ. Geom. 2, 421–446 (1968)

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referee whose comments and suggestions improved the exposition of the paper. We thank Jérémie Brieussel for helpful comments on the preliminary version of the paper.

Author information

Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, CNRS, PSL Research University, 45 Rue d’Ulm, 75005, Paris, France

    Anna Erschler

  2. Department of Mathematics, UC San Diego, 9500 Giman Dr. La Jolla, San Diego, CA, 92093, USA

    Tianyi Zheng

Authors
  1. Anna Erschler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianyi Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianyi Zheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of the authors is supported by the ERC Grant GroIsRan. The first named author also thanks the support of the ANR Grant MALIN.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Erschler, A., Zheng, T. Growth of periodic Grigorchuk groups. Invent. math. 219, 1069–1155 (2020). https://doi.org/10.1007/s00222-019-00922-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-019-00922-0

Search

Navigation