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Self-Similar Groups

  • Marianna C. Bonanome
  • Margaret H. Dean
  • Judith Putnam Dean
Chapter
Part of the Compact Textbooks in Mathematics book series (CTM)

Abstract

Self-similar groups, or automata groups, consist of certain automorphisms of the infinite complete rooted binary tree. We describe them using several different concepts: computers are designed, portraits are drawn, and self-similar rules are written. Some well-known self-similar groups such as Grigorchuk’s group, the Adding Machine, and the Tower of Hanoi are explored.

References

  1. 1.
    127 rect: Cantor set, a fractal, presented in seven iterations, Wikimedia Commons, January 18 2007. https://commons.wikimedia.org/wiki/File:Cantor_set_in_seven_iterations.svg
  2. 3.
    ArEb: The Mandelbrot Set is a mathematical fractal defined by the recursive formula z = z 2 + c, where z and c are complex numbers. This image was calculated for 100,000 iterations using the freeware program Fractal Explorer 2.02. Window boundaries: − 2 < Re(c) < 0.5 and − 0.9375 < Im(c) < 0.9375, Wikimedia Commons, April 23, 2007. https://commons.wikimedia.org/wiki/File:Blue-Gold_Mandelbrot_Set.jpg
  3. 6.
    Bartholdi, L.: FR GAP package: computations with functionally recursive groups, Version 2.4.1 (2017). http://www.gap-system.org/Packages/fr.html
  4. 8.
    Bartholdi, L., Virág, B.: Amenability via random walks. Duke Math. J. 130(1), 39–56 (2005)MathSciNetCrossRefGoogle Scholar
  5. 15.
    Bondarenko, I., Grigorchuk, R., Kravchenko, R., Muntyan, Y., Nekrashevych, V., Savchuk, Z., Šunić, Z.: Classification of groups generated by 3-state automata over a 2-letter alphabet (2008). http://arxiv.org/abs/0803.3555v1
  6. 17.
    Brooks, L.: Curly fern leaf. Wikimedia Commons (2013). https://commons.wikimedia.org
  7. 19.
    Brunner, A.M., Sidki, S.: The generation of GL(n, Z) by finite state automata. Int. J. Alg. Comput. 8(1), 127–139 (1998)MathSciNetCrossRefGoogle Scholar
  8. 22.
    Caponi, L.: On the Classification of groups generated by automata with 4 states over a 2-letter alphabet, University of South Florida, M. A. thesis, advisor: D. Savchuk (2014)Google Scholar
  9. 27.
    Day, M.M.: Amenable semigroups. Ill. J. Math. 1, 509–544 (1957)MathSciNetzbMATHGoogle Scholar
  10. 30.
    Elder, M.: A short introduction to self-similar groups. Asia Pac. Math. Newsl. 3(1), (2013). http://www.asiapacific-mathnews.com/03/0301/0017_0021.pdf
  11. 31.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Published by Jones, Bartlett. CRC Press, Boca Raton (1992)Google Scholar
  12. 32.
    Escher, M.C.: Sketch of Alhambra Tiles (1934). https://www.pinterest.com/pin/418271884112983852/?lp=true
  13. 33.
    Escher, M.C.: Smaller and smaller, woodcut (1956). https://www.pinterest.co.uk/pin/329044316505529408/?lp=true
  14. 34.
    Fijakowski, A.J.: Fractal generated using a finite transformation. Wikimedia Commons (2005). https://commons.wikimedia.org
  15. 36.
    Grigorchuk, R. I.: On Burnside’s problem on periodic groups. Funct. Anal. Appl. 14(1), 41–43 (1980)MathSciNetCrossRefGoogle Scholar
  16. 37.
    Grigorchuk, R.I.: Degrees of growth of finitely generated groups and the theory of invariant means. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 48(5), 939–985 (1984)MathSciNetGoogle Scholar
  17. 38.
    Grigorchuk, R.I.: An example of a finitely presented amenable group that does not belong to the class. EG. Mat. Sb. 189(1), 79–100 (1998)MathSciNetCrossRefGoogle Scholar
  18. 39.
    Grigorchuk, R.I.: Just infinite branch groups. In: New Horizons in Pro-p Groups, 121179. Progress in Mathematics, vol. 184. Birkhauser, Boston (2000)CrossRefGoogle Scholar
  19. 40.
    Grigorchuk, R.I., Šunić, Z.: Self-similarity and branching in group theory. London Math. Soc. Lecture Note Ser. 339, 36–95 (2007)MathSciNetzbMATHGoogle Scholar
  20. 44.
    Gupta, N., Sidki, S.: On the Burnside problem for periodic groups. Math. Z. 182, 385 (1983). 388. MR85g:20075MathSciNetCrossRefGoogle Scholar
  21. 50.
    Kaimanovich, V.: “Münchhausen trick” and amenability of self-similar groups. Int. J. Algebra Comput. 15(5–6), 907–937 (2005)CrossRefGoogle Scholar
  22. 56.
    Lysenok, I.G.: A system of defining relations for a Grigorchuk group. Math. Notes 38(4), 784–792 (1985)MathSciNetCrossRefGoogle Scholar
  23. 62.
    Mohri, M.: Minimization algorithms for sequential transducers. Theor. Comput. Sci. 234, 177–201 (2000)MathSciNetCrossRefGoogle Scholar
  24. 65.
    Muntyan, Y., Savchuk, D.: AutomGrp GAP package for: computations in self-similar groups and semigroups. Version 1.3 (2016). http://www.gap-system.org/Packages/automgrp.html
  25. 66.
    Nekrashevych, V.: Self-Similar Groups, pp. 9–23. American Mathematical Society, Providence (2005)Google Scholar
  26. 74.
    Spitznagel, E.L.: Selected Topics in Mathematics, p. 137. Holt, Rinehart and Winston, New York (1971). ISBN 0-03-084693-5Google Scholar
  27. 77.
    The GAP Group: GAP Groups, Algorithms, and Programming, Version 4.8.7 (2017). http://www.gap-system.org
  28. 84.
    Wikimedia Commons: Nautilus cutaway logarithmic spiral, https://commons.wikimedia.org/wiki/File:NautilusCutawayLogarithmicSpiral.jpg
  29. 87.
    xlibber: A close-up of some arabic tiles in Cartagena in Spain. Wikimedia Commons (2010). https://commons.wikimedia.org
  30. 88.
    Żuk, A.: Automata groups, Institut de Mathematiques Universite Paris 7. http://cms.dm.uba.ar/Members/gcorti/workgroup.GNC/notes.pdf

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Marianna C. Bonanome
    • 1
  • Margaret H. Dean
    • 2
  • Judith Putnam Dean
    • 3
  1. 1.Department of Math and Computer ScienceNew York City College of Technology, The City University of New YorkBrooklynUSA
  2. 2.Department of MathematicsBorough of Manhattan Community College, The City University of New YorkNew YorkUSA
  3. 3.Department of MathematicsMonroe Community CollegeRochesterUSA

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