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.
I try in my prints to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears.
M. C. Escher
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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
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
Bartholdi, L.: FR GAP package: computations with functionally recursive groups, Version 2.4.1 (2017). http://www.gap-system.org/Packages/fr.html
Bartholdi, L., Virág, B.: Amenability via random walks. Duke Math. J. 130(1), 39–56 (2005)
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
Brooks, L.: Curly fern leaf. Wikimedia Commons (2013). https://commons.wikimedia.org
Brunner, A.M., Sidki, S.: The generation of GL(n, Z) by finite state automata. Int. J. Alg. Comput. 8(1), 127–139 (1998)
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)
Day, M.M.: Amenable semigroups. Ill. J. Math. 1, 509–544 (1957)
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
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)
Escher, M.C.: Sketch of Alhambra Tiles (1934). https://www.pinterest.com/pin/418271884112983852/?lp=true
Escher, M.C.: Smaller and smaller, woodcut (1956). https://www.pinterest.co.uk/pin/329044316505529408/?lp=true
Fijakowski, A.J.: Fractal generated using a finite transformation. Wikimedia Commons (2005). https://commons.wikimedia.org
Grigorchuk, R. I.: On Burnside’s problem on periodic groups. Funct. Anal. Appl. 14(1), 41–43 (1980)
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)
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)
Grigorchuk, R.I.: Just infinite branch groups. In: New Horizons in Pro-p Groups, 121179. Progress in Mathematics, vol. 184. Birkhauser, Boston (2000)
Grigorchuk, R.I., Šunić, Z.: Self-similarity and branching in group theory. London Math. Soc. Lecture Note Ser. 339, 36–95 (2007)
Gupta, N., Sidki, S.: On the Burnside problem for periodic groups. Math. Z. 182, 385 (1983). 388. MR85g:20075
Kaimanovich, V.: “Münchhausen trick” and amenability of self-similar groups. Int. J. Algebra Comput. 15(5–6), 907–937 (2005)
Lysenok, I.G.: A system of defining relations for a Grigorchuk group. Math. Notes 38(4), 784–792 (1985)
Mohri, M.: Minimization algorithms for sequential transducers. Theor. Comput. Sci. 234, 177–201 (2000)
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
Nekrashevych, V.: Self-Similar Groups, pp. 9–23. American Mathematical Society, Providence (2005)
Spitznagel, E.L.: Selected Topics in Mathematics, p. 137. Holt, Rinehart and Winston, New York (1971). ISBN 0-03-084693-5
The GAP Group: GAP Groups, Algorithms, and Programming, Version 4.8.7 (2017). http://www.gap-system.org
Wikimedia Commons: Nautilus cutaway logarithmic spiral, https://commons.wikimedia.org/wiki/File:NautilusCutawayLogarithmicSpiral.jpg
xlibber: A close-up of some arabic tiles in Cartagena in Spain. Wikimedia Commons (2010). https://commons.wikimedia.org
Żuk, A.: Automata groups, Institut de Mathematiques Universite Paris 7. http://cms.dm.uba.ar/Members/gcorti/workgroup.GNC/notes.pdf
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bonanome, M.C., Dean, M.H., Putnam Dean, J. (2018). Self-Similar Groups. In: A Sampling of Remarkable Groups. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01978-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-01978-5_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-01976-1
Online ISBN: 978-3-030-01978-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)