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Iterated monodromy groups

Published online by Cambridge University Press:  05 July 2011

By
Volodymyr Nekrashevych
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Volodymyr Nekrashevych
Affiliation:
Texas A&M University, USA
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

The paper is a survey of topics related to the theory of iterated monodromy groups and its applications. We also present a collection of examples illustrating different aspects of the theory.

Introduction

Iterated monodromy groups are algebraic invariants of topological dynamical systems (e.g., rational functions acting on the Riemann sphere). They encode in a computationally efficient way combinatorial information about the dynamical systems. In hyperbolic (expanding) case the iterated monodromy group contains all essential information about the dynamical system. For instance, the Julia set of the system can be reconstructed from the iterated monodromy group).

Besides their applications to dynamical systems (see, for instance [BN06] and [Nek08b]) iterated monodromy groups are interesting from the point of view of group theory, as they often possess exotic properties. In some sense their complicated structure is parallel to the complicated structure of the associated fractal Julia sets. In some cases the relation with the dynamical systems can be used to understand algebraic properties of the iterated monodromy groups.

Even though the main application of the iterated monodromy groups is dynamics, their origins are in algebra (however, some previous works in holomorphic dynamics contained constructions directly related to the iterated monodromy groups, see [HOV95, LM97, Pil00]). They were defined in 2001 in connection with the following construction due to R. Pink. Let F(x) be a rational function over ℂ.

Type
Chapter
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Groups St Andrews 2009 in Bath , pp. 41 - 93
Publisher: Cambridge University Press
Print publication year: 2011

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References

[AB88] Gert, Almkvist and Bruce Berndt, Gauss, Landen, Ramanujan, the arithmeticgeometric mean, ellipses, π, and the ladies diary, Amer. Math. Monthly 95 (1988), no. 7, 585–608.Google Scholar
[AHM05] Wayne, Aitken, Farshid, Hajir, and Christian, Maire, Finitely ramified iterated extensions, Int. Math. Res. Not. (2005), no. 14, 855–880.Google Scholar
[Ale29] P., Alexandroff, Über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension, Ann. of Math. (2) 30 (1928–1929), 101–187.Google Scholar
[Ale72] S. V., Aleshin, Finite automata and the Burnside problem for periodic groups, Mat. Zametki 11 (1972), 319–328, (in Russian).Google Scholar
[Bar03a] Laurent, Bartholdi, Endomorphic presentations of branch groups, J. Algebra 268 (2003), no. 2, 419–443.Google Scholar
[Bar03b] Laurent, Bartholdi, A Wilson group of non-uniformly exponential growth, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), no. 7, 549–554.Google Scholar
[BB98] Jonathan M., Borwein and Peter B., Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, 4, John Wiley & Sons Inc., New York, 1998, A study in analytic number theory and computational complexity, Reprint of the 1987 original, A Wiley-Interscience Publication.Google Scholar
[BG00] Laurent, Bartholdi and Rostislav I., Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Proc. Steklov Inst. Math. 231 (2000), 5–45.Google Scholar
[BGN03] Laurent, Bartholdi, Rostislav, Grigorchuk, and Volodymyr, Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001. Analysis – Dynamics – Geometry – Stochastics (Peter, Grabner and Wolfgang, Woess, eds.), Birkhäuser Verlag, Basel, Boston, Berlin, 2003, pp. 25–118.Google Scholar
[BGŠ03] Laurent, Bartholdi, Rostislav I., Grigorchuk, and Zoran, Šunik, Branch groups, Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989–1112.Google Scholar
[BH99] Martin R., Bridson and André, Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319, Springer, Berlin, 1999.Google Scholar
[BJ07] Nigel, Boston and Rafe, Jones, Arboreal Galois representations, Geom. Dedicata 124 (2007), 27–35.Google Scholar
[BK08] Jim, Belk and Sarah C., Koch, Iterated monodromy for a two-dimensional map, (preprint, to appear in Proceedings of the Ahlfors-Bers Colloquium), 2008.
[BKN08] Laurent, Bartholdi, Vadim, Kaimanovich, and Volodymyr, Nekrashevych, Amenability of automata groups, (preprint, arXiv:0802.2837v1), 2008.
[BN06] Laurent, Bartholdi and Volodymyr V., Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1–51.Google Scholar
[BN08] Laurent, Bartholdi and Volodymyr V., Nekrashevych, Iterated monodromy groups of quadratic polynomials I, Groups Geom. Dyn. 2 (2008), no. 3, 309–336.Google Scholar
[BORT96] Hyman, Bass, Maria Victoria, Otero-Espinar, Daniel, Rockmore, and Charles, Tresser, Cyclic renormalization and automorphism groups of rooted trees, Lecture Notes in Mathematics, vol. 1621, Springer-Verlag, Berlin, 1996.Google Scholar
[BP06] Kai-Uwe, Bux and Rodrigo, Pérez, On the growth of iterated monodromy groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 61–76.Google Scholar
[Bre93] Glen E., Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993.Google Scholar
[BSV99] Andrew M., Brunner, Said N., Sidki, and Ana. C., Vieira, A just-nonsolvable torsion-free group defined on the binary tree, J. Algebra 211 (1999), 99–144.Google Scholar
[BT24] Stefan, Banach and Alfred, Tarski, Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. Math. 6 (1924), 244–277.Google Scholar
[Bul91] Shaun, Bullett, Dynamics of the arithmetic-geometric mean, Topology 30 (1991), no. 2, 171–190.Google Scholar
[Bul92] Shaun, Bullett, Critically finite correspondences and subgroups of the modular group, Proc. London Math. Soc. (3) 65 (1992), no. 2, 423–448.Google Scholar
[BV05] Laurent, Bartholdi and Bálint, Virág, Amenability via random walks, Duke Math. J. 130 (2005), no. 1, 39–56.Google Scholar
[Che54] Pafnuty L., Chebyshev, Théorie des mécanismes connus sous le nom de parallélogrammes, Mémoires présentés à l'Académie Impériale des science de St-Pétersbourg par divers savant 7 (1854), 539–586.Google Scholar
[Cox84] David A., Cox, The arithmetic-geometric mean of Gauss, Enseign. Math. (2) 30 (1984), no. 3-4, 275–330.Google Scholar
[Day49] Mahlon M., Day, Means on semigroups and groups. The summer meeting in Boulder., Bull. Amer. Math. Soc. 55 (1949), no. 11, 1054.Google Scholar
[DH84] Adrien, Douady and John H., Hubbard, Étude dynamiques des polynômes complexes. (Première partie), Publications Mathematiques d'Orsay, vol. 02, Université de Paris-Sud, 1984.Google Scholar
[DH85] Adrien, Douady and John H., Hubbard, Étude dynamiques des polynômes complexes. (Deuxième partie), Publications Mathematiques d'Orsay, vol. 04, Université de Paris-Sud, 1985.Google Scholar
[Ers04] Anna, Erschler, Boundary behaviour for groups of subexponential growth, Ann. of Math. 160 (2004), 1183–1210.Google Scholar
[Eul48] Leonhardo, Eulero, Introductio in analysin infinitorum. Tomus primus., Apud Marcum-Michaelem Bousquet & Socios, Lausannæ, 1748.Google Scholar
[Eul88] Leonhard, Euler, Introduction to analysis of the infinite. Book I, Springer-Verlag, New York, 1988, Translated from the Latin and with an introduction by John D. Blanton.Google Scholar
[FG91] Jacek, Fabrykowski and Narain D., Gupta, On groups with sub-exponential growth functions. II, J. Indian Math. Soc. (N.S.) 56 (1991), no. 1-4, 217–228.Google Scholar
[For81] Otto, Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, New York – Heidelberg – Berlin: Springer-Verlag, 1981.Google Scholar
[Fra70] John M., Franks, Anosov diffeomorphisms, Global Analysis, Berkeley, 1968, Proc. Symp. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 61–93.
[FS92] J. E., Fornæss and N., Sibony, Critically finite rational maps on ℙ2, The Madison Symposium on Complex Analysis (Madison, WI, 1991), Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 245–260.Google Scholar
[Gau66] Carl, Friedrich Gauss, Werke, Dritter Band, Königliche Geselschaft der Wissenschaften zu Göttingen, 1866.Google Scholar
[Gel94] Götz, Gelbrich, Self-similar periodic tilings on the Heisenberg group, J. Lie Theory 4 (1994), no. 1, 31–37.Google Scholar
[GM05] Yair, Glasner and Shahar, Mozes, Automata and square complexes, Geom. Dedicata 111 (2005), 43–64.Google Scholar
[GNS00] Rostislav I., Grigorchuk, Volodymyr V., Nekrashevich, and Vitaliĭ I., Sushchanskii, Automata, dynamical systems and groups, Proc. Steklov Inst. Math. 231 (2000), 128–203.Google Scholar
[GNS01] Piotr W., Gawron, Volodymyr V., Nekrashevych, and Vitaly I., Sushchansky, Conjugation in tree automorphism groups, Internat. J. Algebra Comput. 11 (2001), no. 5, 529–547.Google Scholar
[Gre69] F. P., Greenleaf, Invariant means on topological groups, Van Nostrand Reinhold, New York, 1969.Google Scholar
[Gri80] Rostislav I., Grigorchuk, On Burnside's problem on periodic groups, Funct. Anal. Appl. 14 (1980), no. 1, 41–43.Google Scholar
[Gri83] Rostislav I., Grigorchuk, Milnor's problem on the growth of groups, Soviet Math. Dokl. 28 (1983), 23–26.Google Scholar
[Gri85] Rostislav I., Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR Izv. 25 (1985), no. 2, 259–300.Google Scholar
[Gri98] Rostislav I., Grigorchuk, An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1998), no. 1, 79–100.Google Scholar
[Gri05] R., Grigorchuk, Solved and unsolved problems aroud one group, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects (T., Smirnova-NagnibedaL., Bartholdi, T., Ceccherini-Silberstein and A., Żuk, eds.), Progress in Mathematics, vol. 248, Birkhäuser, 2005, pp. 117–218.Google Scholar
[Gro81a] Mikhael, Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73.Google Scholar
[Gro81b] Mikhael, Gromov, Structures métriques pour les variétés riemanniennes. Redige par J. LaFontaine et P., Pansu, Textes Mathematiques, vol. 1, Paris: Cedic/Fernand Nathan, 1981.Google Scholar
[Gro99] Mikhael, Gromov, Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J., LaFontaine and P., Pansu, Progress in Mathematics (Boston, Mass.), vol. 152, Boston, MA: Birkhäuser, 1999.Google Scholar
[GS83] Narain D., Gupta and Said N., Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385–388.Google Scholar
[GŠ07] Rostislav, Grigorchuk and Zoran, Šunić, Self-similarity and branching in group theory, Groups St. Andrews 2005. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 339, Cambridge Univ. Press, Cambridge, 2007, pp. 36–95.Google Scholar
[GŻ02a] Rostislav I., Grigorchuk and Andrzej, Żuk, On a torsion-free weakly branch group defined by a three state automaton, Internat. J. Algebra Comput. 12 (2002), no. 1, 223–246.Google Scholar
[GŻ02b] Rostislav I., Grigorchuk and Andrzej, Żuk, Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Computational and Statistical Group Theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemp. Math., vol. 298, Amer. Math. Soc., Providence, RI, 2002, pp. 57–82.Google Scholar
[Haï00] Peter, Haïssinsky, Modulation dans l'ensemble de Mandelbrot, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 37–65.Google Scholar
[Har00] Pierre de la, Harpe, Topics in geometric group theory, University of Chicago Press, 2000.Google Scholar
[Har02] Pierre de la, Harpe, Uniform growth in groups of exponential growth, Geom. Dedicata 95 (2002), 1–17.Google Scholar
[Hir70] Morris W., Hirsch, Expanding maps and transformation groups, Global Analysis, Proc. Sympos. Pure Math., vol. 14, American Math. Soc., Providence, RhodeIsland, 1970, pp. 125–131.Google Scholar
[HOV95] John H., Hubbard and Ralph W., Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, Real and complex dynamical systems (Hillerød, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 464, Kluwer Acad. Publ., Dordrecht, 1995, pp. 89–132.Google Scholar
[IS08] Yutaka, Ishii and John, Smillie, Homotopy shadowing, (preprint arXiv:0804.4629v1), 2008.
[Jul18] Gaston, Julia, Mémoire sur l'iteration des fonctions rationnelles, Journal de mathématiques pures et appliquées 4 (1918), 47–245.Google Scholar
[Kap08] M., Kapovich, Arithmetic aspects of self-similar groups, (preprint arXiv:0809.0323), 2008.
[Kat04] Takeshi, Katsura, A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. I. Fundamental results, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4287–4322 (electronic).Google Scholar
[Ken92] Richard, Kenyon, Self-replicating tilings, Symbolic Dynamics and Its Applications (P., Walters, ed.), Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–264.Google Scholar
[Knu69] Donald E., Knuth, The art of computer programming, Vol. 2, Seminumerical algorithms, Addison-Wesley Publishing Company, 1969.Google Scholar
[Lag85] Joseph-Louis, Lagrange, Sur une novelle méthode de calcul intégral pour les différentielles affectées d'un radical carré sous lequel la variable ne passe pas le quatrième degré, Mém. de l'Acad. Roy. Sci. Turin 2 (1784–85), (see Oeuvres, t.2, Gauthier–Villars, Paris, 1868, pp. 251–312).Google Scholar
[Leo98] Yu. G., Leonov, The conjugacy problem in a class of 2-groups, Mat. Zametki 64 (1998), no. 4, 573–583.Google Scholar
[LM97] Mikhail, Lyubich and Yair, Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, 17–94.Google Scholar
[LMU08] Igor G., Lysenok, Alexey G., Myasnikov, and Alexander, Ushakov, The conjugacy problem in the Grigorchuk group is polynomial time decidable, (preprint, arXiv:0808.2502v1), 2008.
[Man80] Benoit B., Mandelbrot, Fractal aspects of the iteration of z ↦ λz (1 – z) for complex λ and z, Nonlinear Dynamics (R.H.G., Helleman, ed.), Annals of the New York Academy of Sciences, vol. 357, 1980, pp. 249–259.Google Scholar
[Mar91] G. A., Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 17. Berlin etc.: Springer-Verlag, 1991.Google Scholar
[Mil99] John, Milnor, Dynamics in one complex variable. Introductory lectures, Wiesbaden: Vieweg, 1999.Google Scholar
[Nek05] Volodymyr, Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence, RI, 2005.Google Scholar
[Nek07a] Volodymyr, Nekrashevych, Free subgroups in groups acting on rooted trees, (preprint arXiv:0802.2554), 2007.
[Nek07b] Volodymyr, Nekrashevych, A minimal Cantor set in the space of 3-generated groups, Geom. Dedicata 124 (2007), no. 2, 153–190.Google Scholar
[Nek07c] Volodymyr, Nekrashevych, Self-similar groups and their geometry, São Paulo J. Math. Sci. 1 (2007), no. 1, 41–96.Google Scholar
[Nek08a] Volodymyr, Nekrashevych, Combinatorial models of expanding dynamical systems, (preprint arXiv:0810.4936), 2008.
[Nek08b] Volodymyr, Nekrashevych, The Julia set of a post-critically finite endomorphism of ℂℙ2, (preprint arXiv:0811.2777), 2008.
[Nek08c] Volodymyr, Nekrashevych, Symbolic dynamics and self-similar groups, Holomorphic dynamics and renormalization. A volume in honour of John Milnor's 75th birthday (Mikhail, Lyubich and Michael, Yampolsky, eds.), Fields Institute Communications, vol. 53, A.M.S., 2008, pp. 25–73.Google Scholar
[Nek09] Volodymyr, Nekrashevych, Combinatorics of polynomial iterations, Complex Dynamics – Families and Friends (D., Schleicher, ed.), A K Peters, 2009, pp. 169–214.Google Scholar
[Nek10] Volodymyr, Nekrashevych, A group of non-uniform exponential growth locally isomorphic toIMG (z2 + i), Trans. Amer. Math. Soc. 362 (2010), 389–398.Google Scholar
[NS04] Volodymyr, Nekrashevych and Said, Sidki, Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms, Groups: Topological, Combinatorial and Arithmetic Aspects (T. W., Müller, ed.), LMS Lecture Notes Series, vol. 311, 2004, pp. 375–404.Google Scholar
[Pen65] Walter, Penney, A “binary” system for complex numbers, J. ACM 12 (1965), no. 2, 247–248.Google Scholar
[Pil00] Kevin M., Pilgrim, Dessins d'enfants and Hubbard trees, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 671–693.Google Scholar
[PN08] Gábor, Pete and Volodymyr, Nekrashevych, Scale-invariant groups, (preprint arXiv:0811.0220), 2008.
[Roz98] A. V., Rozhkov, The conjugacy problem in an automorphism group of an infinite tree, Mat. Zametki 64 (1998), no. 4, 592–597.Google Scholar
[Run02] Volker, Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002.Google Scholar
[San47] I. N., Sanov, A property of a representation of a free group, Doklady Akad. Nauk SSSR (N. S.) 57 (1947), 657–659.Google Scholar
[Ser80] Jean-Pierre, Serre, Trees, New York: Springer-Verlag, 1980.Google Scholar
[Shu69] Michael, Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199.Google Scholar
[Shu70] Michael, Shub, Expanding maps, Global Analysis, Proc. Sympos. Pure Math., vol. 14, American Math. Soc., Providence, RhodeIsland, 1970, pp. 273–276.Google Scholar
[Sid98] Said N., Sidki, Regular trees and their automorphisms, Monografias de Matematica, vol. 56, IMPA, Rio de Janeiro, 1998.Google Scholar
[Ste51] Norman, Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951.Google Scholar
[Šun07] Zoran, Šunić, Hausdorff dimension in a family of self-similar groups, Geom. Dedicata 124 (2007), 213–236.Google Scholar
[UvN47] Stanisław M., Ulam and John von, Neumann, On combination of stochastic and deterministic processes. Preliminary report, Bull. Amer. Math. Soc. 53 (1947), no. 11, 1120.Google Scholar
[Vin95] Andrew, Vince, Rep-tiling Euclidean space, Aequationes Math. 50 (1995), 191–213.Google Scholar
[Vin00] Andrew, Vince, Digit tiling of Euclidean space, Directions in Mathematical Quasicrystals, Amer. Math. Soc., Providence, RI, 2000, pp. 329–370.Google Scholar
[vN29] John von, Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73–116 and 333, Collected works, vol. I, pages 599–643.Google Scholar
[VV07] Mariya, Vorobets and Yaroslav, Vorobets, On a free group of transformations defined by an automaton, Geom. Dedicata 124 (2007), no. 1, 237–249.Google Scholar
[Wag94] Stan, Wagon, The Banach–Tarski paradox, Cambridge University Press, 1994.Google Scholar
[Wil04a] John S., Wilson, Further groups that do not have uniformly exponential growth, J. Algebra 279 (2004), 292–301.Google Scholar
[Wil04b] John S., Wilson, On exponential growth and uniform exponential growth for groups, Invent. Math. 155 (2004), 287–303.Google Scholar

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