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A group of non-uniform exponential growth locally isomorphic to $ IMG(z^2+i)$

Author: Volodymyr Nekrashevych
Journal: Trans. Amer. Math. Soc. 362 (2010), 389-398
MSC (2000): Primary 20F69, 20E08; Secondary 37F20
Published electronically: July 17, 2009
MathSciNet review: 2550156
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Abstract: We prove that a sequence of marked three-generated groups isomorphic to the iterated monodromy group of $ z^2+i$ converges to a group of non-uniform exponential growth, which is an extension of the infinite direct sum of cyclic groups of order 4 by a Grigorchuk group.

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  • [Bar03] Laurent Bartholdi, A Wilson group of non-uniformly exponential growth, C. R. Acad. Sci. Paris. Sér. I Math. 336 (2003), no. 7, 549-554. MR 1981466 (2004c:20051)
  • [BGN03] Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych, From fractal groups to fractal sets, Fractals in Graz 2001. Analysis - Dynamics - Geometry - Stochastics (Peter Grabner and Wolfgang Woess, eds.), Birkhäuser Verlag, Basel, Boston, Berlin, 2003, pp. 25-118. MR 2091700 (2005h:20056)
  • [BN06] Laurent Bartholdi and Volodymyr V. Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1-51. MR 2285317 (2008c:37072)
  • [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. MR 2216706 (2006m:20062)
  • [Cha00] Christophe Champetier, L'espace de groupes de type fini, Topology 39 (2000), 657-680. MR 1760424 (2001i:20084)
  • [Ers04a] Anna Erschler, Boundary behaviour for groups of subexponential growth, Annals of Mathematics 160 (2004), 1183-1210. MR 2144977 (2006d:20072)
  • [Ers04b] -, Not residually finite groups of intermediate growth, commensurability and non-geometricity, Journal of Algebra 272 (2004), 154-172. MR 2029029 (2004j:20066)
  • [Ers05] -, On degrees of growth of finitely generated groups, Functional Analysis and Its Applications 39 (2005), no. 4, 317-320. MR 2197519 (2006k:20056)
  • [Ers07] -, Automatically presented groups, Groups, Geometry and Dynamics 1 (2007), no. 1, 47-60. MR 2294247 (2008c:20080)
  • [FS92] J. E. Fornæss and N. Sibony, Critically finite rational maps on $ \mathbb{P}^2$, The Madison Symposium on Complex Analysis (Madison, WI, 1991), Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 245-260. MR 1190986 (93j:32034)
  • [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. MR 764305 (86h:20041)
  • [Gri87] R. I. Grigorchuk, Superamenability and the occurrence problem of free semigroups, Funktsional. Anal. i Prilozhen. 21 (1987), no. 1, 74-75. MR 888020 (88k:43002)
  • [Gro81a] Mikhael Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53 (1981), 53-73. MR 623534 (83b:53041)
  • [Gro81b] -, Structures métriques pour les variétés riemanniennes. Redige par J. LaFontaine et P. Pansu, Textes Mathematiques, vol. 1, Paris: Cedic/Fernand Nathan, 1981. MR 682063 (85e:53051)
  • [Har02] Pierre de la Harpe, Uniform growth in groups of exponential growth, Geom. Dedicata 95 (2002), 1-17. MR 1950882 (2003k:20031)
  • [Nek05] Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, Amer. Math. Soc., Providence, RI, 2005. MR 2162164 (2006e:20047)
  • [Nek07] -, A minimal Cantor set in the space of $ 3$-generated groups, Geometriae Dedicata 124 (2007), no. 2, 153-190. MR 2318543 (2008d:20075)
  • [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. MR 2073355 (2005d:20043)
  • [PS72] G. Polya and G. Szegö, Problems and theorems in analysis, volume I, Springer, 1972. MR 0344042 (49:8782)
  • [Wil04a] John S. Wilson, Further groups that do not have uniformly exponential growth, Journal of Algebra 279 (2004), 292-301. MR 2078400 (2005e:20066)
  • [Wil04b] -, On exponential growth and uniform exponential growth for groups, Inventiones Mathematicae 155 (2004), 287-303. MR 2031429 (2004k:20085)

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Additional Information

Volodymyr Nekrashevych
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): February 25, 2008
Published electronically: July 17, 2009
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant DMS-0605019.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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