THE RELATION BETWEEN ADDING MACHINE AND p−ADIC INTEGERS

THE RELATION BETWEEN ADDING MACHINE AND p−ADIC INTEGERS

In this paper, we equip Aut(X) with a natural metric and givean elementary proof that the closure of the adding machine group, a subgroupof the automorphism group, is both isometric and isomorphic to the group ofp−adic integers. This also shows that the group of p−adic integers can beisometrically embedded into the metric space Aut(X∗)

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  • G. N. Arzhantseva and Z. Sunic, Construction of elements in the closure of Grigorchuk group, arXiv:math/0607778v2.
  • L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From Fractal Groups to Fractal Sets, Fractals in Graz, (Peter Grabner and Wolfgang Woess, eds.), Birkh¨euser Verlag, Basel, (2003), 25-118.
  • H. Bass, M.V. Otera-Espinar, D. Rockmore and C. Tresser, Cyclic Renormalition and Auto- morphism Groups and Rooted Trees, Lecture Notes in Math., Springer-Verlag, Berlin, 1996.
  • F. Q. Gouvˆea, p−adic Numbers, Springer-Verlag, Berlin, 1997.
  • R. Grigorchuk, V.V. Nekrashevich and V.I. Sushchanskii, Automata, Dynamical systems and infinite groups, Proc. Steklov Inst. Math., 231 (2000), 134–214.
  • J. E. Holly, Pictures of Ultrametric Spaces, the p−adic numbers, and valued fields, Amer. Math. Monthly, 108 (2001), 721–728.
  • V. Nekrashevych, Self-similar Groups, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 117, 2005.
  • A. M. Robert, A Course in p−adic Analysis, Springer, 2000.
  • W. H. Schikhof, Ultrametric Calculus an Introduction to p−adic Calculus, Cambridge Uni- versity Press, New York, 1984.
  • Anadolu University, Faculty of Science, Department of Mathematics, Eskis¸ehir- TURKEY