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Finite Automata of Polynomial Growth do Not Generate A Free Group

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Abstract

A free subgroup of rank 2 of the automorphism group of a regular rooted tree of finite degree cannot be generated by finite-state automorphisms having polynomial growth. This result is in fact proven for rooted trees of infinite degree under some natural additional conditions.

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References

  1. Brunner, A. and Sidki, S.: The generation of GL(n, ℤ) by finite-state automata, Internat. J. Algebra Comput. 8 (1998), 127–139.

    MathSciNet  Google Scholar 

  2. Nekrachevych, V. V. and Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms, In: T. M. Müller (ed.), Groups-Topological, Combinatorial and Arithmetic Aspects, Cambridge University Press, 2003.

  3. Sidki, S.: Automorphisms of one-rooted trees: growth, circuit structure and acyclicity, J. Math. Sci. 100 (2000), 1925–1943.

    MATH  MathSciNet  Google Scholar 

  4. Ufnarovski, V. A.: On the use of graphs for calculating the basis, growth and Hilbert series of associative algebras, Math. USSR-Sb. 68 (1991), 417–428.

    MathSciNet  Google Scholar 

  5. Sidki, S. and Silva, E. F.: A family of just-nonsolvable torsion-free groups defined on n-ary trees, In: Atas da XVI Escola de Álgebra, Brasília, Matemática Contemporânea 21 (2001), 255–274.

    Google Scholar 

  6. Brunner, A., Sidki, S. and Vieira, A. C.: A just-nonsolvable torsion-free group defined on the binary tree, J. Algebra 211 (1999), 99–114.

    Article  MathSciNet  Google Scholar 

  7. Grigorchuk, R. I.: On the Burnside problem on periodic groups, Funct. Anal. Appl. 14 (1980), 41–43.

    MATH  MathSciNet  Google Scholar 

  8. Gupta, N. and Sidki, S.: On the Burnside problem on periodic groups, Math. Z. 182 (1983), 385–388.

    Article  MathSciNet  Google Scholar 

  9. Brunner, A. and Sidki, S.: Wreath operations in the group of automorphisms of the binary tree, J. Algebra 257 (2002), 51–64.

    Article  MathSciNet  Google Scholar 

  10. Sidki, S.: Tree-wreathing applied to generation of groups by finite automata, in press.

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  1. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília DF, Brazil

    Said Sidki

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  1. Said Sidki
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Sidki, S. Finite Automata of Polynomial Growth do Not Generate A Free Group. Geometriae Dedicata 108, 193–204 (2004). https://doi.org/10.1007/s10711-004-2368-0

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  • DOI: https://doi.org/10.1007/s10711-004-2368-0

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