The $n$-ary adding machine and solvable groups

Document Type : Research Paper


1 Instituto Federal de Educacao

2 Universidade De Brasilia


We describe under various conditions abelian subgroups of the automorphism‎ ‎group $\mathrm{Aut}(T_{n})$ of the regular $n$-ary tree $T_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $\tau =(e‎, ‎\dots‎, ‎e,\tau )\sigma _{\tau‎ ‎}$ where $\sigma _{\tau }$ is the $n$-cycle $\left( 0,1‎, ‎\dots‎, ‎n-1\right) $‎. ‎As‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that‎ ‎every soluble subgroup of $\mathrm{Aut}(T_{n})$‎, ‎containing $\tau $ is an extension of a torsion-free metabelian group by a‎ ‎finite group‎.


Main Subjects

H. Bass, Otero-Espinar, D. Rockmore and C. Tresser (1996). Cyclic Renormalization and the Automorphism Groups of Rooted Trees. Lecture Notes in Mathematics, Springer-Verlag, Berlin. 1621 A. M. Brunner, S. N. Sidki and A. C. Vieira (1999). A just-nonsolvable torsion-free group defined on the binary tree. J. Algebra. 211, 99-114 R. I. Grigorchuk, V. V. Nekrachevych and V. I. Sushchanskii (2000). Automata, dynamical systems, and groups. Proc. Steklov Inst. Math.. 231, 128-203 G. A. Jones (2002). Cyclic regular subgroups of primitive permutation groups. J. Group Theory. 5 (4), 403-407 V. V. Nekrashevych (2005). Self-similar groups. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI. 117 S. N. Sidki (1998). Regular Trees and their Automorphisms. Monografias de Matematica, Impa, Rio de Janeiro. 56 S. N. Sidki (2000). Automorphisms of one-rooted trees: growth, circuit structure and acyclicity. J. Math. Sci.. 100 (1), 1925-1943 S. N. Sidki and E. F. Silva (2001). A family of just-nonsolvable torsion-free groups defined on $n$-ary trees. 16th School of Algebra, Part II (Portuguese) (Brasilia, 2000), Mat. Contemp.. 21, 255-274 S. N. Sidki (2003). The binary adding machine and solvable groups. Internat. J. Algebra Comput.. 13 (1), 95-110 S. N. Sidki (2005). Just-Non-(abelian by P-type) Groups. Progr. Math.. 248, 389-402 M. Vorobets and Y. Vorobets (2007). On a free group of transformations defined by an automaton. Geom. Dedicata. 124, 237-249
Volume 2, Issue 4 - Serial Number 4
December 2013
Pages 43-88
  • Receive Date: 20 April 2013
  • Revise Date: 04 June 2013
  • Accept Date: 04 June 2013
  • Published Online: 01 December 2013