Abstract
A natural interpretation of automorphisms of one-rooted trees as output automata permits the application of notions of growth and circuit structure in their study. New classes of groups are introduced corresponding to diverse growth functions and circuit structure. In the context of automorphisms of the binary tree, we discuss the structure of maximal 2-subgroups and the question of existence of free subgroups. Moreover, we construct Burnside 2-groups generated by automorphisms of the binary tree which are finite state, bounded, and acyclic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. V. Aleshin, “A free group of finite automata,”Vestn. Mosk. Univ., Ser. Mat.,38, 10–13 (1983).
M. Bhattacharjee, “The ubiquity of free subgroups in certain inverse limits of groups,”J. Algebra,172, 134–146 (1995).
H. Bass, M. Otero-Espinar, D. Rockmore, and C. P. L. Tresser, “Cyclic renormalization and automorphism groups of rooted trees,” In:Lect. Notes Math., Vol. 1621, Springer, Berlin (1995).
A. M. Brunner and S. Sidki, “On the automorphism group of the one-rooted binary tree,”J. Algebra,195, 465–486 (1997).
A. M. Brunner and S. Sidki, “The generation ofGL(n), ℤ by finite state automata,”Int. J. Algebra Comput.,8, 127–139 (1998).
A. M. Brunner, S. Sidki, and A. C. Vieira, “A just-nonsolvable torsion-free group defined on the binary tree,” to appear inJ. Algebra.
R. I. Grigorchuk, “On the Burnside problem for periodic groups,”Funkts. Anal. Prilozhen.,14, 41–43 (1980).
R. I. Grigorchuk, “On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata,”, In:Proc. Groups St. Andrews/Bath 1997.
R. I. Grigorchuk and P. F. Kurchanov, “Some questions of group theory related to geometry,” In: A. N. Parshin and I. R. Shafarevich, eds.,Algebra VII. Combinatorial Group Theory. Applications to Geometry. Springer-Verlag (1991), pp. 167–240.
N. Gupta and S. Sidki, “Extensions of groups by tree automorphisms,” In:Contributions to Group Theory. Contemp. Math., vol. 33, American Mathematical Society, Providence, Rhode Island (1984), pp. 232–246.
D. S. Passman and W. V. Temple, “Representations of the Gupta-Sidki group,”Proc. Amer. Math. Soc.,124, 1403–1410 (1996).
A. V. Rozhkov, “The lower central series of one group of automorphisms of a tree,”Mat. Zametki,60, 223–237 (1996).
M. Rubin, “The reconstruction of trees from their automorphism groups,” In:Contemp. Math., Vol. 151, American Mathematical Society, Providence, Rhode Island (1993).
S. Sidki, “A primitive ring associated to a Burnside 3-group,”J. London Math. Soc. (2),55, 55–64 (1997).
S. Sidki, “Regular trees and their automorphisms,” In:Notes of a Course, XIV Escola de Álgebra, Rio de Janeiro, July 1996.Monografias de Matemática, Vol. 56, IMPA, Rio de Janeiro (1998).
V. I. Sushschanskii, “Wreath products and periodic factorable groups,”Mat. Sb.,67, No. 2, 535–553 (1990).
A. C. Vieira, “On the lower central series and the derived series of the Gupta-Sidki 3-group,”Commun. Algebra,26, 1319–1333 (1998).
J. S. Wilson and P. A. Zalesskii, “Conjugacy separability of certain torsion groups,”Arch. Math.,68, 441–449 (1997).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 58, Algebra-12, 1998.
Rights and permissions
About this article
Cite this article
Sidki, S. Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity. J Math Sci 100, 1925–1943 (2000). https://doi.org/10.1007/BF02677504
Issue Date:
DOI: https://doi.org/10.1007/BF02677504