×

zbMATH — the first resource for mathematics

Affine automorphisms of rooted trees. (English) Zbl 1387.20021
Summary: We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional modules \(\mathbb Z^\infty_d\). This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group \(\mathbb Z^2_2 \wr \mathbb Z\).

MSC:
20E08 Groups acting on trees
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E22 Extensions, wreath products, and other compositions of groups
20B27 Infinite automorphism groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Software:
AutomGrp
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bondarenko, I., D’Angeli, D., Rodaro, E.: The lamplighter group \(\mathbb{Z}_3≀ \mathbb{Z}\) generated by a bireversible automaton. Preprint: arXiv:1502.07981 (2015) · Zbl 1291.68233
[2] Brunner, AM; Sidki, S, On the automorphism group of the one-rooted binary tree, J. Algebra, 195, 465-486, (1997) · Zbl 0902.20017
[3] Bartholdi, L.I., Šuniḱ, Z.: Some solvable automaton groups. In: Grigorchuk, R., Mihalik, M., Sapir, M., Šuniḱ, Z. (eds.) Topological and Asymptotic Aspects of Group Theory, vol. 394 of Contemp. Math., pp. 11-29. American Mathematical Society, Providence (2006) · Zbl 1106.20021
[4] Berlatto, A; Sidki, S, Virtual endomorphisms of nilpotent groups, Groups Geom. Dyn., 1, 21-46, (2007) · Zbl 1128.20016
[5] Brunner, AM; Sidki, SN, Abelian state-closed subgroups of automorphisms of \(m\)-ary trees, Groups Geom. Dyn., 4, 455-472, (2010) · Zbl 1221.20018
[6] Caponi, L.: On Classification of Groups Generated by Automata with 4 States over a 2-Letter Alphabet. Master’s thesis, University of South Florida, Department of Mathematics and Statistics, Tampa, FL, 33620, USA (2014) · Zbl 1291.68233
[7] Dantas, A., Sidki, S.: On state-closed representations of restricted wreath product of groups of type \({G}_{p,d}={C}_{p} wr {C}^{d}\). Preprint: arXiv:1505.05165 (2015) · Zbl 1221.20018
[8] Grigorchuk, R; Kravchenko, R, On the lattice of subgroups of the lamplighter group, Int. J. Algebra Comput., 24, 837-877, (2014) · Zbl 1317.20032
[9] Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskiĭ, V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231(Din. Sist., Avtom. i Beskon. Gruppy):134-214 (2000) · Zbl 1383.20018
[10] Grigorchuk, R., Savchuk, D.: Self-similar groups acting essentially freely on the boundary of the binary rooted tree. In: Group theory, combinatorics, and computing, vol 611 of Contemp. Math., pp. 9-48. American Mathematical Society, Providence (2014) · Zbl 1308.20025
[11] Grigorchuk, RI; Żuk, A, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedic., 87, 209-244, (2001) · Zbl 0990.60049
[12] Klimann, I; Picantin, M; Savchuk, D, Orbit automata as a new tool to attack the order problem in automaton groups, J. Algebra, 445, 433-457, (2016) · Zbl 1383.20018
[13] Muntyan, Y., Savchuk, D.: AutomGrp—GAP Package for Computations in Self-Similar Groups and Semigroups, Version 1.2.4. http://www.gap-system.org/Packages/automgrp.html (2014) · Zbl 0990.60049
[14] Nekrashevych, V.: Self-Similar Groups, vol. 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005) · Zbl 1128.20016
[15] Nekrashevych, V., Sidki, S.: Automorphisms of the Binary Tree: State-Closed Subgroups and Dynamics of \(1/2\)-Endomorphisms, vol. 311 of London Math. Soc. Lect. Note Ser., pp. 375-404. Cambridge University Press, Cambridge (2004) · Zbl 1144.20305
[16] Silva, PV; Steinberg, B, On a class of automata groups generalizing lamplighter groups, Int. J. Algebra Comput., 15, 1213-1234, (2005) · Zbl 1106.20028
[17] Woryna, A, The concept of self-similar automata over a changing alphabet and lamplighter groups generated by such automata, Theor. Comput. Sci., 412, 96-110, (2013) · Zbl 1291.68233
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.