zbMATH — the first resource for mathematics

The lamplighter group \(\mathbb Z_3 \wr \mathbb Z\) generated by a bireversible automaton. (English) Zbl 1368.20033
Summary: We construct a bireversible self-dual automaton with three states over an alphabet with three letters which generates the lamplighter group \(\mathbb Z_3 \wr \mathbb Z\). In particular, this fact shows that not all groups defined by birevirsible automata are finitely presented.

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E08 Groups acting on trees
20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
68Q70 Algebraic theory of languages and automata
Full Text: DOI
[1] Bartholdi, L., Silva, P. (2010). Groups defined by automata. Available at http://arxiv.org/abs/1012.1531.
[2] DOI: 10.1090/conm/394/07431
[3] DOI: 10.1142/S0218196714500337 · Zbl 1314.20028
[4] DOI: 10.1007/s10711-004-1815-2 · Zbl 1088.20037
[5] DOI: 10.1142/S0218196714500374 · Zbl 1317.20032
[6] DOI: 10.1016/S0764-4442(00)01702-X · Zbl 0969.57022
[7] Grigorchuk R. I., Proc. Steklov Inst. Math 231 pp 128– (2000)
[8] DOI: 10.1023/A:1012061801279 · Zbl 0990.60049
[9] Macedońska O., Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 12 pp 36– (2000)
[10] Muntyan, Y., Savchuk, D. (2008). AutomGrp–GAP package for computations in self-similar groups and semigroups, Version 1.1.2.
[11] DOI: 10.1090/surv/117
[12] Savchuk, D., Sidki, S. (2015). Affine automorphisms of rooted trees. Preprint: arxiv:1510.08434. · Zbl 1387.20021
[13] DOI: 10.1016/j.jalgebra.2011.02.049 · Zbl 1239.20032
[14] DOI: 10.1142/S0218196705002761 · Zbl 1106.20028
[15] DOI: 10.1142/S0218196711006194 · Zbl 1239.20033
[16] DOI: 10.4171/GGD/87 · Zbl 1227.20027
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.