# zbMATH — the first resource for mathematics

On state-closed representations of restricted wreath product of groups $$G_{p,d}=C_{p}wrC^{d}$$. (English) Zbl 1427.20034
Summary: Let $$G_{p, d}$$ be the restricted wreath product $$C_pwrC^d$$ where $$C_p$$ is a cyclic group of order a prime $$p$$ and $$C^d$$ a free abelian group of finite rank $$d$$. We study the existence of faithful transitive state-closed (fsc) representations of $$G_{p, d}$$ on the rooted $$m$$-ary tree for some finite $$m$$. The group $$G_{2, 1}$$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $$d \geq 2$$ there are no fsc representations of $$G_{p, d}$$ on the $$p$$-adic tree. We describe all fsc representations of $$G = G_{p, 1}$$ on the $$p$$-adic tree obtained via virtual endomorphisms, where the first level stabilizer of the image of $$G$$ contains its commutator subgroup. Furthermore, for $$d \geq 2$$, we construct fsc representations of $$G_{p, d}$$ on the $$p^2$$-adic tree and exhibit concretely the representation of $$G_{2, 2}$$ on the 4-tree as a finite-state automaton group.

##### MSC:
 20E08 Groups acting on trees 20E22 Extensions, wreath products, and other compositions of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E36 Automorphisms of infinite groups 05C05 Trees 20F28 Automorphism groups of groups
AutomGrp
Full Text:
##### References:
 [1] Kaimanovich, V.; Vershik, A., Random walks on discrete groups: boundary and entropy, Ann. Probab., 11, 457-490, (1983) · Zbl 0641.60009 [2] Grigorchuk, R.; Zuk, A., The lamplighter group as a group generated by a 2-state automaton and its spectrum, Geom. Dedicata, 87, 209-244, (2001) · Zbl 0990.60049 [3] Grigorchuk, R.; Linnel, P.; Schick, T.; Zuk, A., On a question of Atiyah, C. R. Acad. Sci. Paris, 331, 9, 663-668, (2000) · Zbl 0969.57022 [4] Silva, P.; Steinberg, B., On a class of automata groups generalizing lamplighter groups, Internat. J. Algebra Comput., 15, 1213-1234, (2005) · Zbl 1106.20028 [5] Kambides, M.; Silva, P.; Steinberg, B., The spectra of lamplighter groups and Cayley machines, Geom. Dedicata, 120, 193-227, (2006) · Zbl 1168.20012 [6] Bartholdi, L.; Sunik, Z., Some solvable automata groups, Contemp. Math., 394, 11-30, (2006) · Zbl 1106.20021 [7] Bartholdi, L., Functionally recursive groups [8] Bondarenko, I.; D’Angeli, D.; Rodaro, E., The lamplighter group $$Z_3 \wr Z$$ generated by a bireversible automaton, Comm. Algebra, 44, 12, 5257-5268, (2016) · Zbl 1368.20033 [9] Savchuk, D.; Sidki, S., Affine automorphisms of rooted trees, Geom. Dedicata, 183, 195-213, (2016) · Zbl 1387.20021 [10] Grigorchuk, R.; Leemann, P.-H.; Nagnibeda, T., Lamplighter groups, de Bruijn graphs, spider-web graphs and their spectra, J. Phys. A, 49, 20, (2016) · Zbl 1345.05043 [11] Sidki, S., Tree wreathing applied to the generation of groups by finite automata, Internat. J. Algebra Comput., 15, 1-8, (2005) · Zbl 1108.20025 [12] Dantas, A.; Sidki, S., On self-similarity of wreath products of abelian groups, Groups Geom. Dyn., (2017), in press [13] Robinson, D. J.S., A course in the theory of groups, Grad. Texts in Math., vol. 80, (1996), Springer Verlag New York [14] Nekrashevych, V.; Sidki, S., Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2 endomorphisms, (Groups: Topological, Combinatorial and Arithmetic Aspects, London Mathematical Lecture Notes Series, vol. 311, (2004), Cambridge Univ. Press Cambridge), 375-404 · Zbl 1144.20305 [15] Nekrashevych, V., Virtual endomorphisms of groups, Algebra Discrete Math., 1, 1, 96-136, (2002) [16] Nekrashevych, V., Self-similar groups, Math. Surveys and Monographs, vol. 117, (2005), American Mathematical Society Providence, RI · Zbl 1087.20032 [17] Brunner, A.; Sidki, S., Abelian state-closed subgroups of automorphisms of m-ary trees, Groups Geom. Dyn., 3, 455-472, (2010) · Zbl 1221.20018 [18] Berlatto, A.; Sidki, S., Virtual endomorphisms of nilpotent groups, Groups Geom. Dyn., 1, 21-46, (2007) · Zbl 1128.20016 [19] Kapovich, M., Arithmetic aspects of self-similar groups, Groups Geom. Dyn., 6, 737-754, (2012) · Zbl 1283.20050 [20] Y. Muntyan, D. Savchuk, AutomGrp - GAP package for computations in self-similar groups and semigroups, version 1.2.4, 2014.
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.