On the concept of fractality for groups of automorphisms of a regular rooted tree
- 1 University of the Basque Country, UPV/EHU, Department of Mathematics.
ISSN: 2385-4227
Year of publication: 2016
Volume: 2
Issue: 1
Pages: 33-44
Type: Article
More publications in: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques
Abstract
The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose, we dene three types of fractality. We show that they are not equivalent, by giving explicit counter-examples. Furthermore, we present some tools that are helpful in order to determine the fractality of a given group.
Bibliographic References
- L. Bartholdi and R.I. Grigorchuk, “On parabolic subgroups and Hecke algebras of some fractal groups”,Serdica Math. J.28(1) (2002), 47–90.
- L. Bartholdi, R.I. Grigorchuk, and Z. Sunik,“Branch groups”, inHandbook of algebra 3,989–1112, North-Holland, Amsterdam, 2003.
- A.M. Brunner and S.N. Sidki, “Abelian state-closed subgroups of automorphisms of m-arytrees”,Groups Geom. Dyn.4(3) (2010), 455–472.
- F. Dahmani, “An example of non-contracting weakly branch automaton group”, in Geometric methods in group theory; Contemp. Math.372, 219–224. Amer. Math. Soc., Providence,RI, 2005.
- D. D’Angeli and A. Donno, “Self-similar groupsand finite Gelfand pairs”,Algebra DiscreteMath.2(2007), 54–69.
- A. Donno, “Gelfand Pairs: from self-similar groups to Markov chains”, PhD thesis, Universita degli studi di Roma, La Sapienza, 2008.
- G.A.Fernandez-Alcober and A.Zugadi-Reizabal, “GGS-groups: order of congruence quotients and Hausdorff dimension”, Trans.Amer. Math. Soc.366(4) (2014), 1993–2017.
- R.I. Grigorchuk, “On Burnside’s problem on periodic groups”, Funktsional. Anal. i Prilozhen.14(1) (1980), 53–54.
- R.I. Grigorchuk, “Some topics in the dynamicsof group actions on rooted trees”, Proceedings of the Steklov Institute of Mathematics 273(1) (2011), 64–175.
- R.I. Grigorchuk and Z. Sunic, “Self-similarityand branching in group theory”, in Groups St. Andrews 2005; London Math. Soc. Lecture Note Ser. 339, 36–95, Cambridge Univ. Press,Cambridge, 2007.
- N. Gupta and S.N. Sidki, “On the Burnside problem for periodic groups”, Math. Z.182(3)(1983), 385–388.
- T. Vovkivsky, “Infinite torsion groups arising as generalizations of the second Grigorchuk group”, in Algebra (Moscow, 1998), 357–377, de Gruyter, Berlin, 2000.