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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2017: 0.67

Online
ISSN
1435-5337
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Volume 25, Issue 6

Issues

The topological decomposition of inverse limits of iterated wreath products of finite Abelian groups

Adam Woryna
Published Online: 2013-03-02 | DOI: https://doi.org/10.1515/forum-2012-0132

Abstract.

Let be an arbitrary infinite sequence of nontrivial finite Abelian transitive groups such that the topological rank ρ of the infinite Cartesian product of these groups is finite. We consider the corresponding inverse limit of iterated permutational wreath products. By using the geometric language of automorphisms of a certain rooted tree we show that the group is topologically generated by the union of two sets each containing ρ elements and such that both the group generated by S and the group generated by are free Abelian groups of rank ρ. Moreover, the group G generated by the above union decomposes as a semidirect product of the group generated by one of these sets and the normal closure of the other set, and the semigroup generated by this union is a free product of semigroups generated by S and , respectively. We derive other algebraic properties of the group G. In particular, the condition for the groups to be finitely generated as profinite groups and some nontrivial results concerning topological ranks of their subgroups are given in this way. For example, for every our construction gives an explicit and naturally defined finitely generated subgroup of an inverse limit of iterated wreath products of finite Abelian groups such that the rank of this subgroup and the topological rank of its topological closure differ by n.

Keywords: Iterated wreath product; rooted tree; automorphisms of rooted trees; topological generation

About the article

Received: 2012-09-02

Revised: 2012-12-10

Published Online: 2013-03-02

Published in Print: 2013-11-01


Citation Information: Forum Mathematicum, Volume 25, Issue 6, Pages 1263–1290, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2012-0132.

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