Nullhomologous words in free groups which are not nullhomologous in any proper subgroup
type: Item
instance of: scholarly article
title: Nullhomologous words in free groups which are not nullhomologous in any proper subgroup
zbMATH Open document ID: 0628.20026
DOI: 10.1007/BF01193627
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: A result of \textit{H. Zieschang} [Abh. Math. Semin. Univ. Hamb. 27, 13-31 (1964; Zbl 0135.418)] asserts that the commutator product \(w=\prod ^{g}_{i=1}a_ ib_ ia_ i^{-1}b_ i^{-1}\) in the free group \(F=<a_ 1,b_ 1,...,a_ g,b_ g>\) is not contained in the commutator subgroup [H,H] of any proper subgroup \(H<G\). The present note replaces [H,H] by bigger groups \(H_ k=\ker nel(H\to H/[H,H]\otimes {\mathbb{Z}}/k{\mathbb{Z}})\), and proves the same kind of theorem for more words \(w\in F_ k\). For every \(k>1\), every finitely generated free group F and every w in a certain subset of \(F_ k\) it shows that no proper subgroup \(H<F\) has \(H_ k\ni w\). The proof uses simple homological arguments.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4021438
zbMATH Keywords: commutator product
zbMATH Keywords: commutator subgroup
zbMATH Keywords: words
zbMATH Keywords: finitely generated free group
author: Albrecht Dold
MaRDI profile type: Publication
cites work: Equations in groups
cites work: Q4145882
cites work: Q5534009
cites work: Q4756475
cites work: Alternierende Produkte in freien Gruppen
full work available at URL: https://doi.org/10.1007/bf01193627
OpenAlex ID: W1975914616
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description: scientific article; zbMATH DE number 4021438