Nullhomologous words in free groups which are not nullhomologous in any proper subgroup
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.
scientific article; zbMATH DE number 4021438
scientific article