Free subgroups in groups with few relators.
In a previous paper [Bull. Lond. Math. Soc. 36, No. 1, 1-2 (2004; Zbl 1048.20015)], the author proved the following result. Let \(G\) be an abstract (resp. pro-\(p\)) group which has a presentation with \(n\) generators and \(m\) relators, where \(m<n,\) and let \(Y\) be any generating set for \(G\). Then there are \(n-m\) elements of \(Y\) that freely generate a free abstract (resp. pro-\(p\)) group. In the present paper he gives a new, nice and much simpler proof of the same result. He also proves the following generalization, in the case of pro-\(p\)-groups. Let \(G\) be a finitely generated pro-\(p\) group generated by a family \(\mathcal A\) of \(n\) finitely generated pro-\(p\) subgroups each having \(\mathbb{Z}_p\) as an image, and suppose that the kernel \(R\) of the natural map from the free pro-\(p\) product \(F\) of the groups in \(\mathcal A\) to \(G\) is generated (as a closed normal subgroup) by \(m\) elements, where \(m<n\). Let \(\mathcal B\) be a family of subgroups of \(G\) that generates \(G\). Then \(\bigcup_{B\in\mathcal B}B\) contains \(n-m\) elements that freely generate a free pro-\(p\) group.
scientific article; zbMATH DE number 5788667