Geodesic
A freedom theorem for groups with one defining relation in the varieties of solvable and nilpotent groups of given lengths
Sbornik. Mathematics, Tome 18 (1972) no. 1, pp. 93-99 
N. S. Romanovskii. A freedom theorem for groups with one defining relation in the varieties of solvable and nilpotent groups of given lengths. Sbornik. Mathematics, Tome 18 (1972) no. 1, pp. 93-99. http://geodesic.mathdoc.fr/item/SM_1972_18_1_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The freedom theorem of Magnus is well known: if a group $G$ is given by generators $x_1,x_2,\dots$ and a single defining relation $r=1$, and if $r$ is not conjugate to any word in $x_2,\dots$, then the elements $x_2,\dots$ freely generate in $G$ a free subgroup. In this note analogous theorems of Magnus are established for groups given by one defining relation in the varieties of soluble and nilpotent groups of given lengths. Bibliography: 7 titles.

[1] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, New York–London–Sydney, 1966 | MR

[2] A. I. Maltsev, “Dva zamechaniya o nilpotentnykh gruppakh”, Matem. sb., 37(79) (1955), 567–572

[3] M. I. Kargapolov, V. N. Remeslennikov, “Problema sopryazhennosti dlya svobodnoi razreshimoi gruppy”, Algebra i logika, 5:6 (1966), 15–25 | MR | Zbl

[4] J. P. Labute, “On the descending central series of groups with a single defining relation”, J. Algebra, 14:1 (1970), 16–23 | DOI | MR | Zbl

[5] J. Lewin, T. Lewin, “On ideals of free associative algebras generated by a single element”, J. Algebra, 8 (1968), 248–255 | DOI | MR | Zbl

[6] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, Moskva, 1972 | MR | Zbl

[7] M. Kholl, Teoriya grupp, IL, Moskva, 1962