Free subgroups of one-relator relative presentations
Algebra i logika, Tome 46 (2007) no. 3, pp. 290-298
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Suppose that $G$ is a non-trivial torsion-free group and $w$ is a word over the alphabet $G\cup\{x^{\pm1}_1,\dots,x^{\pm1}_n\}$. It is proved that, for $n\geqslant2$, the group $\widetilde G=\langle G,x_1,x_2,\dots,x_n\,|\,w = 1\rangle$ always contains a non-Abelian free subgroup. For $n=1$, the question whether there exist non-Abelian free subgroups in $\widetilde G$ is amply settled for the unimodular case (i.e., where the exponent sum of $x_1$ in $w$ is one). Some generalizations of these results are discussed.
Keywords:
relative presentations, one-relator groups, free subgroups.
@article{AL_2007_46_3_a1,
author = {A. A. Klyachko},
title = {Free subgroups of one-relator relative presentations},
journal = {Algebra i logika},
pages = {290--298},
publisher = {mathdoc},
volume = {46},
number = {3},
year = {2007},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2007_46_3_a1/}
}
A. A. Klyachko. Free subgroups of one-relator relative presentations. Algebra i logika, Tome 46 (2007) no. 3, pp. 290-298. http://geodesic.mathdoc.fr/item/AL_2007_46_3_a1/