Locally free subgroups of one-relator groups

Abstract

Let $G_1=\langle x_1,\dots x_s; [x_1,x_{n+1}][x_{2},x_{n+2}]\dots [x_{n},x_{2n}]S\rangle $, $G_2=\langle a, x_1,\dots ,x_s; [a,x_1][a,x_2]\dots [a,x_n]S \rangle $ be one-relator groups. We find conditions on $S$ and $n$ under which the normal closure of each $(n-1)$-generated subgroup of $G_1$ and of each 3-generated subgroup of $G_2$ is locally free.