Geodesic
On the groups in which the subgroups with fixed number of generators are free
Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 3, pp. 675-683 
G. N. Arzhantseva. On the groups in which the subgroups with fixed number of generators are free. Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 3, pp. 675-683. http://geodesic.mathdoc.fr/item/FPM_1997_3_3_a2/
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     author = {G. N. Arzhantseva},
     title = {On the groups in which the subgroups with fixed number of generators are free},
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     pages = {675--683},
     year = {1997},
     volume = {3},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1997_3_3_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We prove here that, in a definite statistical meaning, in almost every group with $m$ generators and $n$ relations (we suppose $m$ and $n$ to be fixed) all $\le L$-generated subgroups of infinite index are free ($L$ is an arbitrary preassigned bound, possibly $L\gg m$) and all subgroups of finite index are not free. To prove this fact we found the condition on relations which guarantee that all subgroups of infinite index with fixed number of generators in a finitely presented group are free. This condition is formulated by means of the finite marked graphs.