The class of groups all of whose subgroups with lesser number of generators are free is generic (Q1358513)

The authors show that in a certain statistical sense, in almost every group with \(m\) generators and \(n\) relations, any subgroup generated by less than \(m\) elements is free. Let \(N_{m,n,t}\) be the number of all presentations of groups of the form \(G=\langle x_1,\dots,x_m\mid r_1=1,\dots,r_n=1\rangle\), where \(m>1\) and length \(|r_i|\leq t\) for each cyclically reduced word \(r_i\), \(N_{m,n,t}^f\) be the number of presentations such that any \((m-1)\)-generated subgroup of \(G\) is free. It is proved that there is a number \(c>0\) such that \(N_{m,n,t}^f/N_{m,n,t}\) is greater than \(1-\exp(-ct)\). Let \(N_d\) be the number of all presentations of groups considered such that \(|r_1|+\dots+|r_n|=d\), \(N^f_d\) be the number of such presentations for which any \((m-1)\)-generated subgroup of \(G\) is free. The authors also show that \(\lim_{d\to\infty}(N_d^f/N_d)=1\).