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The class of groups all of whose subgroups with lesser number of generators are free is generic. (English. Russian original) Zbl 0877.20021
Math. Notes 59, No. 4, 350-355 (1996); translation from Mat. Zametki 59, No. 4, 489-496 (1996).
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 \(\lim_{d\to\infty}(N_d^f/N_d)=1\).

MSC:
20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
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[2] Kourov Notebook. (Unsolved Problems of Group Theory) [in Russian], 11th ed., Nauka, Novosibirsk (1990).
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[4] Roger C. Lyndon and Paul E. Shupp,Combinatorial Group Theory, Springer-Verlag, Berlin-Heidelberg-New York (1977). · Zbl 0368.20023
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