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 \(\lim_{d\to\infty}(N_d^f/N_d)=1\).

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 \(\lim_{d\to\infty}(N_d^f/N_d)=1\).

Reviewer: A.I.Budkin (Barnaul)

##### MSC:

20F05 | Generators, relations, and presentations of groups |

20E05 | Free nonabelian groups |

20E07 | Subgroup theorems; subgroup growth |

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\textit{G. N. Arzhantseva} and \textit{A. Yu. Ol'shanskij}, Math. Notes 59, No. 4, 350--355 (1996; Zbl 0877.20021); translation from Mat. Zametki 59, No. 4, 489--496 (1996)

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##### References:

[1] | V. S. Guba, ”Conditions under which 2-generated groups in small cancellation groups are free,”Izv. Vyssh. Uchebn. Zaved. Mat., [Soviet Math. J. (Iz. VUZ)], No. 7, 12–19 (1986). · Zbl 0614.20022 |

[2] | Kourov Notebook. (Unsolved Problems of Group Theory) [in Russian], 11th ed., Nauka, Novosibirsk (1990). |

[3] | S. W. Margolis and J. C. Meakin, ”Free inverse monoids and graph immersions,”Int. J. Algebra and Comput. 3, No. 1, 79–100 (1993). · Zbl 0798.20056 · doi:10.1142/S021819679300007X |

[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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