zbMATH — the first resource for mathematics

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

20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.