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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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##### References:
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