FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1997, VOLUME 3, NUMBER 3, PAGES 675-683

On the groups in which the subgroups with fixed number of generators are free

G. N. Arjantseva

Abstract

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We prove here that, in a definite statistical meaning, in almost every group with $m$ generators and $n$ relations (we suppose $m$ and $n$ to be fixed) all $$£ L-generated subgroups of infinite index are free ($L$ is an arbitrary preassigned bound, possibly $L\; >>\; m$) and all subgroups of finite index are not free. To prove this fact we found the condition on relations which guarantee that all subgroups of infinite index with fixed number of generators in a finitely presented group are free. This condition is formulated by means of the finite marked graphs.

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