FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1997, VOLUME 3, NUMBER 3, PAGES 675-683
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 $\le L$ -generated
subgroups of infinite index are free ($L$ is an arbitrary
preassigned bound, possibly $L\gg m$ ) 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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