 
Summary: A PROPERTY OF SUBGROUPS OF INFINITE INDEX
IN A FREE GROUP
G. N. ARZHANTSEVA
Abstract. We prove that if H is a finitely generated subgroup of infinite
index in a free group Fm , then, in a certain statistical meaning, the normal
subgroup generatedby ``randomly'' chosen elements r 1 ; : : : ; rn of Fm has trivial
intersection with H.
1. Introduction
Let Fm be a free group with free generators x 1 ; : : : ; xm and H a finitely generated
subgroup of Fm . It is known [3] that if H contains a nontrivial normal subgroup
of Fm then H has finite index in Fm . Karrass and Solitar proved in [4] that if H
has nontrivial intersection with every nontrivial normal subgroup of Fm ; then H
has finite index in Fm . This is a stronger result, since any two nontrivial normal
subgroups of a free group have nontrivial intersection. On the other hand, it is
obvious that if H has finite index in Fm then H has nontrivial intersection with
each of the nontrivial subgroups of Fm . Thus if H is a finitely generated subgroup
of Fm then H is of infinite index if and only if there is a normal subgroup K
of Fm such that K `` H = f1g. In the present paper, we study this property of
subgroups of free groups from a statistical point of view. We prove that if H is a
finitely generated subgroup of Fm of infinite index, then a randomly chosen normal
