 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 128, Number 11, Pages 32053210
S 00029939(00)055088
Article electronically published on May 11, 2000
A PROPERTY OF SUBGROUPS
OF INFINITE INDEX IN A FREE GROUP
G. N. ARZHANTSEVA
(Communicated by Ronald M. Solomon)
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 generated by "randomly" chosen elements r1, . . . , rn of Fm has trivial
intersection with H.
1. Introduction
Let Fm be a free group with free generators x1, . . . , 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
