 
Summary: Contemporary Mathematics
A dichotomy for finitely generated subgroups of word
hyperbolic groups
Goulnara N. Arzhantseva
Abstract. Given L > 0 elements in a word hyperbolic group G, there exists
a number M = M(G,L) > 0 such that at least one of the assertions is true:
(i) these elements generate a free and quasiconvex subgroup of G; (ii) they are
Nielsen equivalent to a system of L elements containing an element of length at
most M up to conjugation in G. The constant M is given explicitly. The result
is generalized to groups acting by isometries on Gromov hyperbolic spaces. For
proof we use a graph method to represent finitely generated subgroups of a
group.
1. Introduction
Let H be a subgroup of a word hyperbolic group G. It is known that either H
is elementary (that is, it contains a cyclic subgroup of finite index) or H contains a
nonabelian free subgroup of rank two. In the case G is torsionfree, there are, up
to conjugacy, finitely many Nielsen equivalence classes of nonfree subgroups of G
generated by two elements [3].
Our main result gives a su#cient condition for H to be free and quasiconvex
in G. It is an improvement of a result due to Gromov [4, 5.3.A].
