 
Summary: Contemporary Mathematics
A dichotomy for nitely 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 nitely generated subgroups of a
group.
Universit e de Gen eve, Section de math ematiques, 24, rue du Li evre, CH1211 Gen eve
24
Email address: Goulnara.Arjantseva@math.unige.ch
2000 Mathematics Subject Classication. 20F67;20E07.
Key words and phrases. Word hyperbolic groups, quasigeodesics, Nielsen equivalence.
The work has been supported in part by the Swiss National Science Foundation, No. PP002
68627.
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