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Summary: ON QUASICONVEX SUBGROUPS OF WORD HYPERBOLIC
GROUPS
G. N. ARZHANTSEVA
Abstract. We prove that a quasiconvex subgroup H of infinite index of a
torsion free word hyperbolic group can be embedded in a larger quasiconvex
subgroup which is the free product of H and an infinite cyclic group. Some
properties of quasiconvex subgroups of word hyperbolic group are also dis
cussed.
1. Introduction
Word hyperbolic groups were introduced by M. Gromov as a geometric gener
alization of certain properties of discrete groups of isometries of hyperbolic spaces
H n . Finite groups, finitely generated free groups, classical small cancellation groups
and groups acting discretely and cocompactly on hyperbolic spaces are basic exam
ples of word hyperbolic groups. Any word hyperbolic group is finitely presented.
Finite extensions and free products of finitely many word hyperbolic groups are
also word hyperbolic. A large number of results on word hyperbolic groups as well
as conjectures and research problems are contained in the original article [9].
In this paper, we study properties of quasiconvex subgroups of word hyperbolic
groups (see the next section for the definition). Our main result gives in fact a
method for constructing quasiconvex subgroups of word hyperbolic groups.
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