 
Summary: A combination theorem for relatively
hyperbolic groups
Emina Alibegovic
June 23, 2004
Abstract
In this paper we give suÆcient conditions a graph of Æhyperbolic
spaces has to satisfy in order to be Æhyperbolic itself. As an applica
tion, we give a simple proof that limit groups are relatively hyperbolic.
1 Introduction
In his work on Diophantine equations over free groups Z. Sela introduced
limit groups. He showed that this class of groups coincides with the class of
!residually free groups that had already been extensively studied. One of
the most important results is a structure theorem for limit groups given by
Kharlampovich{Myasnikov and Sela ([8], [7], [10]).
This work introduced a lot of interesting questions one might ask about
limit groups. We were interested in describing the set of homomorphisms
from an arbitrary nitely generated group G into a limit group L, Hom(G;L).
A key tool in studying Hom(G;L) is a Æhyperbolic space on which the given
limit group L acts freely, by isometries. We construct such a space in Section
3. That the space we constructed is Æhyperbolic follows from Theorem 2.3
