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A combination theorem for relatively hyperbolic groups

Summary: A combination theorem for relatively
hyperbolic groups
Emina Alibegovic
June 23, 2004
In this paper we give su∆cient conditions a graph of ∆-hyperbolic
spaces has to satisfy in order to be ∆-hyperbolic 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 ∆-hyperbolic 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 ∆-hyperbolic follows from Theorem 2.3


Source: Alibegovic, Emina - Department of Mathematics, University of Utah


Collections: Mathematics