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A combination theorem for relatively hyperbolic groups. (English) Zbl 1074.57001
The authors adapt the combination theorem of Bestvina and Feighn to give sufficient conditions which ensure for a given graph of $$\delta$$-hyperbolic groups to be itself $$\delta$$-hyperbolic. As an application, for a given limit group, a $$\delta$$-hyperbolic space is constructed on which the group acts freely by isometries. The existence of such a space gives an answer to the question of whether limit groups are hyperbolic relative to their maximal non-cyclic abelian subgroups.

##### MSC:
 57M07 Topological methods in group theory 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups
##### Keywords:
hyperbolic groups; limit groups; graph of groups
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