  This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In the eighth paper we use a modification of the sieve procedure, that was presented as part of the quantifier elimination process, to prove that free and torsion-free (Gromov) hyperbolic groups are stable. 
Diophantine Geometry over Groups VIII: Stability