Diophantine geometry over groups VIII: Stability

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 denable sets over free and hyperbolic groups. In this eighth paper we use a modication of the sieve procedure, which was used in proving quantifier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.