Diophantine geometry over groups IX: Envelopes and Imaginaries

This paper is the ninth 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 ninth paper we associate a Diophantine set with a definable set, and view it as the Diophantine envelope of the definable set. We use the envelope and duo limit groups that were used in proving stability of the theory of free and torsion-free hyperbolic groups [Se9], to study definable equivalence relations, and in particular, to classify imaginaries over these groups. In the first 6 papers in the sequence on Diophantine geometry over groups we studied sets of solutions to systems of equations in a free group, and developed basic techniques and objects that are required for the analysis of sentences and elementary sets defined over a free group. The techniques we developed, enabled us to present an iterative procedure that analyzes EAE sets defined over a free group (i.e., sets defined using 3 quantifiers), and shows that every such set is in the Boolean algebra generated by AE sets ([Se6],41), hence, we obtained a quantifie