Appears in collection : 2014 - T1 - Random walks and asymptopic geometry of groups.

We study and classify all the definable equivalence relations in a free (and a torsion-free hyperbolic) group. To do that we associate a Diophantine set with every definable set, that contains the definable set, and its generic points are contained in the definable set. We then use this Diophantine set to uniformly parametrize the equivalence classes of a definable equivalence relation.