Hyperbolic groupoids and duality

We define hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism, e.t.c.. We show that for every hyperbolic groupoid G there is a naturally defined dual groupoid G' acting on the Gromov boundary of a Cayley graph of G, which is also hyperbolic and such that (G')' is equivalent to G.