Hyperbolic groupoids and duality

The author introduces a notion of 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, etc. The author describes a duality theory for hyperbolic groupoids. He shows that for every hyperbolic groupoid \mathfrak{G} there is a naturally defined dual groupoid \mathfrak{G}^\top acting on the Gromov boundary of a Cayley graph of \