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 $mathfrakG$ there is a naturally defined dual groupoid $mathfrakGtop$ acting on the Gromov boundary of a Cayley graph of $mathfrakG$. The groupoid $mathfrakGtop$ is also hyperbolic and such that $(mathfrakGtop)top$ is equivalent to $mathfrakG$. Several classes of examples of hyperbolic groupoids and their applications are discussed.