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 $\mathfrak{G}$. The groupoid $\mathfrak{G}^\top$ is also hyperbolic and such that $(\mathfrak{G}^\top)^\top$ is equivalent to $\mathfrak{G}$. Several classes of examples of hyperbolic groupoids and their applications are discussed.