Volodymyr Nekrashevych 
Hyperbolic Groupoids and Duality [PDF ebook] 

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.