# Hyperbolic Groupoids And Duality (memoirs Of The American Mathematical Society)

by Volodymyr Nekrashevych /
2015 / English / PDF

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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.

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.