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# memo_has_moved_text();Hyperbolic groupoids and duality

Volodymyr V. Nekrashevych

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1122
ISBNs: 978-1-4704-1544-0 (print); 978-1-4704-2511-1 (online)
DOI: http://dx.doi.org/10.1090/memo/1122
Published electronically: February 16, 2015
Keywords:Hyperbolic groupoids, Smale spaces, Smale quasi-flows

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Chapters

• Introduction
• Chapter 1. Technical preliminaries
• Chapter 2. Preliminaries on groupoids and pseudogroups
• Chapter 3. Hyperbolic groupoids
• Chapter 4. Smale quasi-flows and duality
• Chapter 5. Examples of hyperbolic groupoids and their duals

### Abstract

We introduce 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. We describe a duality theory for hyperbolic groupoids. We show 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.