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Hyperbolic groupoids and duality

About this Title

Volodymyr V. Nekrashevych, Department of Mathematics, Texas A & M University, College Station, Texas 77843-3368

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)
Published electronically: February 16, 2015
Keywords: Hyperbolic groupoids, Smale spaces, Smale quasi-flows, Gromov hyperbolic graphs
MSC: Primary 37D20, 20L05; Secondary 20F67

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Table of Contents


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


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.

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