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. Cover 1 Title page 2 Introduction 8 0.1. Main definitions and results 9 0.2. Structure of the paper 13 Chapter 1. Technical preliminaries 16 Some remarks on notation and terminology 16 1.1. Logarithmic scales 16 1.2. Gromov-hyperbolic metric spaces and graphs 20 1.3. Local product structures 35 Chapter 2. Preliminaries on groupoids and pseudogroups 38 2.1. Pseudogroups and groupoids 38 2.2. Actions of groupoids and equivalence 40 2.3. Compactly generated groupoids and their Cayley graphs 46 2.4. Relations in Hausdorff groupoids 49 2.5. Groupoids with additional structures 50 Chapter 3. Hyperbolic groupoids 56 3.1. Main definition 56 3.2. Busemann quasi-cocycle 59 3.3. Complete generating sets 60 3.4. Uniqueness of the point ωx 61 3.5. Boundaries of the Cayley graphs 62 3.6. The boundary ∂\G 65 3.7. Geodesic quasi-flow 71 Chapter 4. Smale quasi-flows and duality 74 4.1. Definitions 74 4.2. Special generating sets 75 4.3. Geodesic quasi-flow of a hyperbolic groupoid as a Smale quasi-flow 78 4.4. Hyperbolicity of the Ruelle groupoids 79 4.5. Duality theorems 85 4.6. Other definitions of the dual groupoid 88 4.7. Minimal hyperbolic groupoids 91 Chapter 5. Examples of hyperbolic groupoids and their duals 96 5.1. Gromov hyperbolic groups 96 5.2. Expanding self-coverings 99 5.3. Contracting self-similar groups 103 5.4. Smale spaces 105 Bibliography 110 Index 114 Back Cover 120