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This book provides a detailed introduction to the coarse quasi-isometry of leaves of a foliated space and describes the cases where the generic leaves have the same quasi-isometric invariants.




Every leaf of a compact foliated space has an induced coarse quasi-isometry type, represented by the coarse metric defined by the length of plaque chains given by any finite foliated atlas. When there are dense leaves either all dense leaves without holonomy are uniformly coarsely quasi-isometric to each other, or else every leaf is coarsely quasi-isometric to just meagerly many other leaves. Moreover, if all leaves are dense, the first alternative is characterized by a condition on the leaves called coarse quasi-symmetry. Similar results are proved for more specific coarse invariants, like growth type, asymptotic dimension, and amenability. The Higson corona of the leaves is also studied. All the results are richly illustrated with examples.




The book is primarily aimed at researchers on foliated spaces. More generally, specialists in geometric analysis, topological dynamics, or metric geometry may also benefit from it.

Generic Coarse Geometry of Leaves

- Introduction. - Part I Coarse Geometry of Metric Spaces. - Coarse Quasi-Isometries. - Some Classes of Metric Spaces. - Growth of Metric Spaces. - Amenability of Metric Spaces. - Coarse Ends. - Higson Corona and Asymptotic Dimension. - Part II Coarse Geometry of Orbits and Leaves. - Pseudogroups. - Generic Coarse Geometry of Orbits. - Generic Coarse Geometry of Leaves. - Examples and Open Problems. -