Generic Coarse Geometry of Leaves

Description

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. Front Matter ....Pages i-xv Introduction (Jesús A. Álvarez López, Alberto Candel)....Pages 1-9 Front Matter ....Pages 11-11 Coarse Quasi-Isometries (Jesús A. Álvarez López, Alberto Candel)....Pages 13-28 Some Classes of Metric Spaces (Jesús A. Álvarez López, Alberto Candel)....Pages 29-36 Growth of Metric Spaces (Jesús A. Álvarez López, Alberto Candel)....Pages 37-42 Amenability of Metric Spaces (Jesús A. Álvarez López, Alberto Candel)....Pages 43-49 Coarse Ends (Jesús A. Álvarez López, Alberto Candel)....Pages 51-63 Higson Corona and Asymptotic Dimension (Jesús A. Álvarez López, Alberto Candel)....Pages 65-73 Front Matter ....Pages 75-75 Pseudogroups (Jesús A. Álvarez López, Alberto Candel)....Pages 77-89 Generic Coarse Geometry of Orbits (Jesús A. Álvarez López, Alberto Candel)....Pages 91-114 Generic Coarse Geometry of Leaves (Jesús A. Álvarez López, Alberto Candel)....Pages 115-132 Examples and Open Problems (Jesús A. Álvarez López, Alberto Candel)....Pages 133-162 Back Matter ....Pages 163-173