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.
