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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.
This is the second of two volumes on the qualitative theory of
foliations. For this volume, the authors have selected three
special topics: analysis on foliated spaces, characteristic classes
of foliations, and foliated manifolds. Each of these is an example
of deep interaction between foliation theory and some other
highly-developed area of mathematics. In all cases, the authors
present useful, in-depth introductions, which lead to further study
using the extensive available literature. This comprehensive volume
has something to offer a broad spectrum of readers: from beginners
to advanced students to professional researchers. It contains
exercises and many illustrations. The book would make an elegant
supplementary text for a topics course at the advanced graduate
level. ""Foliations I"" is Volume 23 in the AMS series, ""Graduate
Studies in Mathematics"".
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