Several coarse properties of leaves are shown to hold, either for residually many or for meagerly many leaves
New coarse concepts are introduced to study this residual-meager dichotomy
Numerous examples illustrate the results
Includes a variety of open problems
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2223)
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Table of contents (11 chapters)
About this book
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.
- Asymptotic Dimension
- Coarse Quasi-isometry
- Foliated Space
Authors and Affiliations
Department and Institute of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain
Jesús A. Álvarez López
Department of Mathematics, California State University, Northridge, USA
About the authors
Alberto Candel did his undergraduate work in Mathematics at the University of Santiago de Compostela, worked briefly at the Universidad de Oviedo (Spain), and then moved on to the USA do his graduate work at Washington University in St. Louis, obtaining his PhD in Mathematics in 1992 under the direction of L. Conlon. After postdoctoral work at several places (IAS, U of Chicago, and Caltech), he settled at California State University, Northridge in 2000. His research is in geometric analysis and dynamical systems, with particular emphasis foliations.
Book Title: Generic Coarse Geometry of Leaves
Authors: Jesús A. Álvarez López, Alberto Candel
Series Title: Lecture Notes in Mathematics
Publisher: Springer Cham
Copyright Information: Springer Nature Switzerland AG 2018
Softcover ISBN: 978-3-319-94131-8Published: 29 July 2018
eBook ISBN: 978-3-319-94132-5Published: 28 July 2018
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XV, 173
Number of Illustrations: 16 b/w illustrations