Lecture Notes in Mathematics

Generic Coarse Geometry of Leaves

Authors: Álvarez López, Jesús, Candel, Alberto

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  • 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
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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.

About the authors

​Jesús A. Álvarez López, was born in 1962, and studied Mathematics at the University of Santiago de Compostela (Spain), obtaining the PhD in 1987. He visited the Department of Mathematics of the University of Illinois at Urbana-Champaign during 1988-1990, with a Fulbright grant. Nowadays, he is a professor at the University of Santiago de Compostela. His research is about several topics, like cohomology of foliations, description of equicontinuous foliated spaces, quasi-isometric types of leaves, leafwise heat flow, a trace formula for foliated flows, and Witten’s complex on stratified spaces. 
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.

Table of contents (11 chapters)

Table of contents (11 chapters)
  • Introduction

    Pages 1-9

    Álvarez López, Jesús A. (et al.)

  • Coarse Quasi-Isometries

    Pages 13-28

    Álvarez López, Jesús A. (et al.)

  • Some Classes of Metric Spaces

    Pages 29-36

    Álvarez López, Jesús A. (et al.)

  • Growth of Metric Spaces

    Pages 37-42

    Álvarez López, Jesús A. (et al.)

  • Amenability of Metric Spaces

    Pages 43-49

    Álvarez López, Jesús A. (et al.)

Buy this book

eBook $39.99
price for USA in USD (gross)
  • ISBN 978-3-319-94132-5
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA in USD
  • ISBN 978-3-319-94131-8
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Generic Coarse Geometry of Leaves
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2223
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-94132-5
DOI
10.1007/978-3-319-94132-5
Softcover ISBN
978-3-319-94131-8
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XV, 173
Number of Illustrations
16 b/w illustrations
Topics