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Generic Coarse Geometry of Leaves

  • 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

Book
  • 4.1k Downloads

Part of the Lecture Notes in Mathematics book series (LNM, volume 2223)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Jesús A. Álvarez López, Alberto Candel
    Pages 1-9
  3. Part I

    1. Front Matter
      Pages 11-11
    2. Jesús A. Álvarez López, Alberto Candel
      Pages 13-28
    3. Jesús A. Álvarez López, Alberto Candel
      Pages 29-36
    4. Jesús A. Álvarez López, Alberto Candel
      Pages 37-42
    5. Jesús A. Álvarez López, Alberto Candel
      Pages 43-49
    6. Jesús A. Álvarez López, Alberto Candel
      Pages 51-63
    7. Jesús A. Álvarez López, Alberto Candel
      Pages 65-73
  4. Part II

    1. Front Matter
      Pages 75-75
    2. Jesús A. Álvarez López, Alberto Candel
      Pages 77-89
    3. Jesús A. Álvarez López, Alberto Candel
      Pages 91-114
    4. Jesús A. Álvarez López, Alberto Candel
      Pages 115-132
    5. Jesús A. Álvarez López, Alberto Candel
      Pages 133-162
  5. Back Matter
    Pages 163-173

About this book

Introduction

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.

Keywords

Asymptotic Dimension Coarse Quasi-isometry Foliated Space Growth Leaf

Authors and affiliations

  1. 1.Department and Institute of MathematicsUniversity of Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Department of MathematicsCalifornia State UniversityNorthridgeUSA

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.

Bibliographic information

  • Book Title Generic Coarse Geometry of Leaves
  • Authors Jesús A. Álvarez López
    Alberto Candel
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title Lect.Notes Mathematics
  • DOI https://doi.org/10.1007/978-3-319-94132-5
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-94131-8
  • eBook ISBN 978-3-319-94132-5
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages XV, 173
  • Number of Illustrations 16 b/w illustrations, 0 illustrations in colour
  • Topics Manifolds and Cell Complexes (incl. Diff.Topology)
  • Buy this book on publisher's site