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Tree Lattices

  • Hyman Bass
  • Alexander Lubotzky

Part of the Progress in Mathematics book series (PM, volume 176)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Hyman Bass, Alexander Lubotzky
    Pages 1-12
  3. Hyman Bass, Alexander Lubotzky
    Pages 13-16
  4. Hyman Bass, Alexander Lubotzky
    Pages 17-23
  5. Hyman Bass, Alexander Lubotzky
    Pages 25-33
  6. Hyman Bass, Alexander Lubotzky
    Pages 35-65
  7. Hyman Bass, Alexander Lubotzky
    Pages 67-72
  8. Hyman Bass, Alexander Lubotzky
    Pages 73-90
  9. Hyman Bass, Alexander Lubotzky
    Pages 91-102
  10. Hyman Bass, Alexander Lubotzky
    Pages 103-118
  11. Hyman Bass, Alexander Lubotzky
    Pages 119-149
  12. Hyman Bass, Alexander Lubotzky
    Pages 151-165
  13. Back Matter
    Pages 167-233

About this book

Introduction

Group actions on trees furnish a unified geometric way of recasting the chapter of combinatorial group theory dealing with free groups, amalgams, and HNN extensions. Some of the principal examples arise from rank one simple Lie groups over a non-archimedean local field acting on their Bruhat—Tits trees. In particular this leads to a powerful method for studying lattices in such Lie groups.

This monograph extends this approach to the more general investigation of X-lattices G, where X-is a locally finite tree and G is a discrete group of automorphisms of X of finite covolume. These "tree lattices" are the main object of study. Special attention is given to both parallels and contrasts with the case of Lie groups. Beyond the Lie group connection, the theory has application to combinatorics and number theory.

The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Non-uniform tree lattices are much more complicated than uniform ones; thus a good deal of attention is given to the construction and study of diverse examples. The fundamental technique is the encoding of tree action in terms of the corresponding quotient "graphs of groups."

Tree Lattices should be a helpful resource to researcher sin the field, and may also be used for a graduate course on geometric methods in group theory.

Keywords

Graph Group theory Number theory Sim Vertex combinatorics graph theory lie groups

Authors and affiliations

  • Hyman Bass
    • 1
  • Alexander Lubotzky
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsHebrew UniversityJerusalemIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-2098-5
  • Copyright Information Birkhäuser Boston 2001
  • Publisher Name Birkhäuser Boston
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7413-1
  • Online ISBN 978-1-4612-2098-5
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site