Tree Lattices

1. Front Matter

   Pages i-xii

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2. Introduction

   • Hyman Bass, Alexander Lubotzky

   Pages 1-12

3. Lattices and Volumes

   • Hyman Bass, Alexander Lubotzky

   Pages 13-16

4. Graphs of Groups and Edge-Indexed Graphs

   • Hyman Bass, Alexander Lubotzky

   Pages 17-23

5. Tree Lattices

   • Hyman Bass, Alexander Lubotzky

   Pages 25-33

6. Arbitrary Real Volumes, Cusps, and Homology

   • Hyman Bass, Alexander Lubotzky

   Pages 35-65

7. Length Functions, Minimality

   • Hyman Bass, Alexander Lubotzky

   Pages 67-72

8. Centralizers, Normalizers, and Commensurators

   • Hyman Bass, Alexander Lubotzky

   Pages 73-90

9. Existence of Tree Lattices

   • Hyman Bass, Alexander Lubotzky

   Pages 91-102

10. Non-Uniform Lattices on Uniform Trees

    • Hyman Bass, Alexander Lubotzky

    Pages 103-118

11. Parabolic Actions, Lattices, and Trees

    • Hyman Bass, Alexander Lubotzky

    Pages 119-149

12. Lattices of Nagao Type

    • Hyman Bass, Alexander Lubotzky

    Pages 151-165

13. Back Matter

    Pages 167-233

    PDF

[UPDATED 6/6/2000] 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 $\Gamma$, where $X$ is a locally finite tree and $\Gamma$ 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 applications 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 The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems.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 unifrom ones; thus a good deal of attention is given to the construction and study of diverse examples. Some interesting new phenomena are observed here which cannot occur in the case of Lie groups. The fundamental technique is the encoding of tree actions in terms of the corresponding quotient "graph of groups." {\it Tree Lattices} should be a helpful resource to researchers in the field, and may also be used for a graduate course in geometric group theory.

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

"The book is a helpful resource to researchers in the field and students of geometric methods in group theory."

--Educational Book Review

• Department of Mathematics, University of Michigan, Ann Arbor, USA

  Hyman Bass

• Department of Mathematics, Hebrew University, Jerusalem, Israel

  Alexander Lubotzky

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