Overview

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-archimidean local field acting on their Bruhat-Tits trees. In particular this leads to a powerful method for studying lattices in such Lie groups.

Full Product Details

Author:   H. Bass ,  Hyman Bass ,  L. Carbone ,  Alexander Lubotzky
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   2001 ed.
Volume:   176
Dimensions:   Width: 16.10cm , Height: 1.60cm , Length: 24.10cm
Weight:   0.520kg
ISBN:  

9780817641207

ISBN 10:   0817641203
Pages:   233
Publication Date:   17 November 2000
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Awaiting stock  
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Table of Contents

0 Introduction.- 0.1 Tree lattices.- 0.2 X-lattices and H-lattices.- 0.3 Near simplicity.- 0.4 The structure of tree lattices.- 0.5 Existence of lattices.- 0.6 The structure of A = ?\X.- 0.7 Volumes.- 0.8 Centralizers, normalizers, commensurators.- 1 Lattices and Volumes.- 1.1 Haar measure.- 1.2 Lattices and unimodularity.- 1.3 Compact open subgroups.- 1.5 Discrete group covolumes.- 2 Graphs of Groups and Edge-Indexed Graphs.- 2.1 Graphs.- 2.2 Morphisms and actions.- 2.3 Graphs of groups.- 2.4 Quotient graphs of groups.- 2.5 Edge-indexed graphs and their groupings.- 2.6 Unimodularity, volumes, bounded denominators.- 3 Tree Lattices.- 3.1 Topology on G = AutX.- 3.2 Tree lattices.- 3.3 The group GH of deck transformations.- 3.5 Discreteness Criterion; Rigidity of (A, i).- 3.6 Unimodularity and volume.- 3.8 Existence of tree lattices.- 3.12 The structure of tree lattices.- 3.14 Non-arithmetic uniform commensurators.- 4 Arbitrary Real Volumes, Cusps, and Homology.- 4.0 Introduction.- 4.1 Grafting.- 4.2 Volumes.- 4.8 Cusps.- 4.9 Geometric parabolic ends.- 4.10 ?-parabolic ends and ?-cusps.- 4.11 Unidirectional examples.- 4.12 A planar example.- 5 Length Functions, Minimality.- 5.1 Hyperbolic length (cf. [B3], II, §6).- 5.4 Minimality.- 5.14 Abelian actions.- 5.15 Non-abelian actions.- 5.16 Abelian discrete actions.- 6 Centralizers, Normalizers, and Commensurators.- 6.0 Introduction.- 6.1 Notation.- 6.6 Non-minimal centralizers.- 6.9 N/?, for minimal non-abelian actions.- 6.10 Some normal subgroups.- 6.11 The Tits Independence Condition.- 6.13 Remarks.- 6.16 Automorphism groups of rooted trees.- 6.17 Automorphism groups of ended trees.- 6.21 Remarks.- 7 Existence of Tree Lattices.- 7.1 Introduction.- 7.2 Open fanning.- 7.5 Multiple open fanning.- 8 Non-Uniform Latticeson Uniform Trees.- 8.1 Carbone’s Theorem.- 8.6 Proof of Theorem (8.2).- 8.7 Remarks.- 8.8 Examples. Loops and cages.- 8.9 Two vertex graphs.- 9 Parabolic Actions, Lattices, and Trees.- 9.0 Introduction.- 9.1 Ends(X).- 9.2 Horospheres and horoballs.- 9.3 End stabilizers.- 9.4 Parabolic actions.- 9.5 Parabolic trees.- 9.6 Parabolic lattices.- 9.8 Restriction to horoballs.- 9.9 Parabolic lattices with linear quotient.- 9.10 Parabolic ray lattices.- 9.13 Parabolic lattices with all horospheres infinite.- 9.14 A bounded degree example.- 9.15 Tree lattices that are simple groups must be parabolic.- 9.16 Lattices on a product of two trees.- 10 Lattices of Nagao Type.- 10.1 Nagao rays.- 10.2 Nagao’s Theorem: r = PGL2(Fq[t]).- 10.3 A divisible (q + l)-regular grouping.- 10.4 The PNeumann groupings.- 10.5 The symmetric groupings.- 10.6 Product groupings.

Reviews

The book is a helpful resource to researchers in the field and students of geometric methods in group theory. --Educational Book Review

The book is a helpful resource to researchers in the field and students of geometric methods in group theory. --Educational Book Review

<p> The book is a helpful resource to researchers in the field and students of geometric methods in group theory. <p>--Educational Book Review

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