Tree lattices

Author(s)

• Bass, Hyman
• Lubotsky, Alexander

Bibliographic Information

Tree lattices

Hyman Bass, Alexander Lubotzky ; with appendices by H. Bass ... [et al.]

（Progress in mathematics, v. 176）

Birkhäuser, c2001

• : us
• : sz
• : pbk.

Note

Includes bibliographical references (p. [223]-227) and indexes

Description and Table of Contents

Volume

: us ISBN 9780817641207

Description

[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 The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems.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.

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.

Volume

: pbk. ISBN 9781461274131

Description

This monograph extends this approach to the more general investigation of X-lattices, and these "tree lattices" are the main object of study. 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. Tree Lattices should be a helpful resource to researchers in the field, and may also be used for a graduate course on geometric methods in group theory.

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 Lattices on 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.

Volume

: sz ISBN 9783764341206

Description

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.

Table of Contents

• Lattices and volumes
• graphs of groups, and edge-indexed graphs
• tree lattices
• arbitrary real volumes, cups, and homology
• length functions, minimality
• centralizers, normalizers, and commensurators
• existence of tree lattices
• non-uniform latices on uniform trees
• parabolic actions, lattices, and trees. Appendix: the P Neumann groups
• lattices of Nagao Type.

by "Nielsen BookData"

Details

• NCID: BA49854977
• ISBN:
  • 0817641203
  • 3764341203
  • 9781461274131
• LCCN: 00059911
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Boston ; Basel
• Pages/Volumes: xii, 233 p.
• Size: 24 cm
• Classification:
  • LCC : QA166.2
  • DC21 : 511/.52
• Subject Headings:
  • LCSH : Trees (Graph theory)
  • LCSH : Lie groups
• Parent Bibliography ID:
  • BA00018010