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Overview"[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 The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems.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." Full Product DetailsAuthor: H. Bass , Hyman Bass , L. Carbone , Alexander LubotzkyPublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: Softcover reprint of the original 1st ed. 2001 Volume: 176 Dimensions: Width: 15.50cm , Height: 1.30cm , Length: 23.50cm Weight: 0.390kg ISBN: 9781461274131ISBN 10: 1461274133 Pages: 233 Publication Date: 17 September 2011 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of Contents0 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.ReviewsThe 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 The book is a helpful resource to researchers in the field and students of geometric methods in group theory. --Educational Book Review Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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