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and number theory.The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems.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.
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 The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems.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.
Arvustused
"The book is a helpful resource to researchers in the field and students of geometric methods in group theory."
--Educational Book Review
Preface xi Introduction 1(12) Tree lattices 1(1) X-lattices and H-lattices 2(1) Near simplicity 3(1) The structure of tree lattices 4(1) Existence of lattices 4(2) The structure of A = Γ\X 6(1) Volumes 7(1) Centralizers, normalizers, commensurators 8(5) Lattices and Volumes 13(4) Haar measure 13(1) Lattices and unimodularity 13(1) Compact open subgroups 13(1) Discrete group covolumes 14(3) Graphs of Groups and Edge-Indexed Graphs 17(8) Graphs 17(1) Morphisms and actions 18(1) Graphs of groups 18(1) Quotient graphs of groups 19(1) Edge-indexed graphs and their groupings 19(1) Unimodularity, volumes, bounded denominators 20(5) Tree Lattices 25(10) Topology on G = AutX 25(1) Tree lattices 25(1) The group GH of deck transformations 26(1) Discreteness Criterion; Rigidity of (A, i) 27(2) Unimodularity and volume 29(1) Existence of tree lattices 30(1) The structure of tree lattices 31(2) Non-arithmetic uniform commensurators 33(2) Arbitrary Real Volumes, Cusps, and Homology 35(32) Introduction 35(1) Grafting 36(2) Volumes 38(5) Cusps 43(1) Geometric parabolic ends 44(5) Γ-parabolic ends and Γ-cusps 49(2) Unidirectional examples 51(3) A planar example 54(13) Length Functions, Minimality 67(6) Hyperbolic length (cf. (B3), II, §6) 67(1) Minimality 68(3) Abelian actions 71(1) Non-abelian actions 71(1) Abelian discrete actions 71(2) Centralizers, Normalizers, and Commensurators 73(18) Introduction 73(1) Notation 74(3) Non-minimal centralizers 77(3) N/Γ, for minimal non-abelian actions 80(1) Some normal subgroups 81(1) The Tits Independence Condition 82(3) Remarks 85(1) Automorphism groups of rooted trees 86(2) Automorphism groups of ended trees 88(2) Remarks 90(1) Existence of Tree Lattices 91(12) Introduction 91(2) Open fanning 93(5) Multiple open fanning 98(5) Non-Uniform Lattices on Uniform Trees 103(16) Carbones Theorem 103(7) Proof of Theorem (8.2) 110(1) Remarks 110(1) Examples. Loops and cages 111(5) Two vertex graphs 116(3) Parabolic Actions, Lattices, and Trees 119(32) Introduction 119(1) Ends (X) 120(1) Horospheres and horoballs 121(1) End stabilizers 122(1) Parabolic actions 123(2) Parabolic trees 125(1) Parabolic lattices 125(1) Restriction to horoballs 126(1) Parabolic lattices with linear quotient 127(4) Parabolic ray lattices 131(8) Parabolic lattices with all horospheres infinite 139(4) A bounded degree example 143(5) Tree lattices that are simple groups must be parabolic 148(1) Lattices on a product of two trees 149(2) Lattices of Nagao Type 151(16) Nagao rays 151(6) Nagaos Theorem: Γ = PGL2(Fq(t)) 157(3) A divisible (q + 1)-regular grouping 160(1) The PNeumann groupings 161(2) The symmetric groupings 163(1) Product groupings 164(3) Appendix (BCR): The Existence Theorem for Tree Lattices 167(18) Hyman Bass Lisa Carbone Gabriel Rosenberg Appendix (BT): Discreteness Criteria for Tree Automorphism Groups 185(28) Hyman Bass Jacques Tits Appendix (PN): The PNeumann Groups 213(10) Hyman Bass Alexander Lubotzky References 223(6) Index of Notation 229(2) Index 231
