Tree Lattices

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.