Tree Lattices

[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lattices, and a (previously unpublished) development of the theory of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.

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.