Tree Lattices

by Bass, Hyman; Lubotzky, Alexander

• ISBN13: 9780817641207
• ISBN10: 0817641203
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: Springer Verlag

Summary

[UPDATED 6/6/2000]Group actions on trees furnish a unified geometric way of recastingthe chapter of combinatorial group theory dealing with free groups,amalgams, and HNN extensions. Some of the principal examples arisefrom rank one simple Lie groups over a non-archimedean local fieldacting on their Bruhat--Tits trees. In particular this leads to apowerful method for studying lattices in such Lie groups.This monograph extends this approach to the more general investigationof $X$-lattices $\Gamma$, where $X$ is a locally finite tree and$\Gamma$ is a discrete group of automorphisms of $X$ of finitecovolume. These "tree lattices" are the main object of study.Special attention is given to both parallels and contrasts with thecase of Lie groups. Beyond the Lie group connection, the theory hasapplications to combinatorics and number theory.The authors present a coherent survey of the results on uniform treelattices, and a (previously unpublished) development of the theory ofnon-uniform tree lattices, including some fundamental and recentlyproved existence theorems. Non-uniform tree lattices are much morecomplicated than unifrom ones; thus a good deal of attention is givento the construction and study of diverse examples. Some interestingnew phenomena are observed here which cannot occur in the case of Liegroups. The fundamental technique is the encoding of tree actions interms of the corresponding quotient "graph of groups."{\it Tree Lattices} should be a helpful resource to researchers in thefield, and may also be used for a graduate course in geometric grouptheory.

Table of Contents

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)
Carbone's 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)
Nagao's 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