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lattices, and a (previously unpublished) development of the theory of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