1. Graphs of groups, tree actions and edge-indexed graphs
We shall assume that the reader is familiar with the basic aspects of the theory of
group actions on simplicial trees as developed by H. Bass and J.P. Serre ([B], [S]). One
of the essential ideas that emerges from this theory is that an action of a group on a tree
is completely encoded in a 'quotient graph of groups' (cf. 1.2.3). This fundamental fact
will allow us to construct tree lattices by constructing instead the appropriate graph of
groups (cf. section 2.3).
An additional aspect of the Bass-Serre theory is that the 'edge-indexed graph' of a
quotient graph of groups completely determines its universal covering tree up to isomor-
phism (cf. 1.2.3). As we shall see in section 2.3, this suggests a technique for constructing
tree lattices, starting only with the appropriate edge-indexed graph.
To this end, we begin with the necessary preliminaries on graphs of groups and edge-
indexed graphs. Additional references for this section are [BK], [BLJ.
1.1 Graphs of groups
Let A (VA,EA,do,di,—) denote a graph, with vertices VA, oriented edges EA,
initial and terminal functions do and d\ that pick out the endpoints of an edge, and an
involution, —, on the edge set that is fixed point free, and reverses the orientation. For
a vertex v G VA, let
E0(v) = {ee EA \d0e = v}.
Let A = (A, A) be a graph of groups, with vertex groups (AQ)aeVA, edge groups
(Ae = Ae)eeEA and monomorphisms ae:Ae Adoe-
Let r = 7Ti (A, ao) be the fundamental group of A with respect to a base point ao G V A
and let X (A, ao) be its universal covering tree (see [B], [S]).
(1) A fundamental fact of the Bass-Serre theory is that T acts on X without inversion
and with quotient projection p: X A T\X. Moreover, if e 6 EX, x = doe, p(x) a
and p(e) f then af. A/ «— » Aa is isomorphic to the inclusion Te
c
Tx.
(2) We call the graphs of groups A faithful if the action of T on X is faithful; that is, if
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