12 LISA CARBONE

(3) The fundamental theory of Bass and Serre tells us that

r^TT^A,^), a0 G VA,

X = (A,a

0

), a0eVA,

that is, we can naturally identify T with the fundamental group of F\\X, and X with

the universal covering tree of r\\A r .

It follows that every graph of groups A encodes an action of a group 7Ti (A, ao), ao G ^ ^

on a tree X = (A, ao), and conversely, every action of a group T on a tree X without

inversion arises from a quotient graph of groups T\\X.

1.3 Edge-indexed graphs and their groupings

Let A = (A, A) be a graph of groups. For e G EA, doe — a, put

(1) l(e) = [Aa'.OLeAe].

Thus i assigns a positive integer i(e) 0 to each oriented edge e G JEM. We assume that

all indices i(e) are finite.

(2) I f z ( e ) l , we say that e is a ramified edge. Otherwise, we say that e is unramified.

We call /(A) = (A,i) the edge-indexed graph associated to A, and we observe that every

edge-indexed graph (A,i) arises in this way from a graph of groups ([B]).

(3) Given an edge-indexed graph (A, z), a graph of groups A such that 1(A) = (A,i) is

called a grouping of (A,i). We call A a finite grouping if the vertex groups Aa are finite

and a faithful grouping if A is a faithful graph of groups.

(4) Let (A,i) be an edge-indexed graph and A a grouping of {A,i). Let ao G VA and

put X = (A, ao). A fundamental observation is the following: ([B] 1.18, [BL] 2.5)

(5) X = (A, ao) and the projection p: X — » A

depend only on (^,z,ao) and not on the grouping A.