(3) The fundamental theory of Bass and Serre tells us that
r^TT^A,^), a0 G VA,
X = (A,a
), 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.
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