the defining homomorphism p: T Aut(X) is injective. In general, any A (A, A) has a
'faithful quotient' A' = (^4, A') with groups (Afa)aeVA and (Afe)eeEA which are quotients
of (Aa)aeVA and (Ae = Ae)eeEA respectively, so that the diagrams, for doe = a,
i i
commute and induce bijections:
(see [B], 1.24- 1.25).
(3) It follows that A and A' have the same universal covering tree X. We conclude that
if a graph of groups A is not faithful, we can always pass to its faithful quotient A'.
(4) If A is a graph of finite groups Aa, we can define the volume of A by
aeVA l ^
1.2 Group actions on trees and quotient graphs of groups
(1) Let X be a tree and suppose that T acts on X without inversion; that is, for each
e e ^ , r - e / r - e .
In this case, we can form the quotient graph p: X A = T\X, and we build a graph
of groups
A = (A,A) = T\\X
on the quotient graph A = T\X so that for e G EX, x doe, p(x) a, and p(e) = /,
the embedding af-Aj Aa is isomorphic to the inclusion Te ^- Tx.
(2) In particular,
lAa:afAf] = [Tx:Te} = \TX e\ = \p^(f)\-
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