NON-UNIFORM LATTICES ON UNIFORM TREES 11

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

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)\-