NON-UNIFORM LATTICES ON UNIFORM TREES 13

(6) We can therefore denote X as X — (A, z, ao). We recall from 1.2.2 that p determines

[-4aoe:ae.Ae] = 2(e) f° r e a c n e € EA, thus p determines (A,i).

(7) In fact, if x G VX, p(x) = a and e G Eo(a) then we have the local map on vertices

P(xy-Eo{x) -» J50(a) and

z ( e ) = | p - 1

)

( e ) | .

(8) Let (A, i) be an edge-indexed graph and let ao G V A It follows that every grouping

A of (J4,Z) gives rise to a group T = 7Ti(A, ao) that acts on X = (j4,2,ao) without

inversion and with quotient p: X — ^4 = T\X. If we replace A with its faithful quotient

we obtain a subgroup T G = Aut(X) whose stabilizers Tx and Te are isomorphic to

the vertex and edge groups Aa and Ae of A.

For a vertex a G VA, we define the degree of a in (A, i):

(9) deg{Aii)(a)= ^ i(e).

eeEo(a)

If A is a finite grouping of {A,i), then we have ([BL] 2.6 (15)):

(10) Vol(A) = j±-Vola(A,i).

1.4 Existence of finite groupings

In this section, we describe the fundamental ingredient for constructing discrete groups

which will be tree lattices. We shall start with an edge-indexed graph (A,i). A 'finite

grouping' A of (A,i) will give rise to a discrete group 7Ti(A,ao) which acts discretely on

the tree X = (A,i,ao). Our technique for constructing tree lattices will be described

fully in 2.3.

Let (A,i) be an edge-indexed graph. The fundamental problem 'the grouping game'

is the following:

(1) Find A with 1(A) = (A,i) and all groups finite.