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) 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.
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