154 4. PROOF OF THE COMPACTNESS THEOREMThe following proposition about the composition of approximate isome-
tries will be used in the construction of the directed system (see subsection
2.3 of this chapter for the definition) when proving Theorem 3.9.PROPOSITION 4.7 (Composition of approximate isometries, I). Thereexist Gp for p E N such that if i : (Mr, gi) ---+ ( Mr+l, gi+I) are ( Ei, p )-
approximate isometries for i = 0, 1 with Ei :::; 1, then
sup sup l\7 0 ({)!g2 - go) Igo :::; co+ c1Gp,
osrspxEM1sup sup l\72 ((<I>1^1 )* (<I>0-^1 )* go - g2) I :::; EI+ coGp,
Osrsp xEM2 g2
where we have denoted \7 i ~ \7 gi.PROOF. Using (4.1), we estimatel<I>{)<I>!g2 - go Igo :::; l<I>{)<I>!g2 - <I>{)g1lg 0 + l<I>{)g1 - golgo
:::; (1 +co) l<I>!g2 - gilg 1 + c:o
:::; (1 + c:o) c1 +co.
By Corollary 4.6 for 1 :::; r:::; pl\7(i (<I>{)<I>!g2) Igo = l\7(i (<I>{)<I>!g2 - go) Igo
:::; l\7(i (<I>{)<I>!g2 - <I>{)g1) Igo+ l\7(i (<I>{)g1 - go) Igo:::; (1 + co)(r+^2 )/^2 (1\71 (<I>!g2 - gi)lg 1+ coGp,0,2 I: I \7~ ( <I>!g2 - gi) I ) +co
k=O gi
:::; (1 + co)(r+^2 )/^2 (c:1 + c:0Gr,o,2rc1) +co.By symmetry we have
I (1
1)* (o-
1)* go - g21g2 :::; (1 + c1) co+ c1,
l\70 ( ( <I>1^1 )* ( <I> 01 )*go) lg
2:::; (1 + c:1)(r+^2 )/^2 (co+ c1 Gr,o,2rco) + c1.The proposition is proved for Gp = 2CP+^2 )/^2 (1 + pGp,o, 2 ).
COROLLARY 4.8 (Composition of approximate isometries, II). If
i : (Mr, gi) ---+ (Mr+l, gi+1)
Dare (ci,P)-approximate isometries for i = 0, 1, ... k, then k o · · · o 1 o o:
(M(), go) ---+ (Mk+l' gk+I) is a (Gp 2:,~=0 Ei,P )-approximate isometry.
PROOF. We shall induct on the number of compositions. By induction,k-1 o · · · o o is a (Gp I:,~,:;~ ci,P )-approximate isometry. By Proposition