Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS 25
By construction, for every point in B Lr (p) ′ there exists a vector field X t whose
integral curve connects it to a point in B Lρi (p) satisfying (22). For points covered
by several sets B r/2 (q m ) ′′ we pick any one. Then we have
∫B Lr (p) ′ ∫ 1
0
Mx ( ‖∇·Xnew‖ t ) 3 2
2 3
(c q (t)) dt dµ(q) ≤
≤
(13)
≤
≤
l∑
(C 9 (L) + 2 n ¯C2 )(rε i ) vol B r/2 (q m )
m=1
3 n (C 9 (L) + 2 n ¯C2 )(rε i ) vol B rL (p)
¯C 2 · Lrε i · vol B rL (p).
The last inequality holds if L = 9 n ≥ 2 · 2 n · 3 n and ¯C 2 ≥ 3 n · C 9 (L).
Recall that by (21)
vol B r/2 (q m ) − vol B r/2 (q m ) ′′ ≥ (C 8 (L) + 2 n ¯C1 )rε i vol B r/2 (q m )
for any m and thus
vol B Lr (p) − vol B Lr (p) ′
≤
(13)
≤
≤
l∑
(C 8 (L) + 2 n ¯C1 )rε i vol B r/2 (q m )
m=1
provided that L = 9 n ≥ 2 · 6 n and ¯C 1 ≥ 3 n C 8 (L).
This finishes the proof of Sublemma 5.2.
3 n (C 8 (L) + 2 n ¯C1 )rε i vol(B Lr (p))
¯C 1 Lrε i vol B Lr (p)
Observe that for any two good points x i , y i ∈ G 1 (p i ) we can deduce from Sublemma
5.2 that vol(B 2 (x i ) ′ ∩ B 2 (y i ) ′ ) ≥ vol(B 1/2 (p i )) for all large i.
Then it follows, also by Sublemma 5.2, that there is q ∈ B 2 (x i ) ′ ∩ B 2 (y i ) ′ ,
x ′ i ∈ B Lρ i
(x i ) and y i ′ ∈ B Lρ i
(y i ) such that for the integral curve c connecting x ′ i
with q on the first half of the interval and q with y i ′ on the second half we have for
the corresponding time dependent vector field Xi
t
∫ 1
(
Mx(‖∇·X t i ‖ 3/2 ) ) 2/3
(c(t)) dt ≤ ¯C3 ε i
□
0
with a constant ¯C 3 independent of i.
Let f i := φ i1 be the flow of X t i evaluated at time 1. Since λ i = 1 ρ i
, we can
employ Proposition 3.6 to see that
f i : [λ i M i , x i ] → [λ i M i , y i ]
has the zooming in property. Thus, the Gromov–Hausdorff limits of the two sequences
are isometric.
For any lifts ˜x i and ỹ i of x i and y i to the universal cover we can lifts ˜x ′ i and ỹ′ i
of x ′ i and y′ i in the Lρ i neighbourhoods of ˜x i and ỹ i . Let ˜f i : ˜Mi → ˜M i be the lift
f i with ˜f i (˜x ′ i ) = ỹ′ i . Proposition 3.6 ensures that ˜f i : [λ i ˜Mi , ˜x i ] → [λ i ˜Mi , ỹ i ] has the
zooming in property as well.
It remains to check the last part of the Rescaling Theorem concerning the fundamental
group. Assume that π 1 (M i , p i ) is generated by loops of length ≤ R.