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