Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS 13
3. Maps which are on all scales close to isometries.
For a map f : X → Y between metric spaces we define the distance distortion
on scale r by
(7) dt f r (p, q) = min{r, |d(p, q) − d(f(p), f(q))|}.
Definition 3.1. Let (M i , p 1 i ) and (N i, p 2 i ) be two sequences of Riemannian manifolds
with B i (p j i ) being compact and Ric B i(p j i )
> −C for some C, all i and j = 1, 2.
We say that a sequence of diffeomorphisms f i : M i → N i has the zooming in property
if the following holds:
There exist R 0 > 0, sequences r i → ∞, ε i → 0 and subsets B 2ri (p j i )′ ⊂ B 2ri (p j i )
(j = 1, 2) satisfying
a) vol ( B 1 (q) ∩ B 2ri (p j i )′) ≥ (1 − ε i ) vol(B 1 (q)) for all q ∈ B ri (p j i ).
b) For all p ∈ B ri (p 1 i )′ , all q ∈ B ri (p 2 i )′ and all r ∈ (0, 1] we have
∫
− dt fi
r (x, y)dµ(x)dµ(y) ≤ rε i and
B
∫ r(p)×B r(p)
− dt f −1
i
r (x, y)dµ(x)dµ(y) ≤ rε i .
B r(q)×B r(q)
c) There are subsets S j i ⊂ B 1(p j i ) with vol(Sj i ) ≥ 1 2 vol( B 1 (p j i )) (j = 1, 2) and
f(S 1 i ) ⊂ B R 0
(p 2 i ) and f −1 (S 2 i ) ⊂ B R 0
(p 1 i ).
We will call elements of B 2ri (p 1 i )′ good points and sometimes use the convention
B r (q) ′ := B 2ri (p j i )′ ∩ B r (q) for all B r (q) ⊂ B 2ri (p j i ).
In the applications we will always have N i = M i . However, in some instances
d(p 1 i , p2 i ) → ∞. That is why it might be helpful to also think about maps between
two unrelated pointed manifolds. If the choice of the base points is not clear we
will say that f i : [M i , p 1 i ] → [N i, p 2 i ] has the zooming in property. Notice that we
do not require f i (p 1 i ) = p2 i . However, property c) ensures that the base points are
respected in a weaker sense.
Lemma 3.2. Let (M i , p 1 i ), (N i, p 2 i ) and f i be as above. Then, after passing to a
subsequence, (M i , p 1 i ) G−H
−→ (X 1, p 1 ∞), (N i , p 2 i ) G−H
−→ (X 2, p 2 ∞) and f i converges in
i→∞ i→∞
the weakly measured sense to a measure preserving isometry f ∞ : X 1 → X 2 , that is
for each r > 0 there is a sequence δ i → 0 and subsets S i ⊂ B r (p 1 i ) satisfying
• vol(S i ) ≥ (1 − δ i ) vol(B r (p 1 i )) and
• f i|Si is pointwise close to f ∞|Br(p ∞).
Moreover, f −1
i
converges in this sense to f −1
∞ .
First we claim that the following holds.
Sublemma 3.3. There exists C 1 (n) such that for any good point x ∈ B ri (p 1 i )′ and
any r ≤ 1 there is a subset B r (x) ′′ ⊂ B r (x) with
vol B r (x) ′′ ≥ (1 − C 1 ε i ) vol B r (x) and f i (B r (x) ′′ ) ⊂ B 2r (f i (x)).
Proof. The proof proceeds by induction on the size of r as follows. It is clear that
at good points x ∈ B ri (p 1 i )′ the differential of f i has a bilipschitz constant e Cεi
with some universal C provided ε i < 1/2. Therefore it is clear that the statement