Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS 15
where we used that the differential of f i at q ∈ B r (q i ) ′ has a bilipschitz constant
e Cεi and that the ratio of vol(B 2r (f i (q))) and vol(B r (q)) is for large i bounded by
a universal constant.
We are left with the case of r ∈ [1/2, 1]. By Lemma 3.2, f i and g i converge in the
weakly measured sense to isometries. Clearly this implies that f i ◦ g i also converges
in the weakly measured sense to an isometry. And this implies that the distortion
function dt fi◦gi
1 has averaged L 1 norm ≤ δ i on every unit ball in B ρi (p i ) for some
ρ i → ∞ and δ i → 0.
□
The next lemma explains the notion zooming in property.
Lemma 3.5. Let (M i , p 1 i ), (N i, p 2 i ) and f i be as above. Then there is a ρ i → ∞,
δ i → 0 and Ti
1 ⊂ B ρi (p 1 i ) such that the following holds
• vol(B 1 (q) ∩ Ti 1) ≥ (1 − δ i) vol(B 1 (q)) for all q ∈ B ρi/2(p 1 i ).
• For any sequence of real numbers λ i → ∞ and any sequence q i ∈ Ti
1
f i : (λ i M i , q i ) → (λ i N i , f i (q i ))
has the zooming in property. We say that f i is good on all scales at q i .
Proof. Let G j i = B 2r i
(p j i )′ and B j i = B 2r i
(p i ) \ B 2ri (p j i )′ . After adjusting r i → ∞
and ε i → 0 we may assume vol(B j i ) ≤ ε i vol(B 1 (q)) for all q ∈ B 2ri (p j i ). Let χ ij
be the characteristic function of B j i . By the weak 1-1 inequality there exists a
universal C such that the set
H j i := { x ∈ B ri/2(p j i ) | Mx(χ ij) ≥ √ }
ε i
satisfies
vol(H j i ) ≤ C√ ε i vol(B 1 (q))
for all q ∈ B 2ri (p j i ). We put T i 1 := ( ) ( )
B ri/2(p 1 i )\H1 i ∩ f
−1
i Bri/2(p 2 i )\H2 i and
Ti
2 := f i (Ti 1). Using Lemma 3.2 we can find ρ i → ∞ and δ i → 0 such that
vol ( B ρi (p j i ) \ T j )
i ≤ δi vol(B 1 (q)) for all q ∈ B ri (p j i ), j = 1, 2.
By definition of T j i
vol(B r (q) ∩ G j i )
vol(B r (q))
≥ 1 − √ ε i for all q ∈ T j i and all r ≤ 1, j = 1, 2.
Let dt λif
r
denote the distortion on scale r of f i : λ i M i → λ i N i . Clearly
dt λif
r (p, q) = λ i dt fi
r/λ i
(p, q).
Thus for all λ i → ∞ and all q i ∈ T j i
the zooming in property.
the map f i : (λ i M i , q i ) → (λ i N i , f i (q i )) has
□
Proposition 3.6 (First main example). Let α > 1. Consider a sequence of manifolds
(M i , p i ) with a fixed lower Ricci curvature bound and a sequence of time
dependent vector fields Xi
t (piecewise constant in time) with compact support. Let
c i : [0, 1] → B ri (p i ) be an integral curve of Xi t with c i(0) = p i