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