Margulis Lemma

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6 VITALI KAPOVITCH AND BURKHARD WILKING d G−H (B Ni (p i ), B Ni (0)) → 0 as i → ∞. The functions b i j are obtained by solving the Dirichlet problem on B 2 (p i ) with b i i | ∂B 2(p i ) = fi i| ∂B 2(p i ) We will need a weak type 1-1 estimate for manifolds with lower Ricci curvature bounds which is a well-known consequence of the doubling inequality [Ste93, p. 12]. Lemma 1.4 (Weak type 1-1 inequality). Suppose (M n , g) has Ric ≥ −1 and let f : M → R be a nonnegative function. Define Mx f(p) := sup r≤1 − ∫ B r(p) f. Then the following holds (a) If f ∈ L α (M) with α ≥ 1 then Mx f is finite almost everywhere. (b) If f ∈ L 1 (M) then vol { x | Mx f(x) > c } ≤ C(n) ∫ c f for any c > 0. M (c) If f ∈ L α (M) with α > 1 then Mx f ∈ L α (M) and || Mx f|| α ≤ C(n, α)||f|| α . Note that as an immediate corollary we see that if f ∈ L α (M) with α > 1 then for any x ∈ M we have the following pointwise estimate (3) Mx((Mx f) α )(x) ≤ C(n, α) Mx(f α )(x). We will also need the following inequality (for any α > 1) which is an immediate consequence of the definition of Mx and Hölder inequality. (4) [Mx f(x)] α ≤ Mx(f α )(x). Indeed, for any r ≤ 1 we have that Mx(f α )(x) ≥ − ∫ B r(x) f α ≥ ( − ∫ B r(x) f) α and the inequality follows by taking the supremum over all r ≤ 1. Combining the last two inequalities we deduce (5) Mx[Mx(f)](x) ≤ Mx[(Mx(f)) α ] 1/α (x) ≤ C(n, α) 1/α Mx[f α ](x) for α > 1. Finally we note that for α > 1, β > 1 we have (6) Mx[Mx(f β/α )] α (x) ≤ C(n, α) Mx(f β )(x). This inequality immediately follows by applying (3) to g = f β α . We will also make use of the so-called segment inequality of Cheeger and Colding which says that in average the integral of a nonnegative function along all geodesics in ball can be estimated by the L 1 norm of the function. Theorem 1.5 (Segment inequality). [CC96] Given n and r 0 there exists τ = τ(n, r 0 ) such that the following holds. Let Ric(M n ) ≥ −(n−1) and let g : M → R + be a nonnegative function. Then for r ≤ r 0 ∫ − B r(p)×B r(p) ∫ d(z1,z 2) 0 ∫ g(γ z1,z 2 (t))dµ(z 1 )dµ(z 2 ) ≤ τ · r · − g(q) dµ(q), B 2r(p) where γ z1,z 2 denotes a minimal geodesic from z 1 to z 2 . Finally we will need the following observation Lemma 1.6 (Covering Lemma). There exists a constant C(n) such that the following holds. Suppose (M n , g) has Ric g ≥ −(n − 1). Let f : M → R be a nonnegative function, p ∈ M, π : ˜M → M be the universal cover of M, ˜p ∈ ˜M a lift of p, and let ˜f = f ◦ π. Then ∫ ∫ − ˜f ≤ C(n)− f. B 1(˜p) B 1(p)
STRUCTURE OF FUNDAMENTAL GROUPS 7 Proof. We choose a measurable section j : B 1 (p) → B 1 (˜p), i.e. π(j(x)) = x for any x ∈ B 1 (p). Let T = j(B 1 (p)). Then we obviously have that diam(T ) ≤ 2 and ∫ ∫ − f = − ˜f. B 1(p) Let S be the union of g(T ) over all g ∈ π 1 (M) such that g(T ) ∩ B 1 (˜p) ≠ ∅. It’s obvious from the triangle inequality that S ⊂ B 3 (˜p). It is also clear that ∫ ∫ − f = − ˜f B 1(p) and B 1 (˜p) ⊂ S. Lastly, notice that by Bishop–Gromov relative volume comparison vol B 3 (˜p) ≤ C(n) vol B 1 (˜p) for some universal constant C(n). Since B 1 (˜p) ⊂ S ⊂ B 3 (˜p), we also have vol B 3 (˜p) ≤ C(n) vol S and hence ∫ ∫ ∫ − ˜f ≤ C(n)− ˜f = C(n)− f. B 1(˜p) S T S B 1(p) It is easy to see that the above proof generalizes to an arbitrary cover of M. 1.3. Equivariant Gromov–Hausdorff convergence. Let Γ i be a sequence of closed subgroups of the isometry groups of metric spaces X i and ˜p i ∈ X i . If (X i , ˜p i ) converges in the pointed Gromov–Hausdorff topology to (Y, ˜p ∞ ), then after passing to a subsequence one can find a closed subgroup G ⊂ Iso(Y ) such that (X i , Γ i , ˜p i ) → (Y, G, ˜p ∞ ) in the equivariant Gromov–Hausdorff topology. For details we refer the reader to [FY92]. Definition 1.7. A sequence of subgroups Υ i ⊂ Γ i is called uniformly open, if there is some ε > 0 such that Υ i contains the set { g ∈ Γi | d(gq, q) < ε for all q ∈ B 1/ε (˜p i ) } for all i. Γ i is called uniformly discrete if {e} ⊂ Γ i is uniformly open. We say the sequence Υ i is boundedly generated if there is some R such that Υ i is generated by {g ∈ Υ i | d(g˜p i , ˜p i ) < R}. Lemma 1.8. Let Υ j i ⊂ Γ i be uniformly open with Υ j i → Υj ∞ ⊂ G, j = 1, 2. a) Υ 1 i ∩ Υ2 i is uniformly open and converges to Υ1 ∞ ∩ Υ 2 ∞. b) If g i ∈ Γ i converges to g ∞ ∈ G then g i Υ 1 i g−1 i is uniformly open and converges to g ∞ Υ 1 ∞g∞ −1 . c) Suppose in addition that Υ j i is boundedly generated, j = 1, 2. If Υ1 ∞ ∩ Υ 2 ∞ has finite index H in Υ 1 ∞, then Υ 1 i ∩ Υ2 i has index H in Υ1 i for all large i. The proof is an easy exercise. Example 1 (Z l actions converging to a Z 2 -action.). Consider a 2-torus T 2 k given by a Riemannian product S 1 ×S 1 where each of the factors has length k. Put l = k 2 +1 and let Z/lZ act on T 2 k by ( e 2πi k 2 k2πi ) +1 , e k 2 +1 . In the equivariant Gromov–Hausdorff topology the action will converge (for k → ∞) to the standard Z 2 action on R 2 . One can, of course, define a similar sequence of actions on S 3 × S 3 but in this case the actions will not be uniformly cocompact. □
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Margulis Lemma

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