Margulis Lemma

More documents

Recommendations

Info

26 VITALI KAPOVITCH AND BURKHARD WILKING For every good point q ∈ B 1 (p i ) let r i (q) denote the minimal number such that π 1 (M i , q) can be generated by loops of length ≤ r i (q). Let r i denote the supremum of r i (q) over all good points q and choose a good point q i with r i (q i ) ≥ 1 2 r i. It suffices to check that lim sup i→∞ λ i r i ≤ 1. Suppose we can find a subsequence with λ i r i ≥ 1/4 for all i. By the Product Lemma (2.1), the sequence ( 1 r i M i , q i ) subconverges to (R k × K ′ , q ∞ ) with diam(K ′ ) ≤ 4 · 10 −n2 . Combining Lemma 2.2 with Lemma 2.3 we see that π 1 ( 1 r i M i , q i ) can be generated by loops of length ≤ 2 diam(K ′ ) + c i with c i → 0 – a contradiction. □ 6. The Induction Theorem for C-Nilpotency C-nilpotency of fundamental groups of manifolds with almost nonnegative Ricci curvature (Corollary 2) will follow from the following technical result. Theorem 6.1 (Induction Theorem). Suppose (M n i , p i) is a sequence of pointed n-dimensional Riemannian manifolds (not necessarily complete) satisfying (1) Ric Mi ≥ −1/i; (2) B i (p i ) is compact for any i; (3) There is some R > 0 such that π 1 (B R (p i )) → π 1 (M i ) is surjective for all i; (4) (M i , p i ) G−H −→ i→∞ (Rk × K, (0, p ∞ )) where K is compact. Suppose in addition that we have k sequences f j i : [ ˜M i , ˜p i ] → [ ˜M i , ˜p i ] of diffeomorphisms of the universal covers ˜M i of M i which have the zooming in property, where ˜p i is a lift of p i , and which normalize the deck transformation group acting on ˜M i , j = 1, . . . k. Then there exists a positive integer C such that for all sufficiently large i, π 1 (M i ) contains a nilpotent subgroup N ⊳ π 1 (M i ) of index at most C such that N has an (f j i )C! -invariant (j = 1, . . . , k) cyclic nilpotent chain of length ≤ n − k, that is: We can find {e} = N 0 ⊳ · · · ⊳ N n−k = N such that [N, N h ] ⊂ N h−1 and each factor group N h+1 /N h is cyclic. Furthermore, each N h is invariant under the action of (f j i )C! by conjugation and the induced automorphism of N h /N h+1 is the identity. Although we will have f j i = id in the applications, it is crucial for the proof of the theorem by induction to establish it in the stated generality. Proof. The proof proceeds by reverse induction on k. For k = n the result follows immediately from Theorem 1.2. We assume that the statement holds for all k ′ > k and we plan to prove it for k. We argue by contradiction. After passing to a subsequence we can assume that any subgroup N ⊂ π 1 (M i ) of index ≤ i does not have nilpotent cyclic chain of length ≤ n − k which is invariant under (f j i )i! (j = 1, . . . , k). After passing to a subsequence, ( ˜M i , ˜p i ) converges to (R¯k × ˜K, (0, ˜p ∞ )), where ˜K contains no lines. Moreover, we can assume that the action of π 1 (M i ) converges with respect to the pointed equivariant Gromov–Hausdorff topology to an isometric action of a closed subgroup G ⊂ Iso(R¯k × ˜K). It is clear that the metric quotient (R¯k × ˜K)/G is isometric to R k ×K. Using that lines in R k × K can be lifted to lines in R¯k × ˜K, we deduce that R¯k = R k × R l and the action of G on the first Euclidean factor is trivial, (cf. proof of Lemma 2.3). Since the action of G on R l × ˜K and hence on ˜K is cocompact, we can use an
STRUCTURE OF FUNDAMENTAL GROUPS 27 observation of Cheeger and Gromoll [CG72] to deduce that ˜K is compact for there are no lines in ˜K. By passing once more to a subsequence, we can assume that f j i converges in the weakly measured sense to an isometry f∞ j on the limit space, j = 1, . . . , k. The overall most difficult step is essentially a consequence of the Rescaling Theorem: Step 1. Without loss of generality we can assume that K is not a point. We assume K = pt. The strategy is to find a new contradicting sequence converging to R k × K ′ where K ′ ≠ pt. After rescaling every manifold down by a fixed factor we may assume that the limit isometry f∞ j displaces (0, p ∞ ) by less than 1/100, j = 1, . . . , k. We choose the set of ”good” points G 1 (p i ) as in Rescaling Theorem 5.1. We let G 1 (˜p i ) denote those points in B 1 (˜p i ) projecting to G 1 (p i ). By Lemma 1.6, vol(G1(˜pi)) vol(B → 1. 1(˜p i)) By Lemma 3.5, we can remove small subsets H i ⊂ G 1 (p i ) (and the corresponding subsets from G 1 (˜p i )) such that ≤ δ i → 0 and for any choice of points vol H i vol G 1(p i) ˜q i ∈ G 1 (˜p i )\H i the sequence f j i and (f j i )−1 (i ∈ N) is good on all scales at ˜q i , j = 1, . . . , k. To simplify notations we will assume that it is already true for all choices of ˜q i ∈ G 1 (˜p i ). For all large i we can choose a point ˜q i ∈ G 1/2 (˜p i ) with f j i (˜q i) ∈ G 1 (˜p i ), j = 1, . . . , k. Let q i be the image of ˜q i in M i and ¯f j i : M i → M i be the induced diffeomorphism. By Rescaling Theorem 5.1, there is a sequence g j i of isotopic to identity diffeomorphisms g j i : [λ iM i , ¯f j i (q i)] → [λ i M i , q i ] with the zooming in property. Moreover, by the same theorem, we can find lifts ˜g j i of g j i such that ˜g j i : [λ i ˜M i , f j i (˜q i)] → [λ i ˜Mi , ˜q i ] has the zooming in property. Using Lemma 3.4 we see that f j i,new := f j i ◦ ˜gj i : [λ i ˜M i , ˜q i ] → [λ i ˜Mi , ˜q i ] has the zooming in property as well for j = 1, . . . , k. Since g j i is isotopic to the identity, it follows that conjugation by ˜g j i induces an inner automorphism of π 1(M i ). Therefore f j i,new produces the same element αj i ∈ Out(π 1(M i )) as f j i . The Rescaling Theorem also ensures that π 1 (λ i M i , q i ) remains boundedly generated. Finally, it states that (λ i M i , q i ) converges to (R k × K, (0, q ∞ )) with K being a compact space with diam(K) = 10 −n2 . Thus, (λ i M i , q i ) and the maps f j i,new on the universal covers give a new contradicting sequence with the limit satisfying K ≠ pt. From now on we will assume that it’s true for the original contradicting sequence. Step 2. Without loss of generality we can assume that f j i converges in the measured sense to the identity map of the limit space R k × R l × ˜K, j = 1, . . . , k. We prove this by finite induction on j. Suppose we already found a contradicting sequence where fi 1, . . . , f j−1 i converge to the identity. We have to construct one where in addition f i := f j i converges to the identity.
Page 1 and 2: STRUCTURE OF FUNDAMENTAL GROUPS OF
Page 3 and 4: STRUCTURE OF FUNDAMENTAL GROUPS 3 c
Page 5 and 6: STRUCTURE OF FUNDAMENTAL GROUPS 5 s
Page 7 and 8: STRUCTURE OF FUNDAMENTAL GROUPS 7 P
Page 9 and 10: STRUCTURE OF FUNDAMENTAL GROUPS 9 2
Page 11 and 12: STRUCTURE OF FUNDAMENTAL GROUPS 11
Page 25: STRUCTURE OF FUNDAMENTAL GROUPS 25

Margulis Lemma

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?