Margulis Lemma

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28 VITALI KAPOVITCH AND BURKHARD WILKING We first consider the induced diffeomorphisms ¯f i : M i → M i which converge to an isometry ¯f ∞ of R k × K. Thus, there is A ∈ O(k) and v ∈ R k such that the induced isometry of the Euclidean factor pr( ¯f ∞ ): R k → R k is given by (w ↦→ Aw + v). We claim that we can assume without loss of generality that v = 0. In fact, by Proposition 3.8, we can find a sequence of diffeomorphisms g i : M i → M i with the zooming in property such that g i converges to an isometry g ∞ which induces translation by −v on the Euclidean factor. Moreover, there is a lift ˜g i : ˜Mi → ˜M i which also has the zooming in property if we endow ˜M i with the base point ˜p i . By Lemma 3.4, we are free to replace f i by ˜g i ◦ f i and hence v = 0 without loss of generality. Since ¯f ∞ fixes the origin of the Euclidean factor, we obtain that for the limit f ∞ of f i the following property holds. Given any m > 0 we can find g m ∈ G such that d(f∞(g m m (0, ˜p ∞ )), (0, ˜p ∞ )) ≤ diam(K) in R k × R l × ˜K. It is now an easy exercise to find a sequence ν b of natural numbers and a sequence of g b ∈ G for which f ν b ∞ ◦ g b converges to the identity: We consider a finite ε-dense set {a 1 , . . . , a N } in the ball of radius 1 ε around (0, ˜p ∞ ) ∈ R k+l × ˜K. For each integer m choose a g m ∈ G such that d ( f m ∞( gm (0, ˜p ∞ ) ) , (0, p ∞ ) ) ≤ diam(K). The elements (f∞(g m m a 1 ), · · · , f∞(g m m a N )) are contained in B 1/ε+diam(K ′ )(0, ˜p ∞ ). Thus, there are m 1 ≠ m 2 such that d(f∞ m1 (g m1 a j ), f∞ m2 (g m2 a j )) < ε for j = 1, . . . , N. Consequently, d(f∞ m1−m2 (ga j ), a j ) < ε for j = 1, . . . , N with g = f∞ m2−m1 gm −1 1 f∞ m1−m2 g m2 ∈ G. In summary, f∞ m1−m2 ◦ g displaces any point in the ball of radius 1 ε around (0, p ∞) by at most 4ε. Since ε was arbitrary, this clearly proves our claim. Let (ν b , g b ) be as above. For each b we choose a large i = i(b) ≥ 2ν b and g b ∈ π 1 (M i ) such that f ν b i ◦ g b is in the measured sense close to f ν b ∞ ◦ g b . We also choose i so large that (f ν b i(b) ◦ g b) b∈N still has the zooming in property and converges to the identity. This finishes the proof of Step 2. Further Notations. We may replace p i with any other point p ∈ B 1/2 (p i ). Thus we may assume that p ∞ is a regular point in K. Let π 1 (M i , p i , r) = π 1 (M i , p i )(r) denote the subgroup of π 1 (M, p i ) generated by loops of length ≤ r. Similarly, recall that G(r) is the group generated by elements which displace (0, ˜p ∞ ) by at most r. By the Gap Lemma (2.4) there is an ε > 0 and ε i → 0 such that π 1 (M i , p i , ε) = π 1 (M i , p i , ε i ). Moreover, G(r) = G(ε) for all r ∈ (0, 2ε]. We put Γ i := π 1 (M i , p i ) and Γ iε = π 1 (M i , p i , ε). Step 3. The desired contradiction arises if the index [Γ i : Γ iε ] is bounded by some constant C for all large i. If C denotes a bound on the index and d = C!, then (f j i )d leaves the subgroup Γ iε invariant and we can find a new contradicting sequence for the manifolds ˜M i /Γ iε . Thus we may assume that Γ i = Γ iε . As we have observed above, π 1 (M i , p i ) = π 1 (M i , p i , ε i ) for some ε i → 0. We now choose a sequence λ i → ∞ very slowly such that • (λ i M i , p i ) → (R k × C p∞ K, o) = (R k′ , 0) with k ′ > k. • π 1 (λ i M i , p i ) is generated by loops of length ≤ 1.
STRUCTURE OF FUNDAMENTAL GROUPS 29 • f j i : λ ˜M i i → λ i ˜Mi still has the zooming in property and still converges to the identity of the limit space. We put f k+1 i = · · · = fi k′ = id and obtain a new sequence contradicting our induction assumption. Step 4. There is H > 0 and a sequence of uniformly open subgroups (see Definition 1.7) Υ i ⊂ Γ iε such that the index [Γ iε : Υ i ] ≤ H and such that Υ i is normalized by a subgroup of Γ i with index ≤ H for all large i. After passing to a subsequence we may assume that Γ iε converges to the limit action of a closed subgroup G ε ⊂ G. Clearly G ε contains the open subgroup G(ε) ⊂ G. Since the kernel of pr is compact (as a closed subgroup of Iso( ˜K)), the images pr(G) and pr(G ε ) are closed subgroups. Moreover, pr(G ε ) is an open subgroup of pr(G) since it contains pr(G(ε)) ⊃ pr(G) 0 . Using Fukaya and Yamaguchi [FY92, Theorem 4.1] or that the component group of pr(G) is the fundamental group of the nonnegatively curved manifold pr(G)\ Iso(R l ), we see that pr(G)/ pr(G) 0 is virtually abelian. Choose a subgroup G ′ ⊳ G of finite index such that pr(G ′ )/ pr(G) 0 is free abelian. Let G ′ ε = G ′ ∩ G ε . Then pr(G) 0 ⊂ pr(G ′ ε) ⊂ pr(G ′ ). Moreover, pr(G ′ ε) is normal in pr(G ′ ) and pr(G ′ )/ pr(G ′ ε) is a finitely generated abelian group. The inverse image Ĝε := pr −1 (pr(G ε )) ∩ G ′ is a normal subgroup of G ′ with the same quotient, i.e. G ′ /Ĝε = pr(G ′ )/ pr(G ′ ε). Since the kernel of pr is compact, the open subgroup G ′ ε is cocompact in Ĝε and thus its index [Ĝε : G ′ ε] is finite. In particular, we can say that there is some integer H 1 > 0 and a subgroup G ′ with [G : G ′ ] ≤ H 1 such that [G ε : (gG ε g −1 ∩ G ε )] < H 1 for all g ∈ G ′ . We want to check that there are only finitely many possibilities for gG ε g −1 ∩ G ε where g ∈ G ′ . Let g ∈ G ′ . Notice that Γ iε is a uniformly open sequence of subgroups. Choose g i ∈ Γ i with g i → g. By Lemma 1.8, Υ i := g i Γ iε g −1 i ∩ Γ iε converges to gG ε g −1 ∩ G ε and the index of Υ i in Γ iε is ≤ H 1 for all large i. By Theorem 2.5, the number of generators of Γ iε is bounded by some a priori constant. Therefore, Γ iε contains at most h 0 = h 0 (n, H 1 ) subgroups of index ≤ H 1 . Combining these statements we see that {gG ε g −1 ∩ G ε | g ∈ G ′ } = {g h G ε g −1 h ∩ G ε | h = 1, . . . , h 0 } for suitably chosen elements g 1 , . . . , g h0 ∈ G ′ . We choose g hi ∈ Γ i converging to g h ∈ G. By Lemma 1.8, the sequence of subgroups Υ i := is uniformly open and converges to Υ ∞ := ⋂h 0 i=1 ⋂h 0 h=1 g h G ε g −1 h g hi Γ iε g −1 hi = ⋂ g∈G ′ gG ε g −1 . Clearly, Υ ∞ ⊳G ′ . It is not hard to see that we can find elements c 1 i , . . . , cτ i ∈ Γ i that generate a subgroup of index ≤ [G : G ′ ] such that each c j i converges to an element c j ∞ ∈ G ′ . Since c j i Υ i(c j i )−1 is uniformly open and converges to c j ∞Υ ∞ (c j ∞) −1 = Υ ∞ , one can now apply Lemma 1.8 c) to see that c j i normalizes Υ i for large i.
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Margulis Lemma

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