Margulis Lemma

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44 VITALI KAPOVITCH AND BURKHARD WILKING finite index. This in turn implies that π 1 (M i )/N i is virtually cyclic and by a) we can assume that it is a fixed group F. The result (contradiction) will now follow algebraically from b): Let ρ j : F → GL(n j , Q) be a finite collection of irreducible rational representations j = 1, . . . , j 0 . Consider, for all nilpotent torsion free groups N of rank n − 1, all short exact sequences N → Γ → F with the property that there is a chain {e} = N 0 ⊳ · · · ⊳ N h0 = N such that [N, N h ] ⊂ N h−1 , each N h is normal in Γ, N h /N h−1 is free abelian group of positive rank and the induced representation of F in (N h /N h−1 )⊗ Z Q is in the finite collection. Using that F is virtually cyclic, it is now easy to see that this leaves only finitely many possibilities for the isomorphism type of Γ/N h0−1. This shows that if we replace N by N h0−1 we are still left with finitely many possibilities in a). – a contradiction. □ Corollary 9.4. Given n, D there is a constant C such that any finite fundamental group of an n-manifold (M, g) with Ric > −(n − 1), diam(M) ≤ D contains a nilpotent subgroup of index ≤ C which has a nilpotent basis of length ≤ (n − 1). Example 2. Our theorems rule out some rather innocuous families of groups as fundamental groups of manifolds with lower Ricci curvature bounds. a) Consider a homomorphism h: Z → GL(2, Z) whose image does not contain a unipotent subgroup of finite index. Put h d (x) = h(dx) and consider the group Z ⋉ hd Z 2 . By part b) of Theorem 9.3 the following holds: In a given dimension n and for a fixed diameter bound D only finitely many of these groups can be realized as fundamental groups of manifold with Ric > −(n − 1) and diam(M) ≤ D. b) Consider the action of Z p k−1 on Z p k induced by 1 ↦→ 1 + p. In a given dimension n we have that for k ≥ n and p > C Marg (constant in the Margulis Lemma) the group Z p k−1 ⋉Z p k can not be a subgroup of a compact manifold with almost nonnegative Ricci curvature, since it does not have a nilpotent basis of length ≤ n. Similarly, by Corollary 9.4, for a given D there are only finitely many primes p such that Z p n−1 ⋉ Z p n is isomorphic to the fundamental group of an n-manifold with Ric > −(n − 1) and diam(M) ≤ D. Problem. Let D > 0. Can one find a finite collection of 3-manifolds such that for any 3-manifold M with Ric > −1 and diam(M) ≤ D, there is a manifold T in the finite collection and a finite normal covering T → M for which the covering group contains a cyclic subgroup of index ≤ 2? 10. The Diameter Ratio Theorem The aim of this section is to prove Theorem 8. Lemma 10.1. Let X i be a sequence of compact inner metric spaces converging to a torus T. Then π 1 (X i ) is infinite for large i. The proof of the lemma is an easy exercise.
STRUCTURE OF FUNDAMENTAL GROUPS 45 Proof of Theorem 8. Suppose, on the contrary, that (M i , g i ) is a sequence of compact manifolds with diam(M i ) = D, Ric Mi > −(n − 1), #π 1 (M i ) < ∞ and the diameter of the universal cover ˜M i tends to infinity. By Corollary 9.4, we know that π 1 (M i ) contains a subgroup of index ≤ C(n, D) which has a nilpotent basis of length ≤ (n − 1). Thus, we may assume that π 1 (M i ) itself has a nilpotent basis of length u < n. We also may assume that u is minimal with the property that a contradicting sequence exist. Put ˆM i := ˜M i /[π 1 (M i ), π 1 (M i )]. Clearly [π 1 (M i ), π 1 (M i )] has a nilpotent basis of length ≤ u − 1. By construction ˆM i can not be a contradicting sequence and therefore diam( ˆM i ) → ∞. Let A i := π 1 (M i )/[π 1 (M i ), π 1 (M i )] denote the deck transformation group of the normal covering ˆM i → M i . 1 Claim. The rescaled sequence ˆM diam( ˆM i is precompact in the Gromov–Hausdorff i) topology and any limit space has finite Hausdorff dimension. The problem is, of course, that no lower curvature bound is available after rescaling. The claim will follow from a similar precompactness result for certain Cayley graphs of A i . Recall that diam(M i ) = D. Choose a base point ˆp i ∈ ˆM i . Let f 1 , . . . , f ki ∈ A i be an enumeration of all the elements a ∈ A i with d(ˆp i , aˆp i ) < 10D. There is no bound for k i , but clearly f 1 , . . . , f ki is a generator system of A i . We define a weighted metric on the abelian group A i as follows: ⎧ ⎨∑k i d(e, a) := min |ν j | · d(ˆp, f j ˆp) ⎩ j=1 ∣ ∏k i j=1 f νj j ⎫ ⎬ = a ⎭ for all a ∈ A i and d(a, b) = d(ab −1 , e) for all a, b ∈ A i . Note that this metric coincides with the restriction to A i of the inner metric on its Cayley graph where each edge corresponding to f j is given the length d(ˆp, f j ˆp). It is easy to see that the map ι i : A i → ˆM i , a i ↦→ a i ˆp i is a quasi isometry with uniform control on the constants involved. In fact, there is some L independent of i such that 1 L d(a, b) ≤ d(ι i(a), ι i (b)) ≤ Ld(a, b) for all a, b ∈ A i . and the image is of ι i is D-dense. Therefore, it suffices to show that 1 diam(A A i) i is precompact in the Gromov– Hausdorff topology and all limit spaces of convergent subsequences are finite dimensional. For the proof we need Subclaim. There is a R 0 > D (independent of i) such that the homomorphism h: A i → A i , x ↦→ x 2 satisfies that the R 0 neighborhood of h(B R (e)) contains (e), for all R and all i. B 3R 2 Using that B 10D (e) ⊂ A i is isometric to a subset of B 10D (ˆp i ), we can employ the Bishop–Gromov inequality in order to find a universal constant k such that B 10D (e) does not contain k points with pairwise distance ≥ D. Put R 0 := 10D · k.
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Margulis Lemma

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