Margulis Lemma

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8 VITALI KAPOVITCH AND BURKHARD WILKING Remark 1.9. a) The example shows that it is difficult to relate an open subgroup in the limit to corresponding subgroups in the sequence. b) If g1, i . . . , gh i i is a generator system of some subgroup Γ ′ i ⊆ Γ i, then one can consider – after passing to a subsequence – a different limit construction to get a subgroup G ′ ⊂ G: For some fixed R consider all words w in g1, i . . . , gh i i with the property w ⋆ p i ∈ B R (˜p i ). This defines a subset in the Cayley graph of Γ ′ i . Consider now all elements in Γ′ i represented by words in the identity component of this subset. Passing to the limit gives a closed subset S R ⊂ G of the limit group and one can now take the limit of S R for R → ∞. This sort of word limit group can be much smaller than the regular limit group as the above examples shows. Although it depends on the choice of the generator system, there are occasions where this limit behaves more natural than the usual. However, this idea is used only indirectly in the paper. 1.4. Notations and conventions. Throughout the paper we will use the following conventions. • As already mentioned − ∫ S f dµ stands for ∫ 1 vol(S) S f dµ. • Mx(f)(x) denotes the maximum function of f evaluated at x, see Lemma 1.4 for the definition. • A map σ : X → Y between inner metric spaces is called a submetry if σ(B r (p)) = B r (σ(p)) for all r > 0 and p ∈ X. • If S ⊂ F is a subset of a group, then 〈S〉 denotes the subgroup generated by S. • If g 1 and g 2 are elements in a group, [g 1 , g 2 ] := g 1 g 2 g1 −1 g−1 2 denotes the commutator. For subgroups F 1 , F 2 we put [F 1 , F 2 ] := 〈{[f 1 , f 2 ] | f i ∈ F i }〉. • Generators b 1 , . . . , b n ∈ F of a group are called a nilpotent basis if [b i , b j ] ∈ 〈b 1 , . . . , b i−1 〉 for i < j. • N ⊳ F means: N is a normal subgroup of F. • For a Riemannian manifold (M, g) and λ > 0 we let λM denote the Riemannian manifold (M, λ 2 g). • For a limit space Y a tangent cone C p Y at p is some Gromov–Hausdorff limit of (λ i Y, p) for some λ i → ∞. Tangent cones are not always cones and are not necessarily unique. • For a limit space Y of manifolds with lower Ricci curvature bound a regular point p ∈ Y is a point all of whose tangent cones at p are given by R kp . By a result of Cheeger and Colding [CC97] these points have full measure and are thus dense. • For two sequences of pointed metric spaces (X i , p i ), (Y i , q i ) and maps f i : X i → Y i we use the notation f i : [X i , p i ] → [Y i , q i ] to indicate that f i converges in some sense to a map from the pointed Gromov–Hausdorff limit of (X i , p i ) to the pointed Gromov–Hausdorff limit of (Y i , q i ). It usually does not mean that f i (p i ) = q i . However, if this is the case we write f i : (X i , p i ) → (Y i , q i ). • If a group G acts on a metric space we say g ∈ G displaces p ∈ X by r if d(p, gp) = r. We will denote the orbit of p by G ⋆ p. • Gromov’s short generator system (or short basis for short) of a fundamental group is defined at the beginning of section 1.
STRUCTURE OF FUNDAMENTAL GROUPS 9 2. Finite generation of fundamental groups Lemma 2.1 (Product Lemma). Let M i be a sequence of manifolds with Ric Mi > −ε i → 0 satisfying • B ri (p i ) is compact for all i with r i → ∞ and p i ∈ M i , • for every i and j = 1, . . . , k there are harmonic functions b i j : B r i (p i ) → R which are L-Lipschitz and fulfill ∫ k∑ k∑ − | < ∇b i j, ∇b i l > −δ jl | + ‖Hess b i j ‖ 2 dµ → 0 for all R > 0. B R (p i) j,l=1 j=1 Then (B ri (p i ), p i ) subconverges in the pointed Gromov–Hausdorff topology to a metric product (R k ×X, p ∞ ) for some metric space X. Moreover, (b i 1, . . . , b i k ) converges to the projection onto the Euclidean factor. The lemma remains true if one just has a uniform lower Ricci curvature bound and one can also prove a local version of the lemma if r i = R is a fixed number. However, the above version suffices for our purposes. Proof. The main problem is to prove this in the case of k = 1. Put b i = b i 1. After passing to a subsequence we may assume that (B ri (p i ), p i ) converges to some limit space (Y, p ∞ ). We also may assume that b i converges to an L-Lipschitz map b ∞ : Y → R. Step 1. b ∞ is 1-Lipschitz. This is essentially immediate from the segment inequality. Let x, y ∈ Y be arbitrary. Choose R so large that x, y ∈ B R/4 (p ∞ ) and let x i , y i ∈ B R/2 (p i ) be sequences converging to x and y. For a fixed δ
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Margulis Lemma

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