Ricci curvature and the fundamental group

Transcription

1 University of Toronto

2 University of Toronto

3 Structure of the talk 1 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature 2 Bochner s formula Segment Inequality Stability Structure of limit spaces of manifolds with lower bounds Submetries

4 Structure of the talk 3 Main Results 4 of s Gap Lemma 5 Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. 6 Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency

5 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Theorem (Toponogov ) Let (M n, g) be a complete Riemannian manifold of K sec 0. Let ABC be a geodesic triangle in M and let à B C be a triangle in R 2, i.e. AB = à B, AC = à C, CB = C B. Then α α, β β, γ γ. M n, K 0 R 2, K = 0 B B α α, β β, γ γ β β α α à A γ γ C C

6 Gromov s short basis Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Given a complete Riemannian manifold (M, g) and a point p M let Γ = π 1 (M) acting on the universal cover M and let p be a lift of p. For γ Γ we will refer to γ := d( p, γ( p)) as the norm of γ. Choose γ 1 Γ with the minimal norm in Γ. Next choose γ 2 to have minimal norm in Γ\ γ 1. On the i-th step choose γ i to have minimal norm in Γ\ γ 1, γ 2,..., γ i 1. The sequence {γ 1, γ 2,...} is called a short basis of Γ at p. In general, the number of elements of a short basis can be infinite. For any i > j we have γ i γ 1 j γ i. γ j ( p) p γ j γ 1 j γ i γ i γ i ( p) γ i γ 1 j γ i

7 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature If M is closed then γ i 2 diam M for all i and {γ 1, γ 2,...} is finite. γ(t i ) σ i γ(t i+1 ) p σ i+1 γ(t i+2 ) γ(t) γ = i (σ iγ [ti t i+1 ]σ 1 i+1 ) L(σ i ) D L(γ [ti t i+1 ]) ε L(σ i γ [ti t i+1 ]σ 1 i+1 ) 2D + ε

8 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Theorem (Gromov) Let (M n, g) have sec k, diam D. Then π 1 (M) can be generated by C(n, k, D) elements. Proof. Let γ 1,..., γ i,... be a short basis of π 1 (M). Let v i T p M be the direction of a shortest geodesic from p to γ i ( p). Then for any i j by above we have that the angle v i v j = α α π/3. This means that the vectors v 1, v 2,... S n 1 are at least π/3-separated which immediately implies the result.

9 K 0 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature γ j ( p) v j p γ j α v i γ 1 j γ i γ i α α π 3 γ i ( p) γ i γ 1 j γ i

10 Bishop-Gromov volume Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Theorem (Bishop-Gromov s Relative Volume Comparison) Suppose M n has Ric M (n 1)k. Then (1) In particular, (2) (3) Vol ( B r (p)) Vol( B k r (0)) and Vol (B r(p)) Vol(B k r (0)) are nonincreasing in r. Vol (B r (p)) Vol(B k r (0)) for all r > 0, Vol (B r (p)) Vol (B R (p)) Vol(Bk r (0)) Vol(BR k (0)) for all 0 < r R, and equality holds if and only if B r (p) is isometric to B k r (0). Here B k r (0) is the ball of radius r in the n-dimensional simply connected space of constant curvature k. Note that this implies that if the volume of a big ball has a lower bound, then all smaller balls also have lower volume bounds

11 Idea of the proof for K sec 0 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Consider the following contraction map f : B R (p) B r (p). Given x B R (p) let γ : [0, 1] M be a shortest geodesic from p to x. Define f(x) = γ( r R ). At points where the geodesics are not unique we choose any one. By Toponogov we have that f(x)f(y) r R xy Therefore VolB r (p) Volf(B R (p)) ( r R )n VolB R (p)

12 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature f(x) p B(p, r) x f(y) B(p, R) y f(x)f(y) xy f(x) 0 f(x) f(y) xỹ = r R x R 2 f(y) ỹ

13 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Corollary (δ-separated net bound) Let (M n, g) have Ric k(n 1). Let p M, 0 < δ < R/2. Suppose x 1,... x N is a δ-separated net in B R (p). Then Proof. N C(n, R, δ) By Bishop-Gromov we have VolB δ/2 (x i ) c(n, R, δ)volb 2R (p). Since the balls B δ/2 (x i ) are disjoint we have VolB 2R (p) i VolB δ/2(x i ) N c(n, R, δ)volb 2R (p) result follows.

14 VolB δ/2 (x i ) VolB 2R (p) c(n, δ, R) The balls B δ/2 (x i ) are disjoint therefore Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature R x k B δ/2(x k ) p 2R x i x j x i x j δ there can only be so many of them.

15 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Definition Let X, Y be two compact inner metric spaces. A map f : X Y is called an ɛ- approximation if f(p)f(q) pq ɛ for any p, q X; For any y Y there exists p X such that f(p)y ɛ We define the distance between X and Y as d G H (X, Y ) = inf ɛ such that there exist ɛ- approximations from X to Y and from Y to X. distance turns out to be a distance on the set of isometry classes of compact inner metric spaces. Remark If f : X Y is an ɛ- approximation then there exist g : Y X which is a 2ɛ- approximation. In particular, d G H (X, Y ) 2ɛ.

16 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Definition G H Let X i X. Suppose Γ i is a closed sub of Isom(X i ) and i Γis a closed sub of Isom(X). We say that (X i, Γ i ) converges to (X, Γ) in equivariant topology if there is ε i 0 such that Remark For any g Γ there is g i Γ i which is ε i -close to g; For any g i Γ i there is g Γ which is ε i -close to g. One can similarly define pointed (equivariant) of pointed proper spaces.

17 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature The following observation of Gromov is crucial. Theorem (Gromov) Let M be a class of compact inner metric spaces satisfying the following property. There exists a function N : (0, ) (0, ) such that for any δ > 0 and any X M there are at most N(δ) points in X with pairwise distances δ. Then M is precompact in the topology. By the Corollary on δ-separated net bound this immediately implies Corollary The class M Ric (n, k, D) of complete n-manifolds with Ric k(n 1), diam D is precompact in the the topology. The class of complete pointed n-maniolfds (M n, p) with Ric k(n 1) is precompact in the pointed topology.

18 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Remark The Corollary on δ-separated net bound also easily implies that limit points of M Ric (n, k, D) have dimension n. Theorem (Yamaguchi) Let Mi n N m as i in topology. where sec(mi n ) k and N is a smooth manifold. Then for all large i there exists a fiber bundle F i Mi n N. Example Consider S 3 with the metric g ε given by (S 3 S 1 ε )/S 1 where S 1 ε is the circle of radius ε and S 1 acts on S 3 S 1 ε diagonally by the Hopf action on S 3 and by rotations on S 1 ε. Then (S 3, g ε ) has sec 0 and (S 3, g ε ) converges to the round S 2 of radius 1 2 as ε 0.

19 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Theorem (Toponogov alternate formulation) Let (M n, g) be a complete Riemannian manifold of K sec 0. Let ABC be a geodesic triangle in M and let à B C be a triangle in R 2. Let D be a point on the side BC and let B be the point on the side B C with BD = B D and CD = C D. Then AD à D. M n, K 0 B D AD à D R 2, K = 0 B D A C à C

20 How concave is that? Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature This means that distance function to a point is more concave than the distance function to a point in the space of constant curvature. How concave is that? Definition Define md k (r) by the formula r 2 2 if r = 0 1 md k (r) = k (1 cos( kr)) if k > 0 1 k (1 cosh( k r)) if k < 0 Then md k (0) = 0, md k(0) = 1 and md k + kmd k 1

21 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Let f(x) = md k ( xp ) where p S n k - simply connected space form of constant curvature k we have Hess x f = (1 kf(x))id. In particular, for any unit speed geodesic γ(t) we have that f(γ(t)) + kf(γ(t)) = 1 Note that for k = 0 this means that Hess x f = Id and f(γ(t)) = 1

22 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Theorem (Toponogov restated) Let M n have sec k and p M. Let f(x) = md k ( xp ). Then Hess x f (1 kf(x))id and For any unit speed geodesic γ. f(γ(t)) + kf(γ(t)) 1 These inequalities can be understood in the barrier sense or in the following sense. Definition A function f : M R is called λ-concave if for any unit speed geodesic γ(t) we have f(γ(t)) + λt2 2 is concave

23 Why do we care? Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature This means that manifolds with sec k naturally possess a LOT of semiconcave functions. Why do we care? They provide a useful technical tools. Specifically, the following simple observation is of crucial importance. Theorem Gradient flow of a concave function f on a complete Riemannian manifold M n is 1-Lipshitz. Proof. Let p, q M and let γ : [0, d] M be a unit speed geodesic with γ(0) = p, γ(d) = q. Here d = pq. Let φ t be the gradient flow of f. Then φ t (p)φ t (q) +(0) L(φ t (γ)) (0)

24 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Aside. Recall First variation formula Let γ : [0, d] ( ε, ε) M n be a smooth family of curves with c(s) = γ(s, 0) being a unit speed geodesic. Let X = tγ(s, t) be the variation vector field. Then L(γ t ) (0) = X(d, 0), c (d) X(0, 0), c (0) X(0, 0) c (0) γ t (s) γ 0 (s) = c(s) c (d) X(d, 0)

25 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Applying first variation formula gives L(φ t (γ)) (0) = q f, γ (d) p f, γ (0) = f(γ(d)) f(γ(0)) 0 since f(γ(s)) is concave. p γ(s) φ t (p) p f φ t (γ(s)) φ t (q) q q f

26 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature A similar argument shows that if f is λ-concave then φ t is e λt -Lipschitz. Yamaguchi s Fibration Theorem and gradient flows of semi-concave functions are key technical tools for proving topological results about manifolds with. Theorem (Almost splitting theorem) Let (Mi n, p i) G H 1 i. Suppose X contains a (X, p) where sec M i i line. Then X is isometric to Y R for some metric space Y. Remark If X n is an Alexandrov space of curv 0 and X contains a line then X = Y R for some Y.

27 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature If sec M n 0 and M is closed then one can use the Splitting Theorem to show that M = R k K where K is compact. This can be used to show Theorem (Cheeger-Gromoll) If sec M n 0 and M is closed then a finite cover of M is diffeomorphic to T k K where K is simply connected. Theorem (Gromov) Let M n have sec k, diam D. Then n β i (M) C(n, k, D) i=0 Theorem (Perelman s stability theorem) Let Mi n be a sequence of compact manifolds with sec k converging to X where dim X = n. Then M i is homeomorphic to X for all large i.

28 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Example Let sec N k and let f : N R be convex. Let f i : N R be smooth convex functions converging to f. Let c be any value in the range of f different from max f. Then {f i = c} is a smooth manifold of sec k and {f i = c} converges to {f = c} with respect to intrinsic metrics. {f = c} {f i = c} G H i {f i = c} X = {f = c}

29 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Corollary (Grove-Petersen-Wu) The class of n-manifolds with sec k, diam D, Vol v contains C(n, k, D, v) homeomorphism types. Definition A closed smooth manifold M is called almost nonnegatively curved if it admits a sequence of Riemannian metrics {g i } on M whose sectional curvatures and diameters satisfy sec(m, g i ) 1/i and diam(m, g i ) 1/i.

30 Toponogov Short basis. Bishop-Gromov volume Equivariant Yamaguchi s fibration theorem Semiconcave functions Topological results for sectional curvature Example Let N 3 be the space of real 3 3 of the form 1 x z 0 1 y N 3 is a nilpotent Lie. Let Γ = N SL(3, Z). Then M 3 = N/Γ admits almost nonnegative sectional curvature. But it does not admit nonnegative sectional curvature because Γ is not virtually abelian. Theorem (C-Nilpotency Theorem for π 1, K-Petrunin-Tuschmann, 2006) Let M be an almost nonnegatively curved m-manifold. Then π 1 (M) is C(m)-nilpotent, i.e., π 1 (M) contains a nilpotent sub of index at most C(m).

31 Bochner s formula Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries The most basic tool in studying manifolds with bounds is Bochner s formula. Theorem (Bochner s formula) For a smooth function f on a complete Riemannian manifold (M n, g), 1 2 f 2 = Hessf 2 + f, ( f) + Ric( f, f).

32 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Bochner s formula applied to distance functions can be used to prove Theorem (Global Laplacian Comparison) Let Ric M n (n 1)k, p M and let f(x) = md k ( xp ). Then x f (1 kf(x))n in the weak sense. Recall Theorem (Toponogov restated) Let M n have sec k and p M. Let f(x) = md k ( xp ). Then Hess x f (1 kf(x))id

33 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries By applying the Bochner formula to f = log u with an appropriate cut-off function and looking at the maximum point one has Cheng-Yau s gradient estimate for harmonic functions. Theorem (Gradient Estimate, Cheng-Yau 1975) Let Ric M n (n 1)k on B R2 (p) and u: B R2 (p) R satisfying u > 0, u = 0. Then for R 1 < R 2, on B R1 (p), (4) u u c(n, k, R 1, R 2 ).

34 Bishop-Gromov volume Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Theorem (Bishop-Gromov s Relative Volume Comparison) Suppose M n has Ric M (n 1)k. Then (5) In particular, (6) (7) Vol ( B r (p)) Vol( B k r (0)) and Vol (B r(p)) Vol(B k r (0)) are nonincreasing in r. Vol (B r (p)) Vol(B k r (0)) for all r > 0, Vol (B r (p)) Vol (B R (p)) Vol(Bk r (0)) Vol(BR k (0)) for all 0 < r R, and equality holds if and only if B r (p) is isometric to B k r (0). Here B k r (0) is the ball of radius r in the n-dimensional simply connected space of constant curvature k.

35 Segment Inequality Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Theorem (Cheeger Colding Segment inequality) 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) [ xy 0 g(γ x,y (t)) ] dµ x dµ y τ r B 2r (p) g(q) dµ q where γ x,y denotes a minimal geodesic from x to y.

36 γ x,y (t) y x Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries r p 2r

37 Comparison to Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Toponogov Comparison does not hold Fibration Theorem does not hold L 2 -Toponogov holds for long thin triangles (Colding) Almost splitting theorem holds Theorem (Cheeger Colding Almost splitting theorem) Let (Mi n, p i) G H (X, p) with Ric(M i) 1 i i Then X splits isometrically as X = Y R.. Suppose X has a line. As far as I know, this crucial property does not hold for any synthetic definition of metric spaces with lower bounds. For example, Banach spaces satisfy such definitions but fail this property.

38 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Example Topological stability theorem aka Perelman does not hold There exists a sequence of metrics on K3 with Ric 0 converging to T 4 /Z 2 where the Z 2 action is given by complex conjugation (z 1, z 2, z 3, z 4 ) ( z 1, z 2, z 3, z 4 ). The quotient T 4 /Z 2 is not a topological manifold as neighbourhoods of fixed points are homeomorphic to cones over RP 3.

39 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Theorem (Volume and topological stability. Colding, Cheeger-Colding) Let Mi n G H N n with Ric(M i ) k(n 1) and N is a closed i smooth manifold. Then VolM i VolN. For all large i approximations M i N are close to diffeomorphisms. Corollary Let (Mi n, p i) (R n, 0) with Ric Mi > 1. Then B R (p i ) is contractible in B R+ε (p i ) for all i i 0 (R, ε).

40 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Theorem Suppose (Mi n, p i) (R k, 0) with Ric Mi 1/i. Then there exist harmonic functions b i 1,..., b i k : B 2(p i ) R such that b i j C(n) for all i and j and B(p i,1) b i j, b i l δ j,l + Hess b i j 2 dµ 0 as i. j,l j Moreover, the maps Φ i = (b i 1,..., b i k ): M i R k provide ε i -Gromov approximations between B 1 (p i ) and B 1 (0) with ε i 0.

41 x j i R k M i f i j(x) = xx j i R Re j p i r = 2 f j (x) = x Re j R x j Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries x 1 i 0 r = 2 Re 1

42 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries The functions b i j in the above theorem are constructed as follows. Fix a large R. Approximate Busemann functions f j in R k given by f j = d(, Re j )) R are lifted to M i using approximations to corresponding functions fj i. Here e j is the j-th coordinate vector in the standard basis of R k. The functions b i j are obtained by solving the Dirichlet problem on B(p i, 2) with b i j B(p i,2) = fj i B(p i,2). Note that fj i(x) 3 on B(p i, 2) refore b i j is uniformly bounded on B(p i, 1) by Chen-Yau gradient estimate.

43 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Lemma (Weak type 1-1 inequality) Suppose (M n, g) has Ric (n 1) and let f : M R be a nonnegative function. Define Mx f(p) := sup r 1 B r f. Then the (p) 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 M c > 0. (c) If f L α (M) with α > 1 then Mx f L α (M) and Mx f α C(n, α) f α. If f L α (M) with α > 1 then we have pointwise Mx((Mx f) α )(x) C(n, α) Mx(f α )(x).

44 What is known about limit spaces? Recall that the class of pointed n-manifolds with Ric k(n 1) is precompact in pointed Gromov topology. Let (X, q) be a limit point of this class. Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Definition A tangent cone, T p X, at p X is the pointed limit of a sequence of the rescaled spaces (λ i X, p), where λ i as i. Tangent cones need not be unique and need not be metric cones. Definition A point, y X, is called regular if for some k, every tangent cone at y is isometric to R k. Theorem (Cheeger-Colding) The set of regular points in X has full measure and in particular is dense.

45 Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Definition A map π : X Y is called a submetry if π(b r (x)) = B r (π(y)) for any p X, r > 0. Example Let G act on X by isometries. Then π : X X/G is a submetry. Submetries are 1-Lipschitz. Fibers of submetries are equidistant. Exercise Let X be proper and let π : X Y be a submetry. Let γ(t) be a shortest geodesic in Y and let p X be a lift of γ(0), i.e. π(p) = γ(0). Then there exists a lift of γ starting at p, i.e. there exists a geodesic γ in X such that γ(0) = p and π( γ(t)) = γ(t).

46 F x F y p γ(t) X Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries π x γ(t) y Y

47 Fundamental results Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries Sectional curvature If sec M > δ > 0 then π 1 (M) is finite. (Myers) If sec M 0 and M is compact then π 1 (M) is virtually abelian. (Cheeger-Gromoll) If sec M n k, diam D then π 1 (M) is generated by C(n, k, D) elements. If RicM > δ > 0 then π 1 (M) is finite. (Myers) If RicM 0 and M is compact then π 1 (M) is virtually abelian. (Cheeger-Gromoll)???

48 Fundamental results Bochner s formula Segment Inequality Stability Structure of limit spaces Submetries If M n admits almost nonnegative sectional curvature then π 1 (M) is C(n)-nilpotent. (KPT) If M n admits almost nonnegative sectional curvature and π 1 (M) is diam M finite then diam M C(n). Fukaya-Yamaguchi (incorrect proof) Conjecture (Gromov): If M n admits almost nonnegative Ricci curvature then π 1 (M) is virtually nilpotent.??? If sec M n 0 then π 1 (M) is finitely generated. Conjecture (Milnor): If Ric M n 0 then π 1 (M) is finitely generated.

49 joint with B. Wilking Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Theorem ( of ) Given n, k, D there exists C(n, k, D) such that for any n-manifold with Ric k(n 1) and diam(m, g) D, the π 1 (M) can be generated by at most C elements. We call a generator system b 1,..., b n of a N a nilpotent basis if the commutator [b i, b j ] is contained in the sub b 1,..., b i 1 for 1 i < j n. Having a nilpotent basis of length n implies in particular that N is nilpotent of rank(n) n.

50 joint with B. Wilking Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Theorem (Generalized Margulis Lemma) In each dimension n there are positive constants C(n), ε(n) such that the following holds for any complete n-dimensional Riemannian manifold (M, g) with Ric > (n 1) on a metric ball B 1 (p) M. The image of the natural homomorphism π 1 ( Bε (p), p ) π 1 ( B1 (p), p ) contains a nilpotent sub N of index C(n). Moreover, N has a nilpotent basis of length at most n. We will also show that equality in this inequality can only occur if M is homeomorphic to an infranilmanifold.

51 joint with B. Wilking Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Corollary (π 1 of Ricci almost nonnegatively curved manifolds) Let (M, g) be a compact manifold with Ric > (n 1) and diam(m) ε(n) then π 1 (M) contains a nilpotent sub N of index C(n). Moreover, N has a nilpotent basis of length n. Conjecture (Milnor) If M n is open with Ric 0 then π 1 (M) is finitely generated. Corollary Let (M, g) be an open n-manifold with nonnegative. Then π 1 (M) contains a nilpotent sub N of index C(n) such that any finitely generated sub of N has a nilpotent basis of length n.

52 joint with B. Wilking Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Milnor: if Ric M 0 and π 1 (M) is finitely generated then it has polynomial growth. Gromov: If Γ is finitely generated and has polynomial growth then it is virtually nilpotent. Problem Rule out M n with Ric 0 and π 1 (M) = Q. Theorem (Compact Version of the Margulis Lemma) Given n and D there are positive constants ε 0 and C such that the following holds: If (M, g) is a compact n-manifold M with Ric > (n 1) and diam(m) D, then there is ε ε 0 and a normal sub N π 1 (M) such that for all p M: 1 the image of π 1 (B ε/1000 (p), p) π 1 (M, p) contains N, 2 the index of N in the image of π 1 (B ε (p), p) π 1 (M, p) is C and 3 N is a nilpotent which has a nilpotent basis of length n.

53 joint with B. Wilking Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Theorem (Finiteness of π 1 mod nilpotent sub) For each D > 0 and each dimension n there are finitely many s F 1,..., F k such that the following holds: If M is a compact n-manifold with Ric > (n 1) and diam(m) D, then there is a nilpotent normal sub N π 1 (M) with a nilpotent basis of length n 1 and rank(n) n 2 such that π 1 (M)/N = F i for suitable i. Theorem (Diameter Ratio Theorem) For n and D there is a D such that any compact manifold M with Ric (n 1) and diam(m) = D satisfies: If π 1 (M) is finite, then the diameter of the universal cover M of M is bounded above by D.

54 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Lemma (Product Lemma) Let M i be a sequence of complete manifolds with Ric Mi > ε i 0 satisfying 1 Ric Mi > ε i 0 2 for every i and j = 1,..., k there are harmonic functions b i j : M i R which are L-Lipschitz and fulfill B(p i,r) k j,l=1 b i j, b i l δ jl + k Hess b i j 2 dµ 0 j=1 For all R > 0 Then (M i, p i ) subconverges in the pointed Gromov 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.

55 Main Results of s Gap Lemma Maps which are on all scales close to isometries. 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 (M 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. Indeed, fix x, y B R (p) and a small δ xy. Let x i, y i M i be a sequence of points converging to x and y respectively. By the Segment Inequality we have that Margulis Lemma. ( b i 1 ) τ(r, δ, n) B δ (x i ) B δ (y i ) γ z1,z 2 as i. B R (p i ) b i 1 0

56 Main Results of s Gap Lemma Therefore, for some z 1 B δ (x i ), z 2 B δ (y i ) we have γ z1,z 2 b i 1 ) h i 0 Therefore Maps which are on all scales close to isometries. Margulis Lemma. and b i (z 1 ) b i (z 2 ) z 1 z 2 + h i b i (x i ) b i (y i ) 2Lδ + b i (z 1 ) b i (z 2 ) 2Lδ + z 1 z 2 + h i 4Lδ + x i y i + h i Passing to the limit b (x) b (y) 2Lδ + xy Since δ > 0 was arbitrary this gives the claim.

57 γ z1 z 2 z 2 y i Main Results δ of s Gap Lemma Maps which are on all scales close to isometries. z 1 δ x i Margulis Lemma. p i R

58 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Step 2. b is a submetry. Fix r > 0, t 0 > 0. Let φ i t be the gradient flow of b i on M i. (b i (φ i t 0 (x)) b i (x)) = B r (p i ) B r (p i ) B r (p i ) t0 0 b i (φ t (x) 2 = t0 0 φ t (B r (p i )) t0 0 t0 0 B(p i,1) b i (x) 2 = t 0 ± ε i d dt bi (φ i t(x)) = b i (φ t (x) 2 = where ε i 0. b i is harmonic and hence φ i t is measure preserving. For most points x B r (p i ) (8) b i (φ i t 0 (x)) b i (x) = t 0 ± ε i

59 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. By the first variation formula B(p i,1) = t0 φ i t 0 (x)x b i (φ t (x) = B(p i,1) 0 b i (φ t (x) = t 0 ± ε i φ t (B(p i,1)) t0 0 by the same argument as above. This means that for most x B(p i, 1) we have (9) φ i t 0 (x) B t0 +ε i (x) Combining (8) and (9) we get that in the limit space Y φ t 0 (x) B t0 (x) and b (φ t 0 (x)) b (x) = t 0 i.e. b is a submetry.

60 Main Results of s Gap Lemma Maps which are on all scales close to isometries. x t 0 φ i t 0 (x) Margulis Lemma. b i b i (x) b i (φ i t 0 (x)) = b i (x) + t 0 ± ε i

61 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Lemma A Let (Y i, G i, p i ) G H (Y, G, p ). i Let G i (ε) denote the sub generated by those elements that displace p i by at most ε, i N { }. Suppose G (ε) = G ( a+b 2 ) for all ε (a, b) and 0 a < b. Then there is some sequence ε i 0 such that G i (ε) = G i ( a+b 2 ) for all ε (a + ε i, b ε i ). Proof Suppose on the contrary we can find g i G i (ε 2 ) \ G i (ε 1 ) for fixed ε 1 < ε 2 (a, b). Without loss of generality d(p i, g i p i ) ε 2. Because of g i G i (ε 1 ) it follows that for any finite sequence of points p i = x 1,..., x h = g i p i G i p i there is one j with d(x j, x j+1 ) ε 1. Clearly this property carries over to the limit and implies that g G (ε 2 ) is not contained in G((ε 1 + a)/2) a contradiction.

62 x n = g i p i Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. x j x j+1 ε 1 x j+1 = g j+1 i p i x j = g j i p i x 2 p i = x 1

63 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Lemma B Suppose (Mi n, q i) converges to (R k K, q ) where Ric Mi 1/i and K is compact. Assume the action of π 1 (M i ) on the universal cover ( M i, q i ) converges to a limit action of a G on some limit space (Y, q ). Then G(r) = G(r ) for all r, r > 2 diam(k). Proof Since Y/G is isometric to R k K, it follows that there is a submetry σ : Y R k. It is immediate from the splitting theorem that this submetry has to be linear, that is, for any geodesic c in Y the curve σ c is affine linear. Hence we get a splitting Y = R k Z such that G acts trivially on R k and on Z with compact quotient K. We may think of q as a point in Z. For g G consider a mid point x Z of q and g q. Because Z/G = K we can find g 2 G with d(g 2 q, x) diam(k).

64 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Clearly d( q, g 2 q ) 1 2 d( q, g q ) + diam(k) d( q, g 1 2 g q ) = d(g 2 q, g q ) 1 2 d( q, g q ) + diam(k). This proves G(r) G ( r/2 + diam(k) ) lemma follows. g 2 q Margulis Lemma. q g 1 2 g r/2 d diam K x g 1 2 g r/2 g q g 1 2 g q

65 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Lemma (Gap Lemma) Suppose we have a sequence of manifolds (M i, p i ) with a lower Ricci curvature bound converging to some limit space (X, p ) and suppose that the limit point p is regular. Then there is a sequence ε i 0 and a number δ > 0 such that the following holds. If γ 1,..., γ li is a short basis of π 1 (M i, p i ) then either γ j δ or γ j < ε i. Moreover, if the action of π 1 (M i ) on the universal cover ( M i, p i ) converges to an action of the limit G on (Y, p ), then the orbit G p is locally path connected. This means there is a gap in lengths of short generators. They are either very short or longer than δ. Proof Idea. By rescaling reduce to the case X = R k. Apply Lemma B (with K = {pt}) to conclude G(r) = G(r ) for any r, r > 0. Apply Lemma A to get the result.

66 of π 1 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Theorem ( of π 1 ) Given n and R there is a constant C such that the following holds. Suppose (M n, g) has Ric (n 1), p M, and π 1 (M, p) is generated by loops of length R. Then there is a point q B R/2 (p) such that any Gromov short generator system of π 1 (M, q) has at most C elements. Observation If Ric M n (n 1) then the number of short generators of π 1 (M, p) with 0 < r 1 < γ i < r 2 is bounded above by C(n, r 1, r 2 ). This immediately follows from Bishop-Gromov volume. Proof of theorem Arguing by contradiction we get a sequence (M n i, p i) satisfying Ric Mi (n 1). For all q i B 1 (p i ) the number of short generators of π 1 (M i, q i ) of length 4 is larger than 2 i.

67 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. By precompactness we may assume that (M i, p i ) converges to some limit space (X, p ). We put dim(x) = max { k there is a regular p B 1/4 (p ) with T x X = R k} Reverse induction on dim(x). Base of induction. dim(x) n + 1. It is well known that this can not happen so there is nothing to prove. Induction step. Step 1. For any contradicting sequence (M i, p i ) converging to (X, p ) there is a new contradicting sequence converging to (R dim(x), 0). Can assume p is regular. Then use the observation above to find a slow rescaling λ i such that after passing to a subsequence (λ i M i, p i ) (R dim(x), 0) number of short generators of length 4 in π 1 (λ i M i, q i ) is still 2 i for any q i B λ im i 1 (p i ) = B M i λ i (p i ).

68 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. Step 2. If there is a contradicting sequence converging to R k, then we can find a contradicting sequence converging to a space whose dimension is larger than k. WLOG Ric Mi 1/i. By Cheeger-Colding we can find harmonic functions (b i 1,..., b i k ): B 1(q i ) R k with and B 1 (q i ) k j,l=1 b i l, b i j δ lj + Hess(b i l) 2 = ε 2 i 0 b i j C(n). By the weak (1,1) inequality we can find z i B 1/2 (q i ) with B r (z i ) for all r 1/4. k j,l=1 b i l, b i j δ lj + Hess(b i l) 2 Cε i 0

69 Main Results of s Gap Lemma Maps which are on all scales close to isometries. Margulis Lemma. By the Product Lemma, for any sequence µ i the spaces (µ i M i, z i ) subconverge to a metric product (R k Z, z ) for some Z depending on the rescaling. Choose r i 1 maximal with the property that there is y i B ri (z i ) such that the short generator system of π 1 (M i, y i ) contains one generator of length r i. By the Gap Lemma, r i 0 By the Product Lemma, (N i, z i ) subconverges to a product (R k Z, z ). Lemma B impies imply that Z can not be a point claim is proved.

70 Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. For a map f : X Y between metric spaces we define the dt f r (p, q) = min{r, d(p, q) d(f(p), f(q)) }. we call dt f r (p, q) the distortion on scale r.

71 Maps close to isometries on all scales Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. Definition Let (M i, p 1 i ) and (N i, p 2 i ) be two sequences of complete Riemannian manifolds with Ric Mi > C for some C, all i and j = 1, 2. We say that a sequence of diffeomorphisms f i : M i N i is close to isometry on all scales if the following holds: There exist R 0 > 0, sequences r i, ε i 0 and subsets B 2ri (p j i ) B 2ri (p j i ) (j = 1, 2) satisfying a) Vol ( B 1 (q) B 2ri (p j i ) ) (1 ε i )Vol(B 1 (q)) for all q B ri (p j i ). b) For all p B ri (p 1 i ), all q B ri (p 2 i ) and all r (0, 1] we have dt f i r (x, y)dµ(x)dµ(y) rε i and B r (p) B r (p) B r (q) B r (q) dt f 1 i r (x, y)dµ(x)dµ(y) rε i. c) There are subsets S j i B 1(p j i ) with Vol(Sj i ) 1 2 Vol( B 1 (p j i )) (j = 1, 2) and f(s 1 i ) B R 0 (p 2 i ) and f 1 (S 2 i ) B R 0 (p 1 i ).

72 Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. Properties if f i : (M i, p 1 i ) (N i, p 2 i ) is close to isometry on all scales and (M i, p 1 i ) G H (X, i p1 ), (N i, p 2 i ) G H (Y, i p2 ) Then f i converges in a weakly measured sense to an isometry f : X Y. If f i : M i N i and g i : N i P i is close to isometry on all scales then f i g i is also close to isometries on all scales. The main source of such maps are gradient flows of modified distance functions with small L 2 norms of hessians. If b: M R is a modified harmonic distance function let X = b and let φ t be the gradient flow. Let p, q M and let γ : [0, d] M be a unit speed geodesic with γ(0) = p, γ(d) = q. Here d = pq. Then φ t (p)φ t (q) +(0) L(φ t (γ)) (0) X = Hess b γ γ

73 Main Example Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. If B R (p i ) Hess b i 2 0 we can use the weak 1-1 inequality, the Segment inequality fact that φ i t is measure preserving to show that φ i t is close to isometry on all scales. Proposition (Main example) Let (M i, g i ) G H i (Rk Y, p ) satisfy Ric Mi > 1/i. Then for each v R k there is a sequence of diffeomorphisms f i : [M i, p i ] [M i, p i ] close to isometries on all scales which converges in the weakly measured sense to an isometry f of R k Y that acts trivially on Y and by w w + v on R k. Moreover, f i is isotopic to the identity re is a lift f i : [ M i, p i ] [ M i, p i ] of f i to the universal cover which is also close to isometry on all scales Remark This is EASY for manifolds with sec 1/i using gradient flows of distance functions.

74 Theorem (Rescaling Theorem) Let (M n, p i ) G H i (Rk, 0) for some k < nwhere Ric Mi > 1/i. Then after passing to a subsequence we can find a compact metric space K with diam(k) = 10 n2, a sequence of subsets G 1 (p i ) B 1 (p i ) with Vol(G 1(p i )) Vol(B 1 (p i ) 1 and λ i such that 1 For all q i G 1 (p i ) the isometry type of the limit of any convergent subsequence of (λ i M i, q i ) is given by the metric product R k K. 2 For all a i, b i G 1 (p i ) we can find a sequence of diffeomorphisms f i : [λ i M i, a i ] [λ i M i, b i ] close to isometries on all scales such that f i is isotopic to the identity. Moreover, for any lift ã i, b i M i of a i and b i to the universal cover Mi we can find a lift f i of f i such that f i : [λ i Mi, ã i ] [λ i Mi, b i ] are close to isometries on all scales as well. Finally, if π 1 (M i, p i ) is generated by loops of length R for all i, then we can find ε i 0 such that π 1 (M i, q i ) is generated by loops of length 1+ε i λ i for all q i G 1 (p i ).

75 Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. The Rescaling Theorem serves as substitute for the Fibration theorem. Idea of the proof of the Rescaling Theorem Let b i : M i R k be the modified harmonic distance functions Put k h i = < b i j, b i l > δ j,l + Hess b i j 2. j j,l=1 After passing to a subsequence by Cheeger Colding we have h i ε 2 i with ε i 0. B 1 (p i ) (10) G 1 (p i ) := { x B 1 (p i ) Mx h i (x) ε i }. We will call elements of G 1 (p i ) good points in B 1 (p i ).

76 Main Results of s Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Most points in B 1 (p i ) are good by the weak 1-1 inequality. Let x be a good point. Use induction on the size of the ball to show that one can get from x to most points in B r (x) with r 1 by composing gradient flows of appropriate modified distance functions which produce maps close to isometries on all scales. Induction step: If true for r/100 then true for r. φ t (x) Margulis Lemma. x r/100 r 100r

77 Given good points x, y B 1 (p i ) can connect them to most points in B 1 (p i ) and hence to each other by composition. Main Results of s z Maps which are on all scales close to isometries. Properties and examples The Rescaling Theorem. Margulis Lemma. x p i y B 1 (p i ) 1

78 Idea of the proof of the Margulis Lemma. Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency Let p i be a sequence of odd primes and let Γ i := Z Z pi be the semidirect product where the homomorphism Z Aut(Z pi ) maps 1 Z to ϕ i given by ϕ i (z + p i Z) = 2z + p i Z. Suppose, contrary to the Margulis Lemma, we have a sequence Mi n with Ric > 1/i and diam(m i ) = 1 and Γ i. G H A typical problem would be that M i i S1 and M G H i R. i We then replace M i by B i = M i /Z pi and in order not to loose information we endow B i with the deck transformation f i : B i B i representing a generator of Γ i /Z pi = Z. Then B i will converge to R as well. S 1 π 1 (M i ) = Z Z pi M i Unwrap Z π R π i B i

79 Main Results of s Then one can find λ i such that the rescaled sequence λ i B i converges to R K with K being compact but not equal to a point. Suppose for illustration that λ i B i R S 1 and λ i Mi R 2 and that the action of Z pi converges to a discrete action of Z on R 2. x i λ i B i Maps which are on all scales close to isometries. Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency x 1 λ i B i R S 1

80 Main Results of s Maps which are on all scales close to isometries. The maps f i : λ i B i λ i B i do not converge, because typically f i would map a base point x i to some point y i = f i (x i ) with d(x i, y i ) = λ i with respect to the rescaled distance. We use gradient flows of modified distance functions to construct diffeomorphisms g i : [λ i B i, y i ] [λ i B i, x i ] with the zooming in property. f i Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency x 1 g i f i (x i ) λ i B i R S 1

81 The composition f new,i := f i g i : [λ i B i, x i ] [λ i B i, x i ] also has the zooming in property and thus converges to an isometry of the limit. Moreover, a lift f new,i : λ i Mi λ i Mi of f new,i has the zooming in property, too. Since g i can be chosen isotopic to the identity, the action of f new,i on the deck transformation Z pi = π 1 (B i ) by conjugation remains unchanged. On the other hand, the Z pi -action on M i converges to a discrete Z-action on R 2 and f new,i converges to an isometry f new, of R 2 normalizing the Z-action. This implies that f new, 2 commutes with

82 Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency the Z-action and it is then easy to get a contradiction. Theorem (Induction Theorem) Suppose (M n i, p i) satisfies 1 Ric Mi 1/i; 2 There is some R > 0 such that π 1 (B R (p i )) π 1 (M i ) is surjective for all i; 3 (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 ] which are close to isometries on all scales where p i is a lift of p i, and which normalize the deck transformation 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 sub 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 N h+1 /N h is cyclic. Furthermore, each N h is invariant under the action of (f j i )C! by conjugation induced automorphism of N h /N h+1 is the identity.

83 Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency We argue by contradiction. After passing to a subsequence we can assume ( M i, Γ i, p i ) G H i (Rk R l K, G, (0, p )), where K is compact action of G on the first R k factor is trivial. Structure of the Proof 1 Without loss of generality we can assume that K is not a point. If K = {pt} use the Rescaling Theorem to get a contradicting sequence with K {pt}. 2 WLOG 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. Use gradient flows of harmonic functions fact that Isom( K) is compact so that a high power of any element is close to identity.

84 Main Results of s Maps which are on all scales close to isometries. Margulis Lemma. Idea of the proof of the Margulis Lemma The Induction Theorem for C-Nilpotency 3 By the Gap Lemma there is an ε > 0 and ε i 0 such that Γ i (ε) = Γ i (ε i ) for large i. Then WLOG [Γ i : Γ i (ε)]. If not can assume after passing to a bounded cover that Γ i = Γ i (ε) = Γ i (ε i ). Then use Rescaling Theorem to get a contradicting sequence converging to a space which splits R k with k > k. 4 Show that after passing to a bounded cover, we can assume that Γ i (ε) Γ i is normal in Γ i. 5 Tle limit G acts on R l cocompactly by ρ. Since [Γ i : Γ i (ε)] the ρ(g)/ρ(g) 0 is infinite and virtually abelian. One can can unwrap one Z from it and find corresponding subs Γ i Γ i with Γ i /Γ i Replace M i by M i = M i /Γ i deck transformation generating Γ i /Γ i. Then M i X which splits off R k+1 and we can use the induction assumption. cyclic of order going to infinity. k+1 and add one new fi given by the