Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS OF MANIFOLDS 

WITH RICCI CURVATURE BOUNDED BELOW 

VITALI KAPOVITCH AND BURKHARD WILKING 

The main result of this paper is the following theorem which settles a conjecture 

of Gromov. 

Theorem 1 (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 subgroup N of index ≤ C(n). Moreover, N has a nilpotent basis 

of length at most n. 

We call a generator system b 1 , . . . , b n of a group N a nilpotent basis if the commutator 

[b i , b j ] is contained in the subgroup 〈b 1 , . . . , b i−1 〉 for 1 ≤ i < j ≤ n. 

Having a nilpotent basis of length n implies in particular rank(N) ≤ n. We will 

also show that equality in this inequality can only occur if M is homeomorphic to 

an infranilmanifold, see Corollary 7.1. 

In the case of a sectional curvature bound the theorem is due to Kapovitch, 

Petrunin and Tuschmann [KPT10], based on an earlier version which was proved 

by Fukaya and Yamaguchi [FY92]. 

In the case of Ricci curvature a weaker form of the theorem was stated by 

Cheeger and Colding [CC96]. The main difference is that – similar to Fukaya and 

Yamaguchi’s theorem – no uniform bound on the index of the subgroup is provided. 

Cheeger and Colding never wrote up the details of the proof, and relied on some 

claims in [FY92] stating that their results would carry over from lower sectional 

curvature bounds to lower Ricci curvature bounds if certain structure results would 

be obtained. But compare also Remark 6.2 below. 

Corollary 2. Let (M, g) be a compact manifold with Ric > −(n−1) and diam(M) ≤ 

ε(n) then π 1 (M) contains a nilpotent subgroup N of index ≤ C(n). Moreover, N 

has a nilpotent basis of length ≤ n. 

One of the tools used to prove these results is 

Theorem 3. Given n and D there exists C such that for any n-manifold with 

Ric ≥ −(n−1) and diam(M, g) ≤ D, the fundamental group π 1 (M) can be generated 

by at most C elements. 

2000 Mathematics Subject classification. Primary 53C20. Keywords: almost nonnegative Ricci 

curvature, Margulis Lemma. 

The first author was supported in part by a Discovery grant from NSERC. 

1

2 VITALI KAPOVITCH AND BURKHARD WILKING 

This estimate was previously proven by Gromov [Gro78] under the stronger 

assumption of a lower sectional curvature bound K ≥ −1. Recall that a conjecture 

of Milnor states that the fundamental group of an open manifold with nonnegative 

Ricci curvature is finitely generated. Although Theorem 3 is far from a solution to 

that problem, the Margulis Lemma immediately implies the following. 

Corollary 4. Let (M, g) be an open n-manifold with nonnegative Ricci curvature. 

Then π 1 (M) contains a nilpotent subgroup N of index ≤ C(n) such that any finitely 

generated subgroup of N has a nilpotent basis of length ≤ n. 

Of course, by work of Milnor [Mil68] and Gromov [Gro81], the corollary is well 

known (without uniform bound on the index) in the case of finitely generated 

fundamental groups. We should mention that by work of Wei [Wei88] (and an 

extension in [Wil00]) every finitely generated virtually nilpotent group appears 

as fundamental group of an open manifold with positive Ricci curvature in some 

dimension. 

The proofs of our results are based on the structure results of Cheeger and 

Colding for limit spaces of manifolds with lower Ricci curvature bounds [Col96, 

Col97, CC96, CC97, CC00a, CC00b]. This is also true for the proof of following 

new tool which can be considered as a Ricci curvature replacement of Yamaguchi’s 

fibration theorem as well as a replacement of the gradient flow of semi-concave 

functions used by Kapovitch, Petrunin and Tuschmann. 

Theorem 5 (Rescaling theorem). Suppose a sequence of Riemannian n-manifolds 

(M i , p i ) with Ric ≥ − 1 i 

converges in the pointed Gromov–Hausdorff topology to the 

Euclidean space (R k , 0) with k < n. Then after passing to a subsequence there is a 

subset G 1 (p i ) ⊂ B 1 (p i ), a rescaling sequence λ i → ∞ and a compact metric space 

K ≠ {pt} such that 

• vol(G 1 (p i )) ≥ (1 − 1 i ) vol(B 1(p i )) 

• For all x i ∈ G 1 (p i ) the isometry type of any limit (λ i M i , x i ) is given by 

K × R k . 

• For all x i , y i ∈ G 1 (p i ) we can find a diffeomorphism f i : M i → M i such 

that f i subconverges in the weakly measured sense to an isometry 

f ∞ : 

lim (λ iM i , x i ) → lim (λ iM i , y i ). 

G-H,i→∞ G-H,i→∞ 

We will prove a technical generalization of this theorem in section 5. It will be 

important for the proof of the Margulis Lemma that the diffeomorphisms f i are 

in a suitable sense close to isometries on all scales. The precise concept is called 

zooming in property and is defined in section 3. 

The diffeomorphisms f i will be composed out of gradient flows of harmonic 

functions arising in the analysis of Cheeger and Colding. Their L 2 -estimates on the 

Hessian’s of these functions play a crucial role. 

Like in Fukaya and Yamaguchi’s paper the idea of the proof of the Margulis 

Lemma is to consider a contradicting sequence (M i , g i ). By a fundamental observation 

of Gromov, the set of complete n-manifolds with Ric ≥ −(n − 1) is 

precompact in Gromov–Hausdorff topology. Therefore one can assume that the 

contradicting sequence converges to a (possibly very singular) space X. One then 

uses various rescalings and normal coverings of (M i , g i ) in order to find contradicting 

sequences converging to higher and higher dimensional spaces. One of the 

differences to Fukaya and Yamaguchi’s approach is that if we pass to a normal

STRUCTURE OF FUNDAMENTAL GROUPS 3 

covering we endow the cover with generators of the deck transformation group in 

order not to loose information. Since after rescaling the displacements of these 

deck transformations converge to infinity, this approach seems bound to failure. 

However, using the rescaling theorem we are able to alter the deck transformations 

by composing them with a sequence of diffeomorphisms which are isotopic to the 

identity and which have the zooming in property. With these alterations we are 

able to keep much more information on the action by conjugation of long homotopy 

classes on very short ones. Since the altered deck transformations still converge in 

a weakly measured sense to isometries, this allows one to eventually rule out the 

existence of a contradicting sequence. 

For the proof of the Margulis Lemma we do not need any sophisticated structure 

results on the isometry group of the limit space except for the relatively elementary 

Gap Lemma 2.4 guaranteeing that generic orbits of the limit group are locally path 

connected. However, for the following application of the Margulis Lemma a recent 

structure result of Colding and Naber [CN10] is key. They showed that the isometry 

group of a limit space with lower Ricci curvature bound has no small subgroups and 

thus, by the small subgroup theorem of Gleason, Montgomery and Zippin [MZ55], 

is a Lie group. 

Theorem 6 (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 subgroup N ⊳ π 1 (M) such that for all p ∈ M: 

• the image of π 1 (B ε/1000 (p), p) → π 1 (M, p) contains N, 

• the index of N in the image of π 1 (B ε (p), p) → π 1 (M, p) is ≤ C and 

• N is a nilpotent group which has a nilpotent basis of length ≤ n. 

If D, n, ε 0 and C are given, there is an effective way to limit the number of 

possibilities for the quotient group π 1 (M, p)/N, see Lemma 9.2. In fact we have the 

following finiteness result. We recall that the torsion elements Tor(N) of a nilpotent 

group N form a subgroup. 

Theorem 7. a) For each D > 0 and each dimension n there are finitely 

many groups 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 subgroup 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. 

b) In addition to a) one can choose a finite collection of irreducible rational 

representations ρ j i : F i → GL(n j i , Q) (j = 1, . . . , µ i, i = 1, . . . , k) such that 

for a suitable choice of the isomorphism π 1 (M)/N ∼ = F i the following holds: 

There is a chain of subgroups Tor(N) = N 0 ⊳ · · · ⊳ N h0 = N which are all 

normal in π 1 (M) such that [N, N h ] ⊂ N h−1 and N h /N h−1 is free abelian. 

Moroever, the action of π 1 (M) on N by conjugation induces an action of F i 

on N h /N h−1 and the induced representation ρ: F i → GL ( (N h /N h−1 ) ⊗ Z Q ) 

is isomorphic to ρ j i for a suitable j = j(h), h = 1, . . . , h 0. 

Part a) of Theorem 7 generalizes a result of Anderson [And90] who proved finiteness 

of fundamental groups under the additional assumption of a uniform lower 

bound on volume. 

In the preprint [Wil11] a partial converse of Theorem 7 is proved. In particular 

it is shown there that for a finite collection of finitely presented groups and any


finite collection of rational representations one can find D such that each group Γ 

satisfying the algebraic restrictions described in a) and b) of the above theorem 

with respect to this data contains a finite nilpotent normal subgroup H such that 

Γ/H can be realized as a fundamental group of a n + 2-dimensional manifold with 

diam(M) ≤ D and sectional curvature |K| ≤ 1. 

We can also extend the diameter ratio theorem of Fukaya and Yamaguchi [FY92]. 

Theorem 8 (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. 

In the case of nonnegative Ricci curvature the theorem says that the ratio 

diam( ˜M)/ diam(M) is bounded above. Fukaya and Yamaguchi’s theorem covers 

the case that M has almost nonnegative sectional curvature. The proof of Theorem 

8 has some similarities to parts of the proof of Gromov’s polynomial growth 

theorem [Gro81]. 

Part of the paper was written up while the second named author was a Visiting 

Miller Professor at the University of California at Berkeley. He would like to thank 

the Miller institute for support and hospitality. 

Organization of the Paper 

1. Prerequisites 5 

2. Finite generation of fundamental groups 9 

3. Maps which are on all scales close to isometries 13 

4. A rough idea of the proof of the Margulis Lemma 20 

5. The Rescaling Theorem 21 

6. The Induction Theorem for C-Nilpotency 26 

7. Margulis Lemma 31 

8. Almost nonnegatively curved manifolds with b 1 (M, Z p ) = n 35 

9. Finiteness Results 37 

10. The Diameter Ratio Theorem 44 

We start in section 1 with prerequisites. We have included a subsection on 

notational conventions. Next, in section 2 we will prove Theorem 3. 

Section 3 is somewhat technical. We define the zooming in property, prove the 

needed properties, and provide two somewhat similar construction methods. This 

section serves mainly as a preparation for the proof of the rescaling theorem. 

We have added a short section 4 in which we sketch a rough idea of the proof of 

the Margulis Lemma. 

In section 5 the refined rescaling theorem (Theorem 5.1) is stated and proven. 

In Section 6 we put things together and provide a proof of the Induction Theorem, 

which has Corollary 2 as its immediate consequence. 

The Margulis Lemma follows from the Induction Theorem in two steps which are 

somewhat similar to Fukaya and Yamaguchi’s approach. Nevertheless, we included 

all details in section 7. We also show that the nilpotent group N in the Margulis 

Lemma can only have rank n if the underlying manifold is homeomorphic to an 

infranilmanifold. 

Section 8 uses lower sectional curvature bounds. We give counterexamples to a 

theorem of Fukaya and Yamaguchi [FY92] stating that almost nonnegatively curved 

n-manifolds with first Z p -Betti number equal to n have to be tori, provided p is


sufficiently big. We show that instead these manifolds are nilmanifolds and that 

every nilmanifold covers another nilmanifold with maximal first Z p Betti number. 

Section 9 contains the proof of Theorem 6 and Theorem 7. Finally, Theorem 8 

is proved in section 10. 

1. Prerequisites 

1.1. Short basis. We will use the following construction due to Gromov. 

Given a manifold (M, p) and a group Γ acting properly discontinuously and 

isometrically on M one can define a short basis of the action of Γ at p as follows: 

For γ ∈ Γ we will refer to |γ| = d(p, γ(p)) as the norm or the length 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 n-th step choose γ n to have minimal norm in Γ\〈γ 1 , γ 2 , ..., γ n−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 finite or infinite. In the special case of the action 

of the fundamental group π 1 (M, p) on the universal cover ˜M of M one speaks of 

the short basis of π 1 (M, p). For any i > j we have |γ i | ≤ |γ −1 

j γ i |. 

While a short basis need not be unique the non-uniqueness will not matter in 

any of the proofs in this article and will largely be suppressed. 

If M/Γ is a closed manifold, then the short basis is finite and |γ i | ≤ 2 diam(M/Γ). 

1.2. Ricci curvature. Throughout this paper we will use the notation − ∫ to denote 

the average integral. We will use the following results of Cheeger and Colding. 

Theorem 1.1. [CC96] Let (M n i , p i) G−H 

−→ 

i→∞ (X, p) with Ric(M i) ≥ − 1 i . Suppose X 

has a line. Then X splits isometrically as X ∼ = Y × R. 

We will also need the following corollary of the stability theorem. 

Theorem 1.2. [CC97] 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). 

Even more important for us is the following theorem which is closely linked with 

the proof of the splitting theorem for limit spaces. 

Theorem 1.3. [CC00a] 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 

(1) |∇b i j| ≤ C(n) for all i and j and 

∫ 

(2) − 

B 1(p i) 

∑ 

∣ ∑ 

∣ < ∇b 

i 

j , ∇b i l > −δ j,l + ‖Hess b i 

j 

‖ 2 dµ → 0 as i → ∞. 

j,l 

Moreover, the maps Φ i = (b i 1, . . . , b i k ): M i → R k provide ε i -Gromov–Hausdorff 

approximations between B 1 (p i ) and B 1 (0) with ε i → 0. 

The functions b i j in the above theorem are constructed as follows. Approximate 

Busemann functions f j in R k given by f j = d(·, N i e j )) − N i are lifted to M i using 

Hausdorff approximations to corresponding functions fj i. Here e j is the j-th coordinate 

vector in the standard basis of R k and N i → ∞ sufficiently slowly so that 

j


d G−H (B Ni (p i ), B Ni (0)) → 0 as i → ∞. The functions b i j are obtained by solving 

the Dirichlet problem on B 2 (p i ) with b i i | ∂B 2(p i ) = fi i| ∂B 2(p i ) 

We will need a weak type 1-1 estimate for manifolds with lower Ricci curvature 

bounds which is a well-known consequence of the doubling inequality [Ste93, p. 12]. 

Lemma 1.4 (Weak type 1-1 inequality). Suppose (M n , g) has Ric ≥ −1 and let 

f : M → R be a nonnegative function. Define Mx f(p) := sup r≤1 − ∫ B r(p) 

f. Then 

the 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 c > 0. 

M 

(c) If f ∈ L α (M) with α > 1 then Mx f ∈ L α (M) and || Mx f|| α ≤ C(n, α)||f|| α . 

Note that as an immediate corollary we see that if f ∈ L α (M) with α > 1 then 

for any x ∈ M we have the following pointwise estimate 

(3) Mx((Mx f) α )(x) ≤ C(n, α) Mx(f α )(x). 

We will also need the following inequality (for any α > 1) which is an immediate 

consequence of the definition of Mx and Hölder inequality. 

(4) [Mx f(x)] α ≤ Mx(f α )(x). 

Indeed, for any r ≤ 1 we have that Mx(f α )(x) ≥ − ∫ B r(x) f α ≥ ( − ∫ B r(x) f) α 

and 

the inequality follows by taking the supremum over all r ≤ 1. 

Combining the last two inequalities we deduce 

(5) Mx[Mx(f)](x) ≤ Mx[(Mx(f)) α ] 1/α (x) ≤ C(n, α) 1/α Mx[f α ](x) 

for α > 1. Finally we note that for α > 1, β > 1 we have 

(6) Mx[Mx(f β/α )] α (x) ≤ C(n, α) Mx(f β )(x). 

This inequality immediately follows by applying (3) to g = f β α . 

We will also make use of the so-called segment inequality of Cheeger and Colding 

which says that in average the integral of a nonnegative function along all geodesics 

in ball can be estimated by the L 1 norm of the function. 

Theorem 1.5 (Segment inequality). [CC96] 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) 

∫ d(z1,z 2) 

0 

∫ 

g(γ z1,z 2 

(t))dµ(z 1 )dµ(z 2 ) ≤ τ · r · − g(q) dµ(q), 

B 2r(p) 

where γ z1,z 2 

denotes a minimal geodesic from z 1 to z 2 . 

Finally we will need the following observation 

Lemma 1.6 (Covering Lemma). There exists a constant C(n) such that the following 

holds. Suppose (M n , g) has Ric g ≥ −(n − 1). Let f : M → R be a nonnegative 

function, p ∈ M, π : ˜M → M be the universal cover of M, ˜p ∈ ˜M a lift of p, and 

let ˜f = f ◦ π. Then ∫ 

∫ 

− ˜f ≤ C(n)− f. 

B 1(˜p) 

B 1(p)


Proof. We choose a measurable section j : B 1 (p) → B 1 (˜p), i.e. π(j(x)) = x for any 

x ∈ B 1 (p). Let T = j(B 1 (p)). Then we obviously have that diam(T ) ≤ 2 and 

∫ ∫ 

− f = − ˜f. 

B 1(p) 

Let S be the union of g(T ) over all g ∈ π 1 (M) such that g(T ) ∩ B 1 (˜p) ≠ ∅. It’s 

obvious from the triangle inequality that S ⊂ B 3 (˜p). It is also clear that 

∫ ∫ 

− f = − ˜f 

B 1(p) 

and B 1 (˜p) ⊂ S. Lastly, notice that by Bishop–Gromov relative volume comparison 

vol B 3 (˜p) ≤ C(n) vol B 1 (˜p) for some universal constant C(n). Since B 1 (˜p) ⊂ S ⊂ 

B 3 (˜p), we also have vol B 3 (˜p) ≤ C(n) vol S and hence 

∫ 

∫ ∫ 

− ˜f ≤ C(n)− ˜f = C(n)− f. 

B 1(˜p) 

S 

T 

S 

B 1(p) 

It is easy to see that the above proof generalizes to an arbitrary cover of M. 

1.3. Equivariant Gromov–Hausdorff convergence. Let Γ i be a sequence of 

closed subgroups of the isometry groups of metric spaces X i and ˜p i ∈ X i . If 

(X i , ˜p i ) converges in the pointed Gromov–Hausdorff topology to (Y, ˜p ∞ ), then after 

passing to a subsequence one can find a closed subgroup G ⊂ Iso(Y ) such that 

(X i , Γ i , ˜p i ) → (Y, G, ˜p ∞ ) in the equivariant Gromov–Hausdorff topology. For details 

we refer the reader to [FY92]. 

Definition 1.7. A sequence of subgroups Υ i ⊂ Γ i is called uniformly open, if there 

is some ε > 0 such that Υ i contains the set 

{ 

g ∈ Γi | d(gq, q) < ε for all q ∈ B 1/ε (˜p i ) } for all i. 

Γ i is called uniformly discrete if {e} ⊂ Γ i is uniformly open. We say the sequence 

Υ i is boundedly generated if there is some R such that Υ i is generated by 

{g ∈ Υ i | d(g˜p i , ˜p i ) < R}. 

Lemma 1.8. Let Υ j i ⊂ Γ i be uniformly open with Υ j i → Υj ∞ ⊂ G, j = 1, 2. 

a) Υ 1 i ∩ Υ2 i is uniformly open and converges to Υ1 ∞ ∩ Υ 2 ∞. 

b) If g i ∈ Γ i converges to g ∞ ∈ G then g i Υ 1 i g−1 i is uniformly open and converges 

to g ∞ Υ 1 ∞g∞ −1 . 

c) Suppose in addition that Υ j i is boundedly generated, j = 1, 2. If Υ1 ∞ ∩ Υ 2 ∞ 

has finite index H in Υ 1 ∞, then Υ 1 i ∩ Υ2 i has index H in Υ1 i for all large i. 

The proof is an easy exercise. 

Example 1 (Z l actions converging to a Z 2 -action.). Consider a 2-torus T 2 k 

given by 

a Riemannian product S 1 ×S 1 where each of the factors has length k. Put l = k 2 +1 

and let Z/lZ act on T 2 k by ( e 2πi 

k 2 k2πi ) 

+1 , e k 2 +1 . In the equivariant Gromov–Hausdorff 

topology the action will converge (for k → ∞) to the standard Z 2 action on R 2 . 

One can, of course, define a similar sequence of actions on S 3 × S 3 but in this 

case the actions will not be uniformly cocompact. 

□


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.


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


Lemma 2.2. Suppose a sequence of complete inner metric spaces (Y i , p i ) converges 

to a locally compact metric space (Y ∞ , p ∞ ). Suppose G i acts isometrically on Y i and 

that the action of G i converges to an action of a closed subgroup G ∞ ⊂ Iso(Y ∞ ). 

Let G i (ε) denote the subgroup generated by those elements that displace p i by at 

most ε, i ∈ N ∪ {∞}. Suppose G ∞ (ε) = G ∞ ( a+b 

2 

Then there is some sequence ε i 

(a + ε i , b − ε i ). 

) for all ε ∈ (a, b) and 0 ≤ a < b. 

) for all ε ∈ 

→ 0 such that G i (ε) = G i ( a+b 

2 

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. 

□ 

Lemma 2.3. 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 group 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). 

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) ) and the lemma follows. 

Lemma 2.4 (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 group G on (Y, ˜p ∞ ), then the orbit G ⋆ ˜p ∞ is locally path 

connected. 

Proof. That the orbit is locally path connected is a consequence of the following 

Claim. There is a δ > 0 such that for all points x ∈ G ⋆ ˜p ∞ with d(˜p ∞ , p) < δ we 

can find y ∈ G ⋆ ˜p ∞ with max{d(˜p ∞ , y), d(y, x)} ≤ 0.51 · d(˜p ∞ , x). 

To prove the claim we argue by contradiction and assume that for some δ h > 0 

converging to 0 we can find g h ∈ G with d(g h ˜p ∞ , ˜p ∞ ) = δ h such that for all a ∈ G 

with d(a˜p ∞ , ˜p ∞ ) ≤ 0.51δ h we have d(a˜p ∞ , g˜p ∞ ) > 0.51δ h . 

After passing to a subsequence we can assume that ( 1 

δ h 

Y, ˜p ∞ ) converges to a 

tangent cone (C p∞ Y, o) and that the action of G on 1 

δ h 

Y converges to an action of 

□


some group K on C p∞ Y . Let g ∞ ∈ K be the limit of g h . Clearly d(o, g ∞ o) = 1 and 

for all a ∈ K with d(o, ao) ≤ 0.51 we have d(go, ao) ≥ 0.51. In particular the orbit 

K ⋆ o is not convex. 

Because X = Y/G the quotient C˜p∞ Y/K is isometric to lim h→∞ ( 1 

ε h 

X, p ∞ ) which 

by assumption is isometric to some Euclidean space R k . 

As in the proof of Lemma 2.3 it follows that C˜p∞ Y is isometric to R k × Z and 

the orbits of the action of K are given by v × Z where v ∈ R k . In particular, the K 

orbits are convex – a contradiction. 

Clearly, the claim implies that G ⋆ ˜p ∞ ∩ B r (˜p ∞ ) is path connected for r < δ. In 

fact, it is now easy to construct a Hölder continuous path from ˜p ∞ to any point in 

G ⋆ ˜p ∞ ∩ B r (˜p ∞ ). 

Of course, the claim also implies G(ε) = G(δ) for all ε ∈ (0, δ]. Therefore the 

first part of Lemma 2.4 follows from Lemma 2.2. 

□ 

Theorem 3 will follow from the following slightly more general result. 

Theorem 2.5. Given n and R there is a constant C such that the following holds. 

Suppose (M, g) is a manifold, p ∈ M, B 2R (p) is compact and Ric > −(n − 1) on 

B 2R (p). Furthermore, we assume that π 1 (M, g) is generated by loops of length ≤ R. 

Then π 1 (M, p) can be generated by C loops of length ≤ R. 

Moreover, 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. 

Proof. It is clear that it suffices to show that for a point q ∈ B R/2 (p) the number 

of elements in a Gromov short generator system of π 1 (M, q) is bounded above by 

some a priori constant. 

We consider a Gromov short generator system γ 1 , . . . , γ k of π 1 (M, q). With 

|γ i | ≤ |γ i+1 | for some q ∈ B R/2 (p). 

Clearly |γ i | ≤ 2R and it easily follows from Bishop–Gromov relative volume 

comparison that there are effective a priori bounds (depending only on n, R and r) 

for the number of short generators with length ≥ r. 

We will argue by contradiction. It will be convenient to modify the assumption 

on π 1 being boundedly generated. We call (M i , g i ) a contradicting sequence if the 

following holds 

• B 3 (p i ) is compact and Ric > −(n − 1) on B 3 (p i ). 

• 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 . 

Clearly it suffices to rule out the existence of a contradicting sequence. We may 

assume that (B 3 (p 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 C x X ∼ = R k} . 

We argue by reverse induction on dim(X). We we start our induction at dim(X) ≥ 

n + 1. It is well known that this can not happen so there is nothing to prove. The 

induction step is subdivided in two substeps. 

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). 

Choose q i ∈ B 1/4 (p i ) converging to some point q ∞ ∈ B 1/4 (p ∞ ) with C q∞ X =


R dim(X) . After passing to a subsequence we can assume that for all x i ∈ B 1/4 (q i ) 

the number of short generators of π 1 (M i , x i ) of length ≤ 1 is at least 3 i . 

Since the number of short generators of length ∈ [ε, 4] is bounded by some a priori 

constant, we find a sequence λ i → ∞ very slowly such that for every x ∈ B 1/λi (q i ) 

the number of short generators of π 1 (M i , x) of length ≤ 4/λ i is at least 2 i and 

(λ i M i , q i ) converges to C q∞ X = R dim(X) . Replacing M i by λ i M i and p i and q i 

gives a new contradicting sequence, as claimed. 

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. 

Let (M i , p i ) be a contradicting sequence converging to (R k , 0). We may assume 

without loss of generality that for some r i → ∞ and ε i → 0 the Ricci curvature on 

B ri (p i ) is bounded below by −ε i and that B ri (p i ) is compact. In fact, one can run 

through the arguments of the first step, to see that, after passing to a subsequence, 

a rescaling by λ i → ∞ (very slowly) is always possible. 

By Theorem 1.3 we can find a harmonic map (b i 1, . . . , b i k ): B 1(q i ) → R k with 

and 

∫ 

− 

B 1(q i) 

j,l=1 

k∑ 

|〈∇b i l, ∇b i j〉 − δ lj | + ‖Hess(b i l)‖ 2 = ε i → 0 

|∇b i j| ≤ C(n). 

By the weak (1,1) inequality (Lemma 1.4) we can find z i ∈ B 1/2 (q i ) with 

∫ 

− 

B r(z i) 

j,l=1 

k∑ 

|〈∇b i l, ∇b i j〉 − δ lj | + ‖Hess(b i l)‖ 2 ≤ Cε i → 0 

for all r ≤ 1/4. By the Product Lemma (2.1), for any sequence µ i → ∞ the spaces 

(µ i B r (z i ), z i ) subconverge to a metric product (R k × Z, z ∞ ) for some Z depending 

on the rescaling. 

From Lemmas 2.2 and 2.3 we deduce that there is a sequence δ i → 0 such that 

for all z i ∈ B 2 (p i ) the short generator system of π 1 (M i , z i ) does not contain any 

elements with length in [δ i , 4]. 

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 . We 

have seen above that r i ≤ δ i → 0 

Put N i = 1 r i 

M i . By construction, π 1 (N i , y i ) still has 2 i short generators of 

length ≤ 1 for all y i ∈ B 1 (z i ) ⊂ N i , and there is one with length 1 for a suitable 

y i . By the Product Lemma (2.1), (N i , z i ) subconverges to a product (R k × Z, z ∞ ). 

Lemmas 2.2 and 2.3 imply that Z can not be a point and the claim is proved. □ 

Finally, let us mention that there is a measured version of Theorem 2.5. 

Theorem 2.6. For any n > 1 for any 1 > ε > 0 there exists C(n, ε) such that 

if Ric(M n ) ≥ −(n − 1) on B 3 (p) with B 3 (p) compact. Then there is a subset 

B 1 (p) ′ ⊂ B 1 (p) with vol B 1 (p) ′ ≥ (1 − ε) vol B 1 (p) and for any q ∈ B 1 (p) ′ the short 

basis of π 1 (M, q) has at most C(n, ε) elements of length ≤ 1. 

Since we do not have any applications of this theorem, we omit its proof.


3. Maps which are on all scales close to isometries. 

For a map f : X → Y between metric spaces we define the distance distortion 

on scale r by 

(7) dt f r (p, q) = min{r, |d(p, q) − d(f(p), f(q))|}. 

Definition 3.1. Let (M i , p 1 i ) and (N i, p 2 i ) be two sequences of Riemannian manifolds 

with B i (p j i ) being compact and Ric B i(p j i ) 

> −C for some C, all i and j = 1, 2. 

We say that a sequence of diffeomorphisms f i : M i → N i has the zooming in property 

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 fi 

r (x, y)dµ(x)dµ(y) ≤ rε i and 

B 

∫ r(p)×B r(p) 

− dt f −1 

i 

r (x, y)dµ(x)dµ(y) ≤ rε i . 

B r(q)×B r(q) 

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 ). 

We will call elements of B 2ri (p 1 i )′ good points and sometimes use the convention 

B r (q) ′ := B 2ri (p j i )′ ∩ B r (q) for all B r (q) ⊂ B 2ri (p j i ). 

In the applications we will always have N i = M i . However, in some instances 

d(p 1 i , p2 i ) → ∞. That is why it might be helpful to also think about maps between 

two unrelated pointed manifolds. If the choice of the base points is not clear we 

will say that f i : [M i , p 1 i ] → [N i, p 2 i ] has the zooming in property. Notice that we 

do not require f i (p 1 i ) = p2 i . However, property c) ensures that the base points are 

respected in a weaker sense. 

Lemma 3.2. Let (M i , p 1 i ), (N i, p 2 i ) and f i be as above. Then, after passing to a 

subsequence, (M i , p 1 i ) G−H 

−→ (X 1, p 1 ∞), (N i , p 2 i ) G−H 

−→ (X 2, p 2 ∞) and f i converges in 

i→∞ i→∞ 

the weakly measured sense to a measure preserving isometry f ∞ : X 1 → X 2 , that is 

for each r > 0 there is a sequence δ i → 0 and subsets S i ⊂ B r (p 1 i ) satisfying 

• vol(S i ) ≥ (1 − δ i ) vol(B r (p 1 i )) and 

• f i|Si is pointwise close to f ∞|Br(p ∞). 

Moreover, f −1 

i 

converges in this sense to f −1 

∞ . 

First we claim that the following holds. 

Sublemma 3.3. There exists C 1 (n) such that for any good point x ∈ B ri (p 1 i )′ and 

any r ≤ 1 there is a subset B r (x) ′′ ⊂ B r (x) with 

vol B r (x) ′′ ≥ (1 − C 1 ε i ) vol B r (x) and f i (B r (x) ′′ ) ⊂ B 2r (f i (x)). 

Proof. The proof proceeds by induction on the size of r as follows. It is clear that 

at good points x ∈ B ri (p 1 i )′ the differential of f i has a bilipschitz constant e Cεi 

with some universal C provided ε i < 1/2. Therefore it is clear that the statement


holds for all small r. Hence it suffices to prove that if it holds for r ≤ 1/10, then it 

holds for 10r. By assumption 

∫ 

− 

dt fi 

10r (p, q)dµ(p)dµ(q) ≤ 10rε i. 

B 10r(x)×B 10r(x) 

Furthermore, as long as C 1 ε i ≤ 1/2 our induction assumption implies that there is 

a subset S ⊂ B r (x) with vol(S) ≥ 1 2 vol(B r(x)) and f i (S) ⊂ B 2r (f i (x)). 

By Bishop–Gromov 

∫ 

− dt fi 

10r (p, q)dµ(p)dµ(q) ≤ C 2(n)rε i 

S×B 10r(x) 

with some universal constant C 2 (n). Therefore there is a subset B 10r (x) ′′ ⊂ B 10r (x) 

with vol(B 10r (x)) ′′ ≥ (1 − C 2 (n)ε i /2) vol(B 10r (x)) and 

∫ 

− dt fi 

10r (p, q)dµ(p) ≤ 2r for all q ∈ B 10r(x) ′′ . 

S 

Using that f i (S) ⊂ B 2r (f i (x)) this clearly implies that f i (B 10r (x) ′′ ) ⊂ B 20r (f i (x)). 

Thus the sublemma is valid if we put C 1 (n) = C 2 (n)/2 provided we know in addition 

that C 1 (n)ε i ≤ 1/2. We can remove the upper bound on ε i by just putting C 1 (n) = 

C 2 (n). 

□ 

Of course, a similar inequality holds for f −1 

i . 

Proof of Lemma 3.2. Clearly, we can finish the proof of the lemma by establishing 

Claim. Given δ ∈ (0, 1/10) there exists i 0 such that for all x i , y i ∈ B ri (p 1 i )′ with 

d(x i , y i ) < 1/2 we have |d(f i (x i ), f i (x i )) − d(x i , y i )| ≤ 10δ for all i ≥ i 0 (δ). 

Consider subsets B δ (x i ) ′′ and B δ (y i ) ′′ as in the sublemma. Then vol(B δ (x i ) ′′ ) ≥ 

1 

2 vol(B δ(x i )) for large i and combining with Bishop–Gromov gives 

vol(B 1 (x i )) 2 ≤ C 2 (n, δ) vol(B δ (x i ) ′′ ) vol(B δ (y i ) ′′ ). 

Thus, ∫ 

∫ 

− 

dt fi 

1 (p, q) ≤ C 2− dt fi 

1 (p, q) ≤ C 2ε i . 

B δ (x i) ′′ ×B δ (y i) ′′ B 1(x i) 2 

Choose i 0 so large that C 2 ε i ≤ δ for i ≥ i 0 . Then for such i we can find x ′ i ∈ B δ(x i ) ′′ 

and y i ′ ∈ B δ(y i ) ′′ with dt fi 

1 (x′ i , y′ i ) ≤ δ. Combining with d(f i(x ′ i ), f i(x i )) ≤ 2δ and 

d(f i (y i ′), f i(y i )) ≤ 2δ we deduce dt fi 

1 (x i, y i ) ≤ 7δ as claimed. □ 

Lemma 3.4. Consider (M i , p 1 i ), (N i, p 2 i ), (P i, p 3 i ) like above and two sequences of 

diffeomorphisms f i : M i → N i and g i : N i → P i with the zooming in property. Then 

f i ◦ g i also has the zooming in property. 

Proof. By Sublemma 3.3 there is an i 0 such that for all q i ∈ B ri (p) and r ≤ 1/2 

there is a subset B r (q i ) ′′ ⊂ B r (q i ) ′ with vol(B r (q i ) ′′ ) ≥ (1−C 1 ε i ) and f i (B r (q i ) ′′ ) ⊂ 

B 2r (q i ). Thus, 

∫ 

∫ 

− dt fi◦gi 

r (a, b) ≤ C 2 ε i r + e 2nCεi − dt gi 

r (f i (a), f i (b)) 

B r(q) 2 (B r(q) ′′ ) 

∫ 

2 

≤ C 2 ε i r + C 3 − 

r (x, y) ≤ C 4 ε i , 

B 2r(f i(q)) 2 dt gi


where we used that the differential of f i at q ∈ B r (q i ) ′ has a bilipschitz constant 

e Cεi and that the ratio of vol(B 2r (f i (q))) and vol(B r (q)) is for large i bounded by 

a universal constant. 

We are left with the case of r ∈ [1/2, 1]. By Lemma 3.2, f i and g i converge in the 

weakly measured sense to isometries. Clearly this implies that f i ◦ g i also converges 

in the weakly measured sense to an isometry. And this implies that the distortion 

function dt fi◦gi 

1 has averaged L 1 norm ≤ δ i on every unit ball in B ρi (p i ) for some 

ρ i → ∞ and δ i → 0. 

□ 

The next lemma explains the notion zooming in property. 

Lemma 3.5. Let (M i , p 1 i ), (N i, p 2 i ) and f i be as above. Then there is a ρ i → ∞, 

δ i → 0 and Ti 

1 ⊂ B ρi (p 1 i ) such that the following holds 

• vol(B 1 (q) ∩ Ti 1) ≥ (1 − δ i) vol(B 1 (q)) for all q ∈ B ρi/2(p 1 i ). 

• For any sequence of real numbers λ i → ∞ and any sequence q i ∈ Ti 

1 

f i : (λ i M i , q i ) → (λ i N i , f i (q i )) 

has the zooming in property. We say that f i is good on all scales at q i . 

Proof. Let G j i = B 2r i 

(p j i )′ and B j i = B 2r i 

(p i ) \ B 2ri (p j i )′ . After adjusting r i → ∞ 

and ε i → 0 we may assume vol(B j i ) ≤ ε i vol(B 1 (q)) for all q ∈ B 2ri (p j i ). Let χ ij 

be the characteristic function of B j i . By the weak 1-1 inequality there exists a 

universal C such that the set 

H j i := { x ∈ B ri/2(p j i ) | Mx(χ ij) ≥ √ } 

ε i 

satisfies 

vol(H j i ) ≤ C√ ε i vol(B 1 (q)) 

for all q ∈ B 2ri (p j i ). We put T i 1 := ( ) ( ) 

B ri/2(p 1 i )\H1 i ∩ f 

−1 

i Bri/2(p 2 i )\H2 i and 

Ti 

2 := f i (Ti 1). Using Lemma 3.2 we can find ρ i → ∞ and δ i → 0 such that 

vol ( B ρi (p j i ) \ T j ) 

i ≤ δi vol(B 1 (q)) for all q ∈ B ri (p j i ), j = 1, 2. 

By definition of T j i 

vol(B r (q) ∩ G j i ) 

vol(B r (q)) 

≥ 1 − √ ε i for all q ∈ T j i and all r ≤ 1, j = 1, 2. 

Let dt λif 

r 

denote the distortion on scale r of f i : λ i M i → λ i N i . Clearly 

dt λif 

r (p, q) = λ i dt fi 

r/λ i 

(p, q). 

Thus for all λ i → ∞ and all q i ∈ T j i 

the zooming in property. 

the map f i : (λ i M i , q i ) → (λ i N i , f i (q i )) has 

□ 

Proposition 3.6 (First main example). Let α > 1. Consider a sequence of manifolds 

(M i , p i ) with a fixed lower Ricci curvature bound and a sequence of time 

dependent vector fields Xi 

t (piecewise constant in time) with compact support. Let 

c i : [0, 1] → B ri (p i ) be an integral curve of Xi t with c i(0) = p i


Assume that Xi 

t is divergence free on B r i+100(p i ) and that B ri+100(p i ) is compact. 

Put 

u s,i (x) := ( Mx ‖∇·Xi s ‖ α) 1/α 

and suppose 

∫ 1 

0 

u t,i (c i (t)) = ε i → 0. 

Let f i = φ i1 be the flow of X t i evaluated at time 1. Then for all λ i → ∞ 

f i : (λ i M i , c(0)) → (λ i M i , c(1)) 

has the zooming in property. Moreover, for any lift ˜fi : ˜Mi → ˜M i of f i to the 

universal cover ˜Mi of M i and for any lift ˜p i ∈ ˜M of c(0) = p i the sequence 

˜f i : (λ i ˜Mi , ˜p i ) → (λ i ˜Mi , ˜f i (˜p i )) has the zooming in property as well. 

The proposition remains valid if the assumption on Xi 

t being divergence free 

is removed. However, the proof is easier in this case and we do not have any 

applications of the more general case. We will need the following 

Lemma 3.7. There exists (explicit) C = C(n) such that the following holds. Suppose 

(M n , g) has Ric ≥ −1 and X t is a vector field with compact support, which 

depends on time (piecewise constant). Let c(t) be the integral curve of X t with 

c(0) = p 0 ∈ M and assume that X t is divergence free on B 10 (c(t)) for all t ∈ [0, 1]. 

Let φ t be the flow of X t . Define the distortion function dt r (t)(p, q) of the flow 

on scale r by the formula 

{ 

} 

dt r (t)(p, q) := min r, max ∣ d(p, q) − d(φ τ (p), φ τ (q))| . 

0≤τ≤t 

Put ε := ∫ 1 

0 Mx(‖∇·X t ‖)(c(t)) dt. Then for any r ≤ 1/10 we have 

∫ 

− 

dt r (1)(p, q) dµ(p)dµ(q) ≤ Cr · ε 

B r(p 0)×B r(p 0) 

and there exists B r (p 0 ) ′ ⊂ B r (p 0 ) such that 

vol(B r(p 0) ′ ) 

vol(B r(p 0)) 

≥ (1 − Cε) and φ t (B r (p 0 ) ′ ) ⊂ B 2r (c(t)) for all t ∈ [0, 1]. 

Proof. We prove the statement for a constant in time vector field X t . The general 

case is completely analogous except for additional notational problems. 

Notice that all estimates are trivial if ε ≥ 2 C 

. Therefore it suffices to prove the 

statement with a universal constant C(n) for all ε ≤ ε 0 . We put ε 0 = 1/2C and 

determine C in the process. We again proceed by induction on the size of r. 

Notice that the differential of φ s at c(0) is bilipschitz with bilipschitz constant 

e R s 

0 ‖∇·X‖(c(t))dt ≤ 1 + 2ε. Thus the Lemma holds for very small r. 

Suppose the result holds for some r/10 ≤ 1/100. It suffices to prove that it 

then holds for r. By induction assumption we know that for any t there exists 

B r/10 (c(t)) ′ ⊂ B r/10 (c(t)) such that for any s ∈ [−t, 1 − t] we have 

and 

vol(B r/10 (c(t)) ′ ) ≥ (1 − Cε) vol(B r/10 (c(t))) ≥ 1 2 vol(B r/10(c(t))) 

φ s (B r/10 (c(t)) ′ ) ⊂ B r/5 (c(t + s)),


where we used ε ≤ 1 

2C 

in the inequality. This easily implies that vol(B r/10(c(t))) 

are comparable for all t. More precisely, for any t 1 , t 2 ∈ [0, 1] we have that 

(8) 

1 

C 0 

vol B r/10 (c(t 1 )) ≤ vol B r/10 (c(t 2 )) ≤ C 0 vol B r/10 (c(t 1 )) 

with a computable universal C 0 = C 0 (n). Put 

∫ 

h(s) = − 

dt r (s)(p, q) dµ(p) dµ(q), 

B r/10 (p 0) ′ ×B r(c(0)) 

U := {(p, q) ∈ B r/10 (p 0 ) ′ × B r (c(0)) | dt r (s)(p, q) < r}, 

φ s (U) := {(φ s (p), φ s (q)) | (p, q) ∈ U}, 

and dt ′ dt 

r(s)(p, q) := lim sup r(s+h)(p,q) 

h↘0 h 

. Since dt r is monotonously increasing 

and bounded by r, we deduce that if dt r (s)(p, q) = r, then dt ′ r(s)(p, q) = 0. 

Since φ s is measure preserving, it follows 

∫ 

h ′ d 

(s) ≤ − 

dt |t=0 dt′ r(0)(p, q) 

φ s(U) 

∫ 

≤ 

4 vol(B3r(c(s))2 

vol(B r/10 (c(0))) 

− dt ′ r(0)(p, q), 

2 

B 3r(c(s)) 2 

where we used that φ s (B r/10 (p 0 ) ′ ) 2 ⊂ φ s (U) ⊂ B 3r (c(s)) 2 . If p is not in the cut 

locus of q and γ pq : [0, 1] → M is a minimal geodesic between p and q then 

dt ′ r(0)(p, q) ≤ d(p, q) 

∫ 1 

0 

‖∇·X‖(γ pq (t)) dt. 

Combining the last two inequalities with the segment inequality we deduce 

h ′ (s) ≤ 

∫ 

C 1 (n)r− ‖∇·X‖ 

B 6r(c(s)) 

≤ C 1 (n)r Mx ‖∇·X‖(c(s)) 

and thus h(1) ≤ C 1 (n)rε. Note that the choice of the constant C 1 (n) can be made 

explicit and independent of the induction assumption. 

This in turn implies that that the subset 

{ 

∫ 

} 

B r (p 0 ) ′ := p ∈ B r (p 0 ) ∣ − dt r (1)(p, q) dµ(q) ≤ r/2 

B r/10 (p 0) ′ 

satisfies 

(9) 

It is elementary to check that 

vol(B r (p 0 ) ′ ) ≥ (1 − 2C 1 (n)ε) vol(B r (p 0 )). 

φ t (B r (p 0 ) ′ ) ⊂ B 2r (c(t)) for all t ∈ [0, 1]. 

Then arguing as before we estimate that 

∫ 

− 

dt r (s)(p, q)dµ(p)dµ(q) ≤ C 2 (n) · r · ε. 

B r(c(t)) ′ ×B r(c(t))


Using the definition of dt r (s) and the volume estimate (9) this gives 

∫ 

− dt r (s)(p, q) dµ(p) dµ(q) ≤ C 2 (n) · r · ε + 2rC 1 ε =: C 3 rε. 

B r(c(t)) 2 

This completes the induction step with C(n) = C 3 and ε 0 = 1 

2C 3 

. In order to 

remove the restriction ε ≤ ε 0 one can just increase C 3 by the factor 4, as indicated 

at the beginning. 

□ 

Proof of Proposition 3.6. Put g t,i (x) = ‖∇·Xi t ‖(x). First notice that by Hölder 

∫ 1 

0 

Mx(g t,i (x))(c i (t)) dt ≤ 

∫ 1 

0 

u t,i (c i (t)) dt → 0. 

Let λ i → ∞ and put r i = R λ i 

, where R > 1 is arbitrary. By Lemma 3.7 there is 

S i ⊂ B ri (c i (0)) with 

• vol(S i ) ≥ (1 − δ i ) vol(B ri (c i (0))) with δ i → 0 and 

• φ t (S i ) ⊂ B 2ri (c(t)) for all t. 

In the following we assume that i is so large that δ i ≤ 1/2, and r i ≤ 1/100. As 

in the proof of Lemma 3.7 this easily implies that there is a universal constant 

C = C(n) with 

and 

Using that φ it|Bri (c(0)) ′ 

vol(B 2ri (c i(t))) 

vol(B ri (c i(0))) ≤ C 2 

for all t and all i. 

is measure preserving, we deduce 

∫ 1 

− Mx(g t,i )(φ it (p)) dt dµ(p) ≤ C 

∫S i 0 

≤ 

by (5) 

C 

∫ 1 

0 

∫ 1 

0 

∫ 

− 

B 2ri (c i(t)) 

Mx(g t,i )(q) dt dµ(q) 

Mx(Mx(g t,i ))(c i (t)) dt 

∫ 1 

≤ C · C 2 Mx(gt,i) α 1/α (c i (t)) dt 

0 

∫ 1 

= C · C 2 u t,i (c i (t)) dt → 0. 

Thus we can find ˜δ i → 0 and a subset B ri (c i (0)) ′′ ⊂ S i with 

vol(B ri (c i (0))) − vol(B ri (c i (0)) ′′ ) ≤ ˜δ i min vol(B r 

q∈B ri (c i/R(q)) 

i(0)) 

∫ 1 

0 

Mx(g t,i )(φ it (q)) ≤ ˜δ i for all q ∈ B ri (c(0)) ′′ . 

Recall that r i = R λ i 

with an arbitrary R. By a diagonal sequence argument it is 

easy to deduce that after replacing ˜δ i by a another sequence converging slowly to 

0 we can keep the above estimates for r i = Ri 

λ i 

with R i → ∞ sufficiently slowly. 

Combining this with Lemma 3.7 shows that f i = φ i1 : (λ i M i , c(0)) → (λ i M i , c(1)) 

has the zooming in property. 

Let ˜X i t be a lift of Xt i to the universal covering ˜M i of M i . Consider the integral 

curve ˜c i : [0, 1] → ˜M i of ˜Xt i with ˜c(0) = ˜p i . Clearly ˜c i is a lift of c i and by the 

0


Covering Lemma (1.6) we have 

∫ 1 

0 

Mx(‖∇· ˜Xt i ‖ α ) 1/α (˜c i (t)) dt ≤ C 

∫ 1 

0 

Mx(‖∇·X t i ‖ α ) 1/α (c i (t)) dt 

with some universal constant C. Thus ˜φ i1 : (λ i ˜Mi , ˜p i ) → (λ i ˜Mi , φ i1 (˜p i )) has the 

zooming in property as well. Any other lift ˜f i of f i is obtained by composing ˜φ i1 

with a deck transformation and thus the result carries over to any lift of f i . 

□ 

Proposition 3.8 (Second main example). Let (M i , g i ) be a sequence of manifolds 

with Ric > −1/i on B i (p i ) and B i (p i ) compact. Suppose that (M i , p i ) converges to 

(R k × Y, p ∞ ). 

Then for each v ∈ R k there is a sequence of diffeomorphisms f i : [M i , p i ] → 

[M i , p i ] with the zooming in property 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 and there is a lift ˜f i : [ ˜M i , ˜p i ] → [ ˜M i , ˜p i ] of 

f i to the universal cover which has the zooming in property as well. 

Proof. Using the splitting R k = Rv ⊕(v) ⊥ and replacing Y by Y ×(v) ⊥ we see that 

it suffices to prove the statement for k = 1. 

By the work of Cheeger and Colding [CC96] we can find sequences ρ i → ∞ and 

ε i → 0 and harmonic functions b i : B 4ρi (p i ) → R such that 

|∇b i | ≤ L(n) for all i and 

∫ 

(∣ 

− ∣|∇bi | − 1 ∣ + ‖Hessbi ‖ ) 2 

≤ ε i for any R ∈ [1/4, 4ρ i ]. 

B 4R (p i) 

Let X i be a vector field with compact support with X i = ∇b i on B 3ρi (p i ), and 

let φ it denote the flow of X i . Clearly for any t we can find r i → ∞ such that 

φ it|Bri (p i) is measure preserving. 

Put ψ i := ∣ ∣|∇b i | − 1 ∣ + ‖Hess(b i )‖. We deduce from Lemma 1.4 that 

∫ 

− Mx(ψ i ) 2 ≤ C(n, R)ε i 

and Cauchy Schwarz gives 

∫ 

− 

B 2R (p i) 

B R (p i) 

Mx(ψ i ) ≤ √ C(n, R)ε i . 

Suppose now that t 0 ≤ R 

4L(n) . Then φ t(q) ∈ B 3R (p i ) for all q ∈ B 

4 

R/2 (p i ) 

and all t ∈ [−t 0 , t 0 ]. Combining that φ t is measure preserving and vol(B R (p i )) ≤ 

C 3 vol(B R/2 (p i )), we get that 

∫ ∫ t0 

∫ 

− 

Mx(ψ i )(φ t (p)) ≤ C 3 t 0 − Mx(ψ) 

B R/2 (p i) 

0 

It is now easy to find R i → ∞ and δ i → 0 with 

B R (p i) 

≤ C 4 (t 0 , R, n) √ ε i . 

∫ ∫ t0 

− Mx(ψ i )(φ t (p)) ≤ δ i for all R ∈ [1, R i ]. 

B R (p i) 0


After an adjustment of the sequences δ i → 0 and R i → ∞ we can find B Ri (p i ) ′ ⊂ 

B Ri (p i ) with 

vol(B Ri (p i )) − vol(B Ri (p i ) ′ ) ≤ δ i vol(B 1 (q)) for all q ∈ B Ri (p i ) and 

∫ t0 

0 

Mx(ψ i )(φ t (q)) ≤ δ i for all q ∈ B Ri (p i ) ′ . 

Moreover, the displacement of φ it0 is globally bounded by L(n)t 0 and thus, 

Lemma 3.7 implies that φ it0 has the zooming in property. 

As in the proof of the Product Lemma (2.1) we see that b i converges to the 

projection b ∞ : R × Y → R. In particular b ∞ is a submetry. 

For all q i ∈ B Ri (p i ) ′ the length of the integral curve c i (t) = φ it (q) (t ∈ [0, t 0 ]) is 

bounded above by t 0 +δ i . Moreover, b i (c(t 0 ))−b i (c i (0)) is bounded below by t 0 −δ i . 

This implies that φ ∞t0 satisfies d(p, φ ∞t0 (p)) ≤ t 0 and b ∞ (φ ∞t0 (q)) − b ∞ (p) ≥ t 0 . 

Clearly, equality must hold and φ ∞t (q) is a horizontal geodesic with respect to the 

submetry b ∞ . This in turn implies that φ ∞t0 respects the decomposition Y × R, it 

acts by the identity on the first factor and by translation on the second as claimed. 

Let σ : ˜Mi → M i denote the universal cover, ˜p i ∈ ˜M i a lift of p i , and let ˜X i be a 

lift of X i . By Lemma 1.6 all estimates for X i on B R (p i ) give similar estimates for 

˜X i on B R (˜p i ). Thus the flow ˜φ it of ˜Xi has the zooming in property as well. □ 

4. A rough idea of the proof of the Margulis Lemma. 

The proof of the Margulis Lemma is somewhat indirect and it might not be easy 

for everyone to grasp immediately how things interplay. In this section we want to 

give a rough idea of how the arguments would unwrap in a simple but nontrivial 

case. 

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 the automorphism 

ϕ i given by ϕ i (z + p i Z) = 2z + p i Z. 

Suppose, contrary to the Margulis Lemma, we have a sequence of compact n- 

manifolds M i with Ric > −1/i and diam(M i ) = 1 and fundamental group Γ i . 

A typical problem situation would be that M i converges to a circle and its 

universal cover ˜M i converges to R. 

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 . Then B i will converge to R as well. 

It is part of the rescaling theorem that 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 converges to R × S 1 and λ i ˜Mi converges 

to R 2 and that the action of Z pi converges to a discrete action of Z on R 2 . 

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. 

But a second statement in the rescaling theorem guarantees that we can find a 

sequence of diffeomorphisms g i : [λ i B i , y i ] → [λ i B i , x i ] with the zooming in property. 

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 ˜fnew,i on the deck 

transformation group 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 the Z-action and it is then easy to get a 

contradiction. 

Problem. We suspect that for any given n and D only finitely many of the groups 

in the above family of groups should occur as fundamental groups of compact n- 

manifolds with Ric > −(n − 1) and diameter ≤ D. Note that this is not even 

known under the stronger assumption of K > −1 and diam ≤ D. 

5. The Rescaling Theorem. 

Theorem 5.1 (Rescaling Theorem). Let (M, g i , p i ) be a sequence of n-manifolds 

satisfying Ric Bri (p i) > −µ i and B ri (p i ) is compact for some r i → ∞, µ i → 0. 

Suppose that (M, g i , p i ) → (R k , 0) for some k < n. 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(G1(pi)) 

vol(B → 1 and a sequence λ 1(p i) 

i → ∞ such that 

the following holds 

• 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. 

• 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 ] 

with the zooming in property 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 ˜M i we can 

find a lift ˜f i of f i such that 

˜f i : [λ i ˜Mi , ã i ] → [λ i ˜Mi , ˜b i ] 

has the zooming in property 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+εi 

λ i 

for all 

q i ∈ G 1 (p i ). 

Proof. By Theorem 1.3, after passing to a subsequence we can find a harmonic map 

b i : B i (p i ) → B i+1 (0) ⊂ R k that gives a 1 2 

Gromov–Hausdorff approximation from 

i 

B i (p i ) to B i (0) ⊂ R k . Put 

h i = 

k∑ 

| < ∇b i j, ∇b i l > −δ j,l | + ∑ j 

j,l=1 

‖Hess b i 

j 

‖ 2 . 

After passing to a subsequence we also have by Theorem 1.3 that 

∫ 

− h i ≤ ε 4 i for all R ∈ [1, i] with ε i → 0. 

B R (p i) 

Furthermore, we may assume Ric > −ε i on B i (0). 

(10) G 4 (p i ) := { x ∈ B 4 (p i ) | Mx h i (x) ≤ ε 2 i 

} 

.


We will call elements of G 4 (p i ) ”good points” in B 4 (p i ). We put G r (q) = B r (q) ∩ 

G 4 (p i ) for q ∈ B 2 (p i ) and r ≤ 2. It is immediate from the weak 1-1 inequality 

(Lemma 1.4) that 

vol(G 1(p i)) 

vol(B 1(p i)) → 1. 

By the Product Lemma (2.1), for any q i ∈ G 4 (p i ) and any h ≥ 1 we have that 

hB 1/h (q i ) is ˜ε i close to a ball in a product R k × Y where ˜ε i → 0 can be chosen 

independently of h and q i . Note that the space Y may depend on h and q i . 

For any point q i ∈ G 1 (p i ) let ρ i (q i ) be the largest number ρ < 1 such that the 

map b i : B ρ (q i ) → R k has distance distortion exactly equal to ρ · 10 −n2 . For any 

choice q i ∈ G 1 (p i ) we know that ρ i (q i ) → 0 and by the Product Lemma (2.1) 

( 

1 

ρ M ) 

i(q i) i, q i subconverges to R k × K where K is compact with diam(K) = 10 −n2 . 

Again note, that a priori, the constants ρ i (q i ) and the space K depend on the 

choice of q i ∈ B 1 (p i ) ′ . We will show that, in fact, both are essentially independent 

of the choice of good points in B 1 (p i ). 

For each i we define the number ρ i as the supremum of ρ i (q i ) with q i ∈ G 1 (p i ) 

and put 

λ i = 1 ρ i 

. 

The statement of the theorem will follow easily from the following sublemma. 

Sublemma 5.2. There exist constants ¯C 1 , ¯C 2 independent of i such that the following 

holds. For any good point p ∈ G 1 (p i ) and any r ≤ 2 there exists B r (p) ′ ⊂ B r (p) 

such that 

(11) vol B r (p) ′ ≥ (1 − ¯C 1 rε i ) vol B r (p) 

and such that for any q ∈ B r (p) ′ there exists p ′ ∈ B Lρi (p) (with L = 9 n ) and a time 

dependent (piecewise constant in time) divergence free vector field X t (dependence 

on q is suppressed) and its integral curve c q (t): [0, 1] → B 2 (p i ) such that c(0) = 

p ′ , c(1) = q and 

1 

vol(B r(p)) 

∫B r(p) ′ ∫ 1 

0 

( 

Mx(‖∇·X t ‖ 3/2 ) ) 2/3 

(cq (t)) dt dµ(q) ≤ ¯C 2 rε i . 

Proof. Despite the fact that L can be chosen as L = 9 n we treat it for now as a 

large constant L ≥ 10 which needs to be determined. Throughout the proof we will 

denote by C j various constants depending on n. Sometimes these constants will 

also depend on L in which case that dependence will always be explicitly stated. 

Although we consider a fixed i for the major part of the proof, we take the 

liberty of assuming that i is large. This way we can ensure that for any r ≥ ρ i 

1 

and any good point p ∈ G 1 (p i ) the ball 

Lr B 2Lr(p) is by the Product Lemma in the 

measured Gromov–Hausdorff sense arbitrarily close to a ball in a product R k × K 

where by our choice of ρ i , K has diameter ≤ 10 −n2 . This implies, for example, that 

we can assume that 

( 

1 

(12) 

r1 

) k vol(B 

2 r 2 

≤ 

r1 (q 1)) 

vol(B r2 (q ≤ 2( ) 

r 1 k 

2)) r 2 

for all q1 , q 2 ∈ B Lr (p), r 1 , r 2 ∈ [r/20, 2Lr]. 

The sublemma is trivially true for r ≤ Lρ i with X ≡ 0. By induction it suffices 

to show that if the sublemma holds for some r ∈ (ρ i , 2 L 

] then it holds for Lr.


Let q 1 , . . . q l be a maximal r/2-separated net in B Lr (p). Note that 

l⋃ 

B Lr (p) ⊂ B r/2 (q m ). 

m=1 

By a standard volume argument we can deduce from (12) that 

(13) 

∑ l 

m=1 vol B r/2(q m ) 

≤ 3 k+1 ≤ 3 n . 

vol B Lr (p) 

Fix q m and consider the following vector field 

k∑ 

X(x) = (b i α(p) − b i α(q m ))∇b i α(x). 

α=1 

Since b i α are Lipschitz with a universal Lipschitz constant by (1), X satisfies 

(14) |X(x)| ≤ C(n) · Lr. 

Also, by construction, X satisfies 

(15) Mx ‖∇·X‖ 2 (p) ≤ C 4 ε 2 i L 2 r 2 . 

Note that we get an extra L 2 r 2 factor as compared to (10) because |b i (p)−b i (q m )| ≤ 

C(n)Lr. By applying (6) we get 

( 

Mx [Mx(‖∇·X‖ 3/2 )] 4/3) (p) ≤ C(n) Mx(‖∇·X‖ 2 )(p) ≤ (C 5 Lε i r) 2 . 

In particular, for R 1 = 2C(n)Lr 

∫ 

− [Mx(‖∇·X‖ 3/2 )] 4/3 (p) ≤ (C 5 Lε i r) 2 

B R1 (p) 

provided R 1 ≤ 1. However, in the case of R 1 ∈ [1, 4C(n)] the same follows more 

directly from our initial assumptions combined with Lemma 1.4 for all large i. By 

Cauchy inequality the last estimate gives 

∫ 

( 

(16) − Mx(‖∇·X‖ 3/2 ) ) 2/3 

(p) ≤ C5 Lε i r. 

B R1 (p) 

Consider the measure preserving flow φ of X on [0, 1]. Because of (14) the flow 

lines φ t (q) with q ∈ B r (p) stay in the ball B R1 (p) for t ∈ [0, 1]. We choose a 

universal constant C 6 (L) with C 5 L · vol(B 2C(n)Lr(p)) 

vol(B r/10 (q m) 

≤ C 6 (L) 

Combining that the flow is measure preserving, inequality (16) and our choice 

of C 6 we see that 

(17) 

∫ 

1 

vol(B r (q 10 m)) 

B r(q m) 

∫ 1 

0 

( 

Mx(‖∇·X‖ 3/2 ) ) 2/3 

(φt (x)) dt dµ(x) ≤ C 6 (L)rε i . 

By the Product Lemma (2.1) applied to the rescaled balls 

1 

Lr B Lr(p) we know 

that 1 Lr B Lr(p) is measured Gromov–Hausdorff close to a unit ball in R k ×K 3 where 

diam K 3 ≤ 10 −n2 . Moreover, φ 1 is measured close to a translation by b i (p)−b i (q m ) 

in R k , see proof of Proposition 3.8. Since we only need to argue for large i, we may


assume that by volume 3/4 of the points in B r/10 (q m ) are mapped by φ 1 to points 

in B r/9 (p). 

We choose such a point q ∈ B r/10 (q m ). In view of (17) we may assume in 

addition that 

∫ 1 

By Lemma 3.7 this implies 

0 

Mx(‖∇·X‖ 3/2 ) 2/3 (φ t (q)) dt ≤ 4 3 C 6(L)rε i . 

∫ 

(18) − dt r (1)(x, y) dµ(x) dµ(y) ≤ C 7 (L)r · ε i . 

B r(q)×B r(q) 

Note that by definition, dt r (1)(x, y) ≥ min{r, |d(φ 1 (x), φ 1 (y)−d(x, y)|}. Combining 

this inequality with the knowledge that 3/4 of the points in B r/10 (q m ) end up in 

B 1/9r (p), implies that we can find a subset B r/2 (q m ) ′ ⊂ B r/2 (q m ) with 

(19) 

(20) 

vol(B r/2 (q m ) ′ ) ≥ (1 − C 8 (L)ε i r) vol(B r/2 (q m )) and 

φ 1 (B r/2 (q m ) ′ ) ⊂ B r (p). 

Set B r/2 (q m ) ′′ := B r/2 (q m ) ′ ∩ φ −1 

1 (B r(p) ′ ). Then we get 

φ 1 (B r/2 (q m ) ′′ ) ⊂ B r (p) ′ 

and 

vol(B r/2 (q m ) ′′ ) ≥ vol(B r/2 (q m ) ′ ) − (vol B r (p) − vol B r (p) ′ ) 

(21) 

by (19) and (11) 

≥ (1 − C 8 (L)rε i ) vol B r/2 (q m ) − ¯C 1 rε i vol B r (p) 

by (12) 

≥ (1 − C 8 (L)rε i ) vol B r/2 (q m ) − 2 k+1 ¯C1 rε i vol B r/2 (q m ) 

≥ (1 − (C 8 (L) + 2 n ¯C1 )rε i ) vol B r/2 (q m ), 

where we used that φ 1 is volume-preserving in the first inequality. 

Using the induction assumption, we can prolong each integral curve from any 

q ∈ B r/2 (q m ) ′′ to a point x ∈ B r (p) ′ by extending it by a previously constructed 

integral curve from x to p ′ ∈ B Lρi (p) of a vector field Xold t (which depends on p′ ). 

We set Xnew t = 2Xold 2t for 0 ≤ t ≤ 1/2 and Xt new = −2X for 1/2 ≤ t ≤ 1. Let c q (t) 

be the integral curve of Xnew t with c q (1) = q and c q (0) = p ′ . 

By the induction assumption and (17), we get 

(22) 

∫B r/2 (q m) ′′ ∫ 1 

0 

(Mx ‖∇·X t new‖ 3/2 ) 2/3 (c q (t)) dt dµ(q) ≤ 

≤ C 6 (L)(rε i ) vol B r/2 (q m ) + ¯C 2 (rε i ) vol B r (p) 

(12) 

≤ C 6 (L)(rε i ) vol B r/2 (q m ) + 2 n ¯C2 (rε i ) vol B r/2 (q m ) 

= ( C 9 (L) + 2 n ¯C2 

) 

(rεi ) vol B r/2 (q m ). 

Recall that the balls B r/2 (q m ) cover B Lr (p). We put 

B Lr (p) ′ := B Lr (p) ∩ ⋃ m 

B r/2 (q m ) ′′ .


By construction, for every point in B Lr (p) ′ there exists a vector field X t whose 

integral curve connects it to a point in B Lρi (p) satisfying (22). For points covered 

by several sets B r/2 (q m ) ′′ we pick any one. Then we have 

∫B Lr (p) ′ ∫ 1 

0 

Mx ( ‖∇·Xnew‖ t ) 3 2 

2 3 

(c q (t)) dt dµ(q) ≤ 

≤ 

(13) 

≤ 

≤ 

l∑ 

(C 9 (L) + 2 n ¯C2 )(rε i ) vol B r/2 (q m ) 

m=1 

3 n (C 9 (L) + 2 n ¯C2 )(rε i ) vol B rL (p) 

¯C 2 · Lrε i · vol B rL (p). 

The last inequality holds if L = 9 n ≥ 2 · 2 n · 3 n and ¯C 2 ≥ 3 n · C 9 (L). 

Recall that by (21) 

vol B r/2 (q m ) − vol B r/2 (q m ) ′′ ≥ (C 8 (L) + 2 n ¯C1 )rε i vol B r/2 (q m ) 

for any m and thus 

vol B Lr (p) − vol B Lr (p) ′ 

≤ 

(13) 

≤ 

≤ 

l∑ 

(C 8 (L) + 2 n ¯C1 )rε i vol B r/2 (q m ) 

m=1 

provided that L = 9 n ≥ 2 · 6 n and ¯C 1 ≥ 3 n C 8 (L). 

This finishes the proof of Sublemma 5.2. 

3 n (C 8 (L) + 2 n ¯C1 )rε i vol(B Lr (p)) 

¯C 1 Lrε i vol B Lr (p) 

Observe that for any two good points x i , y i ∈ G 1 (p i ) we can deduce from Sublemma 

5.2 that vol(B 2 (x i ) ′ ∩ B 2 (y i ) ′ ) ≥ vol(B 1/2 (p i )) for all large i. 

Then it follows, also by Sublemma 5.2, that there is q ∈ B 2 (x i ) ′ ∩ B 2 (y i ) ′ , 

x ′ i ∈ B Lρ i 

(x i ) and y i ′ ∈ B Lρ i 

(y i ) such that for the integral curve c connecting x ′ i 

with q on the first half of the interval and q with y i ′ on the second half we have for 

the corresponding time dependent vector field Xi 

t 

∫ 1 

( 

Mx(‖∇·X t i ‖ 3/2 ) ) 2/3 

(c(t)) dt ≤ ¯C3 ε i 

□ 

0 

with a constant ¯C 3 independent of i. 

Let f i := φ i1 be the flow of X t i evaluated at time 1. Since λ i = 1 ρ i 

, we can 

employ Proposition 3.6 to see that 

f i : [λ i M i , x i ] → [λ i M i , y i ] 

has the zooming in property. Thus, the Gromov–Hausdorff limits of the two sequences 

are isometric. 

For any lifts ˜x i and ỹ i of x i and y i to the universal cover we can lifts ˜x ′ i and ỹ′ i 

of x ′ i and y′ i in the Lρ i neighbourhoods of ˜x i and ỹ i . Let ˜f i : ˜Mi → ˜M i be the lift 

f i with ˜f i (˜x ′ i ) = ỹ′ i . Proposition 3.6 ensures that ˜f i : [λ i ˜Mi , ˜x i ] → [λ i ˜Mi , ỹ i ] has the 

zooming in property as well. 

It remains to check the last part of the Rescaling Theorem concerning the fundamental 

group. Assume that π 1 (M i , p i ) is generated by loops of length ≤ R.


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


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.


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.


• 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.


Thus, the normalizer of Υ i has finite index ≤ [G : G ′ ] in Γ i for all large i and 

Step 4 is established. 

Step 5. The desired contradiction arises if, after passing to a subsequence, the 

indices [Γ i : Γ iε ] ∈ N ∪ {∞} converge to ∞. 

By Step 4 there is a subgroup Υ i ⊂ Γ iε of index ≤ H which is normalized by a 

subgroup of Γ i of index ≤ H. Similarly to the beginning of the proof of Step 3, one 

can reduce the situation to the case of Υ i ⊳ Γ i . 

Moreover, Υ i ⊳ Γ i is uniformly open and hence the action of Γ i /Υ i on M i /Υ i is 

uniformly discrete and converges to a properly discontinuous action of the virtually 

abelian group G/Υ ∞ on the space R k × (R l × ˜K)/Υ ∞ . It is now easy to see that 

Γ iε /Υ i contains an abelian subgroup of controlled finite index for large i. 

After replacing Γ i once more by a subgroup of controlled finite index we may 

assume that Γ i /Υ i is abelian. This also shows that without loss of generality 

Υ i = Γ iε . 

Recall that by Step 2 we have assumed that f j i converges to the identity in the 

weakly measured sense for every j = 1, . . . , k. Therefore, the same is true for the 

commutator [f j i , γ i] ∈ Γ i if γ i ∈ Γ i has bounded displacement. This in turn implies 

that [f j i , γ i] ∈ Γ iε for all large i and j = 1, . . . , k. 

By Theorem 2.5, we can find for some large σ elements d i 1, . . . , d i σ ∈ Γ i with 

bounded displacement generating Γ i . We also assume that the first τ elements 

generate Γ iε for some τ < σ. 

Let ˆΓ i := 〈d i 1, . . . , d i σ−1〉. Note that ˆΓ i ⊳ Γ i since ˆΓ i contains Γ iε = Υ i ⊳ Γ i and 

Γ i /Γ iε is abelian. 

If for some subsequence and some integer H the group ˆΓ i has index ≤ H in Γ i 

then we may replace Γ i by ˆΓ i . Thus, without loss of generality, the order of the 

cyclic group Γ i /ˆΓ i tends to infinity. The element f k+1 

i := d i σ ∈ Γ i represents a 

generator of this factor group which has bounded displacement. 

We have seen above that f j i normalizes ˆΓ i and [f k+1 

i , f j i ] ∈ Γ iε ⊂ ˆΓ i , j = 1, . . . , k. 

Clearly f k+1 

i , being an isometry with bounded displacement, has the zooming in 

property. 

We replace M i by B i := ˜M i /ˆΓ i . Notice that (B i , ˆp i ) converges to (R k × (R l × 

˜K)/Ĝ, ˆp ∞) where Ĝ is the limit group of ˆΓi . Since G/Ĝ is a noncompact group 

acting cocompactly on (R l × ˜K)/Ĝ, we see that (Rl × ˜K)/Ĝ splits as Rp × K ′ with 

K ′ compact and p > 0. If p > 1, we put f j i = id for j = k + 2, . . . , k + p. 

Thus the induction assumption applies: There is a positive integer C such that 

the following holds for all large i: 

We can find a subgroup N i of π 1 (B i ) of index ≤ C and a nilpotent chain of 

normal subgroups {e} = N 0 ⊳ . . . ⊳ N n−k−1 = N i such that the quotients are cyclic. 

Moreover, the groups are invariant under the automorphism induced by conjugation 

of (f j i )C! and the induced automorphism of N h+1 /N h is the identity, j = 1, . . . , k+1. 

We put d = C!, and consider the group ¯N i ⊂ π 1 (M i ) generated by (f k+1 

i ) d and 

N i . Clearly, the index of ¯N i in π 1 (M i ) is bounded by C · d. 

Moreover, the chain {e} = N 0 ⊳ . . . ⊳ N n−k−1 ⊳ N n−k := ¯N i is normalized by the 

elements (f j i )d (j = 1, . . . , k) and the action on the cyclic quotients is trivial. 

In other words, the sequence fulfills the conclusion of our theorem – a contradiction. 

□


Remark 6.2. If one was only interested in proving that the fundamental group 

contains a polycylic subgroup of controlled index, the proof would simplify considerably. 

First of all, one would not need any diffeomorphisms f j i to make the 

induction work. In the proof of Step 1 the use of the rescaling theorem could be 

replaced with the more elementary Product Lemma 2.1. Step 2 would become unnecessary. 

The core of the remaining arguments would be the same although they 

would simplify somewhat. 

7. Margulis Lemma 

Proof of Theorem 1. Let B 1 (p) be a metric ball in complete n-manifold with Ric > 

−(n − 1) on B 1 (p), Ñ the universal cover of B 1 (p) and let ˜p ∈ Ñ be a lift of p. 

Step 1. There are universal positive constants ε 1 (n) and C 1 (n) such that the group 

〈 {g 

Γ := ∈ π1 (B 1 (p), p) ∣ d(q, gq) ≤ ε1 (n) for all q ∈ B 1/2 (˜p) }〉 

has a subgroup of index ≤ C 1 (n) which has a nilpotent basis of length at most n. 

Put N = Ñ/Γ and let ˆp denote the image of ˜p. By the definition of Γ, for each 

point q ∈ B 1/2 (ˆp), the fundamental group π 1 (N, q) is generated by loops of length 

≤ ε 1 (n). Moreover, B 3/4 (ˆp) is compact. 

Assume, on the contrary, that the statement is false. Then we can find a sequence 

of pointed n-dimensional manifolds (N i , p i ) satisfying 

• Ric Ni ≥ −(n − 1), 

• B 3/4 (p i ) is compact, 

• for each point q ∈ B 1/2 (p i ) the fundamental group π 1 (N i , q) is generated 

by loops of length ≤ 2 −i , and 

• π 1 (N i , p i ) does not contain a subgroup of index ≤ 2 i which has a nilpotent 

basis of length ≤ n. 

After passing to a subsequence we can assume that (N i , p i ) converges to (X, p ∞ ). 

We choose q i ∈ B 1/4 (p i ) such that q i converges to a regular point q ∈ X. 

Choose λ i → ∞ very slowly such that π 1 (λ i N i , q i ) is still generated by loops of 

length ≤ 1 and such that (λ i N i , q i ) converges to C q X ∼ = R k for some k ≥ 0. 

Notice that the rescaled sequence M i = λ i N i satisfies Ric Mi ≥ − n−1 → 0 

λ 2 i 

and B ri (q i ) is compact with r i = λi 

2 

→ ∞. Thus the existence of this sequence 

contradicts the Induction Theorem 6.1 with fi 1 = · · · = f i k = id. 

We can now finish the proof of the theorem by establishing. 

Step 2. Consider ε 1 (n) > 0 and Γ from Step 1. Then there are ε 2 (n), C 2 (n) > 0 

such that the group 

〈 {g 

H := ∈ π1 (B 1 (p), p) | d(˜p, g˜p) ≤ ε 2 (n) }〉 

satisfies: Γ ∩ H has index at most C 2 (n) in H. 

We will provide (in principle) effective bounds on ε 2 and C 2 depending on n and 

the (ineffective) bound ε 1 (n). Put 

Γ ′ := 

〈 {g 

∈ π1 (B 1 (p), p) | d(q, gq) ≤ ε 1 (n) for all q ∈ B 3/4 (˜p) }〉 ⊂ Γ.


By Theorem 2.5, there exists h = h(n) such that H can be generated by some 

b 1 , . . . , b h ∈ H satisfying d(˜p, b i ˜p) ≤ 4ε 2 (n) for any i = 1, . . . h. We can obviously 

assume that the generating set {b 1 , . . . , b h } contains inverses of all its elements. We 

proceed in three substeps. 

Claim 1. There is a positive integer L(n, ε 1 (n)) such that for all ε 2 (n) < 1 

100L and 

any choice of g i ∈ {b 1 , . . . , b h }, i = 1, . . . , L we can find l, k ∈ {1, . . . , L} with l < k 

and g l · g l+1 · · · g k ∈ Γ ′ . 

We assume ε 2 (n) < 1 

100L 

and we will show that there is an a priori estimate for 

L. Notice that d(˜p, g 1 · · · g l ˜p) < 4ε 2 (n)l ≤ 1 25 holds for l = 1, . . . , L. Thus g 1 · · · g l 

maps the ball of radius B 2/3 (p) into the ball B 3/4 (p). 

We choose a maximal collection of points {a 1 , . . . , a m } in B 2/3 (p) with pairwise 

distances ≥ ε 1 (n)/4. It is immediate from Bishop–Gromov that m can be estimated 

just in terms of ε 1 and n. 

Assume we can choose g i ∈ {b 1 , . . . , b h }, i = 1, . . . , L, such that g l · · · g k ∉ Γ ′ 

for all 1 ≤ l < k ≤ L. This implies that d(g l · · · g k a u , a u ) ≥ ε 1 (n)/4 for some 

u = u(l, k) ∈ {1, . . . , m}. Hence, 

d(g 1 · · · g k a u(l,k) , g 1 · · · g l a u(l,k) ) ≥ ε 1 (n)/4, for 1 < k < l < L. 

We consider the diagonal action of Γ on the m-fold product Ñ i 

m and the point 

a = (a 1 , . . . , a m ). We just showed d(g 1 · · · , g k a, g 1 · · · g l a) ≥ ε 1 (n)/4 for all 1 ≤ 

k < l ≤ L. Thus there L points in the m-fold product B 3/4 (˜p) m which are ε 1 (n)/4 

separated. Now the Bishop–Gromov inequality provides an a priori bound for L. 

Claim 2. Choose L as in Claim 1 and assume ε 2 (n) ≤ 1 

w = b ν1 . . . b νl of length l ≤ L we have wΓ ′ w −1 ⊂ Γ. 

100L 

. Then for every word 

Let γ ′ ∈ π 1 (B 1 (p)) be an element that displaces every point in B 2/3 (˜p) by at most 

ε 1 (n). By the definition of Γ ′ it suffices to show that wγ ′ w −1 ∈ Γ. Let q ∈ B 1/2 (˜p). 

Since ε 2 (n) < 1 

100L , it follows that w−1 ˜p ∈ B 1/25 (˜p) and hence w −1 q ∈ B 2/3 (˜p). 

Therefore 

d(q, wγ ′ w −1 q) = d(w −1 q, γ ′ w −1 q) < ε 1 (n) for all q ∈ B 1/2 (˜p). 

This proves Claim 2. Step 2 now follows from 

Claim 3. Every element h ∈ H is of the form h = wγ with γ ∈ Γ and w = b ν1 · · · b νl 

with l ≤ L. 

We write h = w · γ, where w is a word of length l. We may assume l is chosen 

minimal. It suffices to prove that l ≤ L. Suppose l > L. Then we can apply 

Claim 1 to the tail of w and obtain w = w 1 γ ′ w 2 with γ ′ ∈ Γ ′ , w 2 a word of length 

< L and the word-length of w 1 · w 2 is smaller than the length of w. 

By Claim 2, w2 −1 γ′ w 2 ∈ Γ. Putting γ 2 = w2 −1 γ′ w 2 γ gives h = w 1 · w 2 γ 2 – a 

contradiction, as the length of w 1 w 2 is smaller. 

□ 

Corollary 7.1. In each dimension n there exists ε > 0 such that the following 

holds for any complete n-manifold M and p ∈ M with Ric > −(n − 1) on B 1 (p). 

If the image of π 1 (B ε (p)) → π 1 (B 1 (p)) contains a nilpotent group of rank n, then 

M is homeomorphic to a compact infranilmanifold.


Actually, we will only show that M is homotopically equivalent to an infranilmanifold. 

By work of Farrell and Hsiang [FH83] this determines the homeomorphism 

type in dimensions above 4. The 4 -dimensional case follows from work of 

Freedman–Quinn [FQ90]. Lastly, the 3-dimensional case follows from Perelman’s 

solution of the geometrization conjecture. 

Proof of Corollary 7.1. Notice that it suffices to prove that M i is aspherical for 

large i: In fact, since the cohomological dimension of a rank n nilpotent group 

is n, the group π 1 (M i ) must then be a torsion free virtually nilpotent group of 

rank n and thus, by a result of Lee and Raymond [LR85], it is isomorphic to the 

fundamental group of an infranilmanifold. Therfore M i is homotopically equivalent 

to an infranilmanifold and as explained above this then gives the result. 

We argue by contradiction and assume that we can find ε i → 0 and a sequence 

of complete manifolds M i with Ric > −1 on B 1 (p i ) such that the image of 

π 1 (B εi (p i ), p i ) → π 1 (B 1 (p i ), p i ) contains a nilpotent group of rank n and M i is not 

aspherical. 

By the Margulis Lemma (Theorem 1), the nilpotent group can be chosen to 

have index ≤ C(n) in the image. Arguing on the universal cover of B 1 (p i ) it is not 

hard to deduce that there is δ i → 0 such that for all q i ∈ B 1/10 (p i ) the image of 

π 1 (B δi (q i ), q i ) → π 1 (B 1 (p i ), p i ) also contains a nilpotent subgroup of rank n. 

Next we observe that diam(M i , p i ) → 0. In fact, otherwise we could, after passing 

to a subsequence, assume that (M i , p i ) converges in the Gromov–Hausdorff sense 

to (Y, p ∞ ) with Y ≠ pt. Choose q i ∈ B 1/10 (p i ) converging to a regular point q ∞ 

in the limit Y . Similarly, choose λ i ≤ 1 √ δi 

slowly converging to infinity such that 

(λ i M i , q i ) → (C q∞ Y, o) = (R k , 0) for some k > 0. Let Ñi denote the universal cover 

of B 1 (p i ) and ˜q i a lift of q i . Let Γ i be the subgroup of the deck transformation group 

generated by those elements which displace q i by at most δ i . By construction, Γ i 

contains a nilpotent group of rank n. Put N i = Ñi/Γ i . Since (λ i N i , q i ) → (R k , 0) 

with k > 0, we get a contradiction to the Induction Theorem (6.1). 

Thus, diam(M i ) → 0 and in particular, M i = B 1 (p i ) is a closed manifold for all 

large i. After a slow rescaling we may assume in addition that Ric Mi ≥ −h i → 0. 

Next we plan to show that the universal cover of M i converges to R n . This will 

follow from 

Claim 1. Suppose a sequence of complete n-manifolds (N i , p i ) with lower Ricci 

curvature bound > −h i → 0 converges to (R k × K, p ∞ ) with K compact. Also 

assume that π 1 (N i ) is generated by loops of length ≤ R and contains a nilpotent 

subgroup of rank n − k. Then the universal cover of (Ñi, ˜p i ) converges to (R n , 0). 

We argue by reverse induction on k. The base of induction k = n is obvious. 

By the Induction Theorem, after passing to a bounded cover, we may assume 

that π 1 (N i ) itself is a torsion free nilpotent group of rank n − k. 

Choose N i ⊳ π 1 (N i ) with π 1 (N i )/N i 

∼ = Z. Then ( Ñ i /N i , ˆp i ) converges to (R l × 

K ′ , (0, p ∞ )) for some l > k and K ′ compact and the claim follows from the induction 

assumption. 

Therefore ( ˜M i , ˜p i ) converges to (R n , 0). From the Cheeger–Colding Stability Theorem 

(see Theorem 1.2) it follows that for any R > 0 the ball B R (p) is contractible 

in B R+1 (p) for all p ∈ B R (˜p i ) and i ≥ i 0 (R). Since we have a cocompact deck


transformation group with nearly dense orbits, the result actually holds for all 

p ∈ ˜M i . 

In order to show that M i is aspherical we may replace M i by a bounded cover and 

thus, by Theorem 1 without loss of generality, π 1 (M i ) has a nilpotent basis of length 

≤ n. Because rank(π 1 (M i )) ≥ n it follows that π 1 (M i ) is torsion free. Therefore 

we can choose subgroups {e} = N i 0 ⊳ · · · ⊳ N i n = π 1 (M i ) with N i j /Ni j−1 ∼ = Z. 

In the rest of the proof we will slightly abuse notations and sometimes drop 

basepoints when talking about pointed Gromov–Hausdorff convergence when the 

base points are clear. 

Note that the above claim easily implies that ˜M i /N i j → Rn−j for all j = 0, . . . n. 

Claim 2. N i j ⋆ B r(˜p i ) is contractible in N i j ⋆ B 4 4j r (˜p i) for j = 1, · · · , n and r ∈ 

[ 

1, 4 

4 (2n−j)2 ] and all large i. 

We want to prove the statement by induction on j. For j = 0 it holds as was 

pointed out above. Suppose it holds for j < n and we need to prove it for j + 1. 

Choose g ∈ N i j+1 representing a generator of Ni j+1 /Ni j . 

Notice that N i j+1 ⋆ B R(˜p i )/N i j converges to R × B R(0) ⊂ R n−j as i → ∞ where 

B R (0) is the ball in R n−j−1 . Moreover, the action of N i j+1 /Ni j on the set converges 

to the R action on R × B R (0) given by translations. 

We can also find finite index subgroups ¯N i j+1 ⊂ Ni j+1 with Ni j ⊂ ¯N i j+1 such that 

N i j+1 ⋆ B R(˜p i )/¯N i j+1 converges to S1 × B R (0) ⊂ S 1 × R n−j where S 1 has diameter 

10 −n . It is easy to construct a smooth map ¯σ : N i j+1 ⋆ B R(˜p i )/¯N i j+1 → S1 that is 

arbitrary close to the projection map S 1 × B R (0) → S 1 in the Gromov–Hausdorff 

sense. 

We can lift ¯σ to a map σ : N i j+1 ⋆ B R(˜p i ) → R. Notice that σ commutes with the 

action of ¯N i j+1 where ¯N i j+1 /Ni j can be thought of as the deck transformation group 

of the covering R → S 1 . 

It suffices to show that N i j+1 ⋆ B r(˜p i ) → N i j+1 ⋆ B 4 4j+1 r (˜p i) induces the trivial 

map on the level of homotopy groups. Thus, we have to show that for any k > 0 

any map ι: S k → N i j+1 ⋆ B r(˜p i ) is null homotopic in N i j+1 ⋆ B 4 4j+1 r (˜p i). 

We can assume that ι is smooth. The image of σ ◦ ι is given by an interval 

[a, b] in R. We choose a 10 −n -fine finite subdivision a < t 1 < · · · < t h 

interval such that the t α are regular values. Thus ι −1 (σ −1 (t α )) = H α is a smooth 

hypersurface in S k for every α. 

Notice that by construction, the image ι(H α ) is contained in gN j ⋆ B 2r (˜p i ) for 

some g ∈ ¯N i j+1 . By induction assumption, ι |H α 

is homotopic to a point map in 

gN i j ⋆ B 4 4j 2r (˜p i). We homotope ι into ˜ι such that ˜ι(H α ) is a point for all α and ˜ι is 

4 4j 4r close to ι. 

Consider now all components of H α for all α. They divide the sphere into 

connected regions such that each boundary component of a region is mapped by ˜ι 

to a point and the whole region is mapped to a set gN i j ⋆ B 4 4j 8r (˜p i) for some g. 

Thus the map ˜ι restricted to a region with crushed boundary components is null 

homotopic in g ⋆B 4 2·4 j 8r (˜p i) by the induction assumption. Clearly, this implies that 

ι is null homotopic in N i j+1 ⋆ B 4 4j+1 r (˜p i). 

This finishes the proof of Claim 2. Notice that N i n ⋆ B 1 (˜p i ) = ˜M i for large i since 

diam(M i ) → 0. Thus ˜M i is contractible by Claim 2. 

□


8. Almost nonnegatively curved manifolds with maximal first Betti 

number 

This is the only section where we assume lower sectional curvature bounds. The 

main purpose is to prove 

Corollary 8.1. In each dimension there are positive constants ε(n) and p 0 (n) such 

that for all primes p > p 0 (n) the following holds. 

Any manifold with diam(M, g) 2 K sec ≥ −ε(n) and b 1 (M, Z p ) ≥ n is diffeomorphic 

to a nilmanifold. Conversely, for every p, every compact n-dimensional nilmanifold 

covers another (almost flat) nilmanifold M with b 1 (M, Z p ) = n. 

The second part of the Corollary is fairly elementary but it provides counterexamples 

to a theorem of Fukaya and Yamaguchi [FY92, Corollary 0.9] which asserted 

that only tori should show up. The second part follows from the following Lemma 

and the fact that every nilmanifold is almost flat, i.e. for any i > 0 it admits a 

metric with diam(M, g) 2 |K sec | ≤ 1 i . 

Lemma 8.2. Let Γ be a torsion free nilpotent group of rank n, and let p be a prime 

number. 

a) There is a subgroup of finite index Γ ′ ⊂ Γ for which we can find a surjective 

homomorphism Γ ′ → (Z/pZ) n . 

b) We can find a torsion free nilpotent group ˆΓ containing Γ as finite index 

subgroup such that there a surjective homomorphism ˆΓ → (Z/pZ) n . 

Proof. a) We argue by induction on n. The statement is trivial for n = 1. Assume 

it holds for n − 1. Choose a subgroup Λ ⊳ Γ with Γ/Λ ∼ = Z. By the induction 

assumption, we can find a subgroup Λ ′ ⊂ Λ of finite index and Λ ′′ ⊳ Λ ′ with 

Λ ′ /Λ ′′ ∼ = (Z/pZ) n−1 . Let g ∈ Γ represent a generator of Γ/Λ. Since Λ contains 

only finitely many subgroups with of index ≤ [Λ : Λ ′′ ], it follows that g l is in the 

normalizer of Λ ′ and Λ ′′ for a suitable l > 0. After increasing l further we may 

assume that the automorphism of Λ ′ /Λ ′′ ∼ = (Z/pZ) n−1 induced by the conjugation 

by g l is the identity. 

Define Γ ′ as the group generated by Λ ′ and g l . It is now easy to see that Γ ′ /Λ ′′ 

is isomorphic to (Z/pZ) n−1 × Z and thus Γ ′ has a surjective homomorphism to 

(Z/pZ) n . 

b). An analysis of the proof of a) shows that in a) the index of Γ ′ in Γ can 

be bounded by a constant C only depending on n and p. Let L be the Malcev 

completion of Γ, i.e. L is the unique n-dimensional nilpotent Lie group containing 

Γ as a lattice. Let exp: l → L denote the exponential map of the group. It is 

well known that the group ¯Γ generated by exp( 1 C! exp−1 (Γ)) contains Γ as a finite 

index subgroup. By part a) we can assume now that ¯Γ contains a subgroup ˆΓ of 

index ≤ C such that there is a surjective homomorphism to ˆΓ → (Z/pZ) n . By 

construction, any subgroup of ¯Γ of index ≤ C must contain Γ and hence we are 

done. 

□ 

Lemma 8.3. For any n there exists p(n) such that if Γ is a torsion free virtually 

nilpotent of rank n which admits an epimorphism onto (Z/pZ) n for some p ≥ p(n) 

then Γ is nilpotent. 

Proof. By [LR85], Γ is an almost crystallographic group, that is, there is an n- 

dimensional nilpotent Lie group L, a compact subgroup K ⊂ Aut(L) such that Γ


is isomorphic to a lattice in K ⋉ L. By a result of Auslander (see [LR85] for a 

proof), the projection of Γ to K is finite and we may assume that the projection 

is surjective. Thus K is a finite group and it is easy to see that the action of K 

on L/[L, L] is effective. It is known that N ′ := Γ ∩ [L, L] is a lattice in [L, L]. Thus, 

the image ¯Γ := Γ/N ′ of Γ in K ⋉ L/[L, L] is discrete and cocompact. Since K acts 

effectively on L/[L, L], we can view K ⋉ L/[L, L] as a cocompact subgroup of Iso(R k ) 

where k = dim(L/[L, L]). 

In particular, ¯Γ is a crystallographic group. As the projection of Γ to K is 

surjective, ¯Γ is abelian if and only if K is trivial. If it is not abelian, then 

rank(¯Γ/[¯Γ, ¯Γ]) < rank(¯Γ) = k. 

By the third Bieberbach theorem there are only finitely many crystallographic 

groups of any given rank, and consequently there are only finitely many possibilities 

for the isomorphism type of ¯Γ/[¯Γ, ¯Γ]. Obviously, any homomorphism ¯Γ → (Z/pZ) k 

factors through ¯Γ → ¯Γ/[¯Γ, ¯Γ] → (Z/pZ) k ; this easily implies that there is p 0 such 

that that there is no surjective homomorphism ¯Γ → (Z/pZ) k for p ≥ p 0 . 

Therefore, there is no surjective homomorphism Γ → (Z/pZ) n for p > p 0 . If we 

put p(n) = p 0 , then ¯Γ is abelian, K = {e} and hence Γ is nilpotent. 

□ 

Corollary 8.1 follows from Lemma 8.2, Lemma 8.3 and the following result 

Corollary 8.4. In each dimension there are positive constants C(n) and ε(n) such 

that for any complete manifold with K sec > −1 on B 1 (0) one of the following holds 

a) The image π 1 (B ε(n) (p)) → B 1 (p) contains a subgroup N of index less than 

C(n) such that N has a nilpotent basis of length ≤ n − 1 or 

b) M is compact and homeomorphic to an infranilmanifold. Moreover, any 

finite cover with a nilpotent fundamental group is diffeomorphic to a nilmanifold. 

Proof. Consider a sequence of manifolds (M i , p i ) with K sec > −1 such that M i 

does not satisfy a) with C = i and ε(n) = 1 i 

. As in proof of Corollary 7.1, it is clear 

that diam(M i ) → 0. 

By the Margulis Lemma or by work of [KPT10], we may assume that π 1 (M i ) is 

nilpotent. We plan to show that π 1 (M i ) has rank n for large i. 

The fact that π 1 (M i ) does not contain a subgroup of bounded index with a 

nilpotent basis of length ≤ n − 1, guarantees a certain extremal behavior if we run 

through the proof of the Induction Theorem. It will be clear that the blow up 

limits R k × K, that occur if we go through the procedure provided in the proof of 

the Induction Theorem, are flat – more precisely each K has to be a torus. This 

follows from the fact that we can never run into a situation where Step 3 of the 

proof of the Induction Theorem applies. 

In particular, if we rescale the manifolds to have diameter 1 they must converge 

to a torus K = T h . We can now use Yamaguchi’s fibration theorem [Yam91] to get 

a fibration F i → M i → T h . 

Let Γ i denote the kernel of π 1 (M i ) → π 1 (T h ). In the next h steps of the procedure 

in the proof of the Induction Theorem we replace M i by ˜M i /Γ i endowed with 

h deck transformations fi 1, . . . , f i h projecting to generators of π 1 (T h ). 

( ˆM i := ˜M i /Γ i , ˆp i ) converges to (R h , 0). The next step in the Induction Theorem 

would be to rescale ˆM i so that (λ i ˆMi , ˆp i ) → (R h × K ′ , ˆp ∞ ) with K ′ not a point 

and it will be clear again that K ′ ∼ = T h′ is a torus.


It is easy to see that 1 λ i 

is necessarily comparable to the size of the fiber F i of 

the Yamaguchi fibration. 

From the exact homotopy sequence we can deduce that Γ i = π 1 ( ˆM i ) ∼ = π 1 (F i ). 

The fibration theorem ensures that F i fibers over a new torus T h′ . Let Γ ′ i denote 

the kernel of π 1 (F i ) → π 1 (T h′ ). Clearly π 1 (M i ) has rank n if and only if Γ ′ i has 

rank n − h − h ′ . After finitely many similar steps this shows that π 1 (M i ) has rank 

n. By Corollary 7.1 the first part of b) follows. To get the second part of part b) 

observe that if π 1 (M i ) is nilpotent and torsion free then by the proof of [KPT10, 

Theorem 5.1.1], for all large i we have that M i smoothly fibers over a nilmanifold 

with simply connected factors. The fibers obviously have to be trivial as M i is 

aspherical which means that M i is diffeomorphic to a nilmanifold. 

□ 

Problem. a) We suspect that in the case of almost nonnegatively curved manifolds 

Corollary 8.4 remains valid if one replaces the n − 1 in a) by n − 2. 

b) The result of this section remain valid if one just has a uniform lower bound 

on sectional curvature and lower Ricci curvature bounds close enough to 0. 

However, it would be nice to know whether or whether not they remain valid 

without sectional curvature assumptions. 

9. Finiteness results 

In the proof of the following theorem the work of Colding and Naber [CN10] is 

key. 

Theorem 9.1 (Normal Subgroup Theorem). Given n, D, ε 1 > 0 there are positive 

constants ε 0 < ε 1 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 , ε 1 ] and a normal subgroup N ⊳ π 1 (M) such that for all p ∈ M 

we have: 

• the image of π 1 (B ε/1000 (p), p) → π 1 (M, p) contains N and 

• the index of N in the image of π 1 (B ε (p), p) → π 1 (M, p) is ≤ C. 

To avoid confusion, we should recall that a normal subgroup N ⊳ π 1 (M, p) naturally 

induces a normal subgroup N ⊳ π 1 (M, q) for all p, q ∈ M. 

If we put ε 1 = ε Marg (n), the constant in the Margulis Lemma, then by Theorem 

1, the group N in the above theorem contains a nilpotent subgroup of index 

≤ C Marg which has a nilpotent basis of length ≤ n. After replacing N by a characteristic 

subgroup of controlled index Theorem 6 follows. 

Proof of Theorem 9.1. We will argue by contradiction. It suffices to rule out the 

existence of a sequence (M i , g i ) with the following properties. 

(1) Ric Mi > −(n − 1), diam(M i ) ≤ D; 

(2) for all ε ∈ (2 −i , ε 1 ) and any normal subgroup N⊳π 1 (M i ) one of the following 

holds 

(i) N is not contained in the image of π 1 (B ε/1000 (p)) for some p ∈ M i or 

(ii) the index of N in the image of π 1 (B ε (q i ), q i ) → π 1 (M i , q i ) is ≥ 2 i for 

some q i ∈ M i . 

Since any subsequence of (M i , g i ) also satisfies this condition, we can assume that 

the universal covers ( ˜M i , π 1 (M i ), ˜p i ) converge to (Y, Γ ∞ , ˜p ∞ ). Put Γ i := π 1 (M i ).


For δ > 0 define 

S i (δ) := { g ∈ Γ i | d(p, gp) ≤ δ for all p ∈ B 1/δ (˜p i ) } and Γ iδ := 〈S i (δ)〉 ⊂ Γ i 

for i ∈ N ∪ {∞}. 

By Colding and Naber [CN10], Γ ∞ is a Lie group. In particular, the component 

1 

group Γ ∞ /Γ ∞0 is discrete and hence there is some positive δ 0 ≤ min{ε 1 , 

2D } such 

that Γ ∞δ is given by the identity component of Γ ∞ for all δ ∈ (0, 2δ 0 ). 

It is easy to see that this property carries over to the sequence in the following 

sense. We fix a large constant R ≥ 10, to be determined later, and put ε 0 := δ 0 /R. 

Then Γ iδ = Γ iε0 for all δ ∈ [ε 0 /1000, Rε 0 ] and all i ≥ i 0 (R). 

We claim that Γ iε0 is normal in Γ i . In fact, Γ i can be generated by elements 

which displace ˜p i by at most 2D ≤ 1 ε 0 

. It is straightforward to check that for any 

g i ∈ Γ i satisfying d(g i ˜p i , ˜p i ) ≤ 1 ε 0 

we have g i S(ε 0 /2)g −1 

i ⊂ S(ε 0 ). Thus, 

g i Γ iε0/2g −1 

i ⊂ Γ iε0 = Γ iε0/2 

and clearly the claim follows. 

In summary, Γ iε0/1000 = Γ iRε0 ⊳ Γ i with ε 0 ≤ ε 1 for all i ≥ i 0 (R). By the choice 

of our sequence this implies that for some q i ∈ M the index of Γ iε0 in the image of 

π 1 (B ε0 (q i ), q i )) → π 1 (M i ) is larger than 2 i . 

Similarly to Step 2 in the proof of the Margulis Lemma in section 7, this gives a 

contradiction provided we choose R sufficiently large. In fact, the present situation 

is quite a bit easier as Γ iRε0 is normal in π 1 (M i ). □ 

Lemma 9.2. a) Let ε 0 , D, H, n > 0. Then there is a finite collection of groups 

such that the following holds: 

Let (M, g) be a compact n-dimensional manifold with diam(M) < D, 

Ric(M) > −(n−1) and a normal subgroup N⊳π 1 (M) and ε ≥ ε 0 . Suppose 

that 

• The image I of π 1 (B ε (q)) → π 1 (M, q) satisfies that the index [I : I ∩N] 

is bounded by H, for all q ∈ M. 

• The group N is in the image of π 1 (B ε/1000 (q 0 )) → π 1 (M, q 0 ) for some 

q 0 ∈ M. 

Then π 1 (M)/N is isomorphic to a group in the a priori chosen finite collection. 

b) (Bounded presentation) Given a manifold as in a) one can find finitely 

many loops in M based at p ∈ M whose length and number is bounded by 

an a priori constant such that the following holds: the loops project to a 

generator system of π 1 (M, p)/N and there are finitely many relations (with 

an a priori bound on the length of the words involved) such that these words 

provide a finite presentation of the group π 1 (M, p)/N. 

Instead of compact manifolds one could also consider manifolds satisfying the 

condition that the image of π 1 (B r0 (p 1 )) → π 1 (B R0 (p 1 )) is via the natural homomorphism 

isomorphic to π 1 (M i ), where r 0 and R 0 are given constants. In that 

case one would only need the Ricci curvature bound in the ball B R0+1(p 1 ). In fact, 

the only purpose of the Ricci curvature bound is to guarantee the existence of an 

ε/100–dense finite set in M with an a priori bound on the order of the set. 

Of course, b) implies a). However, we prove a) first and then see that b) follows 

from the proof. In the context of equivariant Gromov–Hausdorff convergence a


bounded presentation can often be carried over to the limit. That is why part b) 

has some advantages over a). 

Proof. a). The idea is to define for each such manifold a 2-dimensional finite CWcomplex 

C whose combinatorics is bounded by some a priori constants and which 

has the fundamental group π 1 (M)/N. 

Choose a maximal collection of points p 1 , . . . , p k in M with pairwise distances 

≥ ε/100. Clearly, there is an a priori bound on k depending only on n, D and ε 0 . 

We also may assume that p 1 = q 0 . 

The points p 1 , . . . , p k will represent the 0-skeleton C 0 of our CW -complex. 

For each point p i define F i as the finite group given by im ε/5 (p i )/(N ∩ im ε/5 (p i )) 

where im δ (p i ) denotes the image of π 1 (B δ (p i ), p i ) → π 1 (M, p i ). For each of the at 

most H elements in F i we choose a loop in B ε/5 (p i ) representing the class modulo 

N. We attach to p i an oriented 1-cell for each loop and call the loop the model path 

of the cell. 

For any two points p i , p j with d(p i , p j ) < ε/20 we choose in the manifold a 

shortest path from p i to p j . In the CW -complex we attach an edge (1-cell) between 

these two points. We call the chosen path the model path of the edge. 

This finishes the definition of the 1-skeleton C 1 of our CW -complex. 

For each point p i ∈ C 0 we consider the part A i of the 1-skeleton C 1 containing 

all points p j with d(p i , p j ) < ε/2 and also all 1-cells connecting them including the 

cells attached initially. If A i has more than one connected component we replace 

A i by the connected component containing p i . 

We choose loops in A i based at p i representing free generators of the free group 

π 1 (A i , p i ). We now consider the homomorphism π 1 (A i , p i ) → π 1 (M, p i ) induced 

by mapping each loop to its model path in M. The induced homomorphism 

α i : π 1 (A i ) → π 1 (M, p i )/N has finite image of order ≤ H since the model paths 

are contained in B ε (p i ). 

The kernel of α i is a normal subgroup of finite index ≤ H in π 1 (A i , p i ). Thus, 

it is finitely generated and there is an a priori bound on the number of possibilities 

for the kernel. We choose a free finite generator system of the kernel and attach 

the corresponding 2-cells to the CW -complex for each generator of the kernel. 

This finishes the definition of the CW -complex C. 

Note that by construction, the total number of cells in our complex is bounded 

by a constant depending only on n, D, H and ε 0 and that the attaching maps of the 

two cells are under control as well. Hence there are only finitely many possibilities 

for the homotopy type of C. Thus, we can finish the proof by establishing that 

π 1 (M)/N is isomorphic to π 1 (C). 

First notice that there is a natural surjective map π 1 (C) → π 1 (M)/N: Consider 

the 1-skeleton C 1 of C. Its fundamental group is free and there is a homomorphism 

π 1 (C 1 ) → π 1 (M) → π 1 (M)/N induced by mapping a path onto its model 

path. The two cells that where attached to the 1-skeleton only added relations to 

the fundamental group contained in the kernel of this homomorphism. Thus, the 

homomorphism induces a homomorphism π 1 (C, p 1 ) → π 1 (M, p 1 )/N. Moreover, it 

is easy to see that the homomorphism is surjective. 

In order to show that the homomorphism is injective we need to show that 

homotopies in M can be lifted in some sense to C. 

Suppose we have a closed curve γ in the 1-skeleton C 1 which is based at p 1 and 

whose model curve c 0 in M is homotopic to a curve in N ⊂ π 1 (M). We parametrize


γ on [0, j 0 ] for some large integer j 0 . We will assume that γ(t) = p j for some j 

implies t ∈ [0, j 0 ] ∩ Z – we do not assume that the converse holds. 

Using that p 1 = q 0 we can find a homotopy H : [0, 1] × [0, j 0 ] → M, (s, t) ↦→ 

H(s, t) = c s (t) such that each c s is a closed loop at p 1 , c 0 is the original curve and 

c 1 is a curve in the ball B ε/1000 (p 1 ). If we choose j 0 large enough, we can arrange 

for L(c s[i,i+1] ) < ε/1000 for all i = 0, . . . , j 0 − 1, s ∈ [0, 1]. 

After an arbitrary small change of the homotopy we can assume that c s (i) intersects 

the cut locus of the 0-dimensional submanifold {p 1 , . . . , p k } only finitely 

many times for every i = 1, . . . , j 0 − 1. We can furthermore assume that at any 

given parameter value s there is at most one i such that c s (i) is in the cut locus of 

{p 1 , . . . , p k }. 

For each c s (i) we choose a point ˜c s (i) := p li (s) with minimal distance to c s (i). 

By construction ˜c s (i) is piecewise constant in s and at each parameter value s 0 

at most one of the chosen points is changed. Moreover, ˜c s (0) = ˜c s (1) = p 1 and 

˜c 1 (i) = p 1 . 

We now define ˜c s|[i,i+1] in three steps. In the middle third of the interval we run 

through c s|[i,i+1] with triple speed. In the last third of the interval we run through 

a shortest path from c s (i + 1) to p li+1 (s). In the first third of the interval we run 

through a shortest path from p li (s) to c s (i) and if there is a choice we choose the 

same path that was chosen in the last third of the previous interval. 

Each curve ˜c s is homotopic to c s . Moreover, ˜c s (i) is piecewise constant equal to 

a vertex. Clearly, s ↦→ ˜c s is only piecewise continuous. 

The ’jumps’ for ˜c occur exactly at those parameter values at which c s (i) is the 

cut locus of the finite set {p 1 , . . . , p k } for some i. 

We introduce the following notation: for two continuous curves c 1 , c 2 in M from 

p to q ∈ M we say that c 1 and c 2 are homotopic modulo N if the homotopy class of 

the loop obtained by prolonging c 1 with the inverse parametrized curve c 2 is in N. 

We define a corresponding curve γ s lifting ˜c s to our CW -complex as follows: 

γ s|[i,i+1] should run in the one skeleton of our CW complex from ˜c s (i) to ˜c s (i+1) ∈ 

C 0 and the model path of γ s|[i,i+1] should be modulo N homotopic to ˜c s|[i,i+1] . We 

choose γ s[i,i+1] in such a way that on the first half of the interval it runs through 

the edge from ˜c s (i) to ˜c s (i + 1) and in order to get the right homotopy type for the 

model path, in the second half of the interval it runs through one of the 1-cells that 

were attached initially to the vertex ˜c s (i + 1) ∈ C 0 . 

Thus, the model curve of γ s|[i,i+1] is homotopic to ˜c s|[i,i+1] modulo N. In particular, 

the model curve of γ s and the loop c s are homotopic modulo N. 

The map s ↦→ γ s is piecewise constant in s. We have to check that at the 

parameter where γ s jumps the homotopy type of γ s does not change. 

However, first we want to check that the original curve γ is homotopic to γ 0 in C. 

By our initial assumption, there are integers 0 = k 0 < . . . < k h = j 0 such that γ(t) 

is in the 0-skeleton if and only if t ∈ {k 0 , . . . , k h }. Going through the definitions 

above it is easy to deduce that γ(k i ) = c 0 (k i ) = ˜c 0 (k i ) = γ 0 (k i ). Thus, it suffices to 

check that γ |[ki,k i+1] is homotopic to γ 0|[ki,k i+1]. We have just seen that the model 

curve of γ 0|[ki,k i+1] is homotopic to ˜c 0|[ki,k i+1] modulo N. This curve is homotopic 

to c 0|[ki,k i+1], which is the model curve of γ |[ki,k i+1]. 

Moreover, the model curve of γ |[ki,k i+1] is in an ε/5-neighbourhood of γ(k i ). By 

the definition of ˜c 0 this implies that ˜c 0|[ki,k i+1] is in an ε/4-neighbourhood of γ(k i ). 

Therefore, γ |[ki,k i+1] as well as γ 0|[ki,k i+1] only meet points of the zero skeleton


with distance < ε/4 to γ(k i ). Thus, they are contained in A j , where j is defined 

by p j = γ(k i ). Since their model curves are homotopic modulo N, our rules for 

attaching 2-cells imply that γ 0|[ki,k i+1] is homotopic to γ [ki,k i+1] in C. 

It remains to check that the homotopy type of γ s does not change at a parameter 

s 0 at which s ↦→ γ s is not continuous. By construction there is an i such that 

γ s|[0,i−1] and γ s|[i+1,j0] are independent of s ∈ [s 0 − δ, s 0 + δ] for some δ > 0. 

The model curve of γ s0−δ|[i−1,i+1] is modulo N homotopic to ˜c s0−δ|[i−1,i+1] which 

in turn is homotopic to ˜c s0+δ|[i−1,i+1] and, finally, this curve is modulo N homotopic 

to the model curve of γ s0+δ|[i−1,i+1]. 

Furthermore, the model curves of γ s0±δ|[i−1,i+1] are in an ε/2-neighbourhood of 

˜c s0 (i − 1). Since they are homotopic modulo N, this implies by definition of our 

complex that γ s0−δ|[i−1,i+1] is homotopic to γ s0+δ|[i−1,i+1] in C . 

Thus, the curve γ is homotopic to γ 1 , which is the point curve by construction. 

b) Since the number of CW -complexes constructed in a) is finite, we can think 

of C as fixed. We choose loops in C 1 representing a free generator system of the 

free group π 1 (C 1 , p 1 ). The model curves of these loops represent generators of 

π 1 (M, p 1 )/N and the lengths of the loops are bounded. 

For each of the attached 2-cells we consider the loop based at some vertex p i in 

the one skeleton given by the attaching map. We choose a path in C 1 from p 1 to 

p i and conjugate the loop back to π 1 (C 1 , p 1 ). 

We can express this loop as word in our free generators and if collect all these 

words for all 2-cells, we get a finite presentation of π 1 (C, p 1 ) ∼ = π 1 (M)/N. □ 

It will be convenient for the proof of Theorem 7 to restate it in a slightly different 

form. 

Theorem 9.3. a) Given D and n there are finitely many groups F 1 , . . . , F k 

such that the following holds: For any compact n-manifold M with Ric > 

−(n − 1) and diam(M) ≤ D we can find a nilpotent normal subgroup N ⊳ 

π 1 (M) which has a nilpotent basis of length ≤ n − 1 such that π 1 (M)/N is 

isomorphic to one of the groups in our collection. 

b) In addition to a) one can choose a finite collection of irreducible rational 

representations ρ j i : F i → GL(n j i , Q) (j = 1, . . . , µ i, i = 1, . . . , k) such that 

for a suitable choice of the isomorphism π 1 (M)/N ∼ = F i the following holds: 

There is a chain of subgroups Tor(N) = N 0 ⊳ · · · ⊳ N h0 = N which are all 

normal in π 1 (M) such that [N, N h ] ⊂ N h−1 and N h /N h−1 is free abelian. 

Moroever, the action of π 1 (M) on N by conjugation induces an action of F i 

on N h /N h−1 and the induced representation ρ: F i → GL ( (N h /N h−1 ) ⊗ Z Q ) 

is isomorphic to ρ j i for a suitable j = j(h), h = 1, . . . , h 0. 

In addition one can assume in a) that rank(N) ≤ n − 2. 

Proof of Theorem 9.3. We consider a contradicting sequence (M i , g i ), that is, we 

have diam(M i , g i ) ≤ D and Ric Mi ≥ −(n − 1) but the theorem is not true for any 

subsequence of (M i , g i ). 

We may assume that diam(M i , g i ) = D and (M i , g i ) converges to some space X. 

Choose p i ∈ M i converging to a regular point p ∈ X. By the Gap Lemma (2.4) 

there is some δ > 0 and ε i → 0 with π 1 (M i , p i , ε i ) = π 1 (M i , p i , δ). 

Claim. There is C and i 0 such that π 1 (M i , p i , δ) contains a subgroup of index ≤ C 

which has a nilpotent basis of length ≤ n − dim(X) for all large i ≥ i 0 .


We choose λ i < 1 ε i 

with λ i → ∞ such that (λ i M i , p i ) converges to (R u , 0) with 

u = dim(X). It is easy to see that (λ i ˜Mi /π 1 (M i , p i , ε i ), ˜p i ) converges to (R u , 0) as 

well. The claim now follows from the Induction Theorem (6.1) with f j i = id. 

We now apply the Normal Subgroup Theorem (9.1) with ε 1 = δ. There is 

a normal subgroup N i ⊳ π 1 (M i , g i ), some positive constants ε and C such that 

N i is contained in π 1 (B εi (p i )) → π 1 (M i , g i ) and N i is contained in the image of 

π 1 (B ε (q i ), q i ) → π 1 (M i , q i ) with index ≤ C for all q i ∈ M. 

By the Claim above, N i has a subgroup of bounded index with a nilpotent basis 

of length ≤ n − dim(X) ≤ n − 1. 

We are free to replace N i by a characteristic subgroup of bounded index and 

thus, we may assume that N i itself has a nilpotent basis of length ≤ n − dim(X). 

By Lemma 9.2 a), the number of possibilities for π 1 (M i )/N i is finite. In particular, 

part a) of Theorem 9.3 holds for the sequence and thus either b) or the 

addendum is the problem. 

By Lemma 9.2 b), we can assume that π 1 (M i , p i )/N i is boundedly presented. By 

passing to a subsequence we can assume that we have the same presentation for all i. 

Let g i1 , . . . , g iβ ∈ π 1 (M i ) denote elements with bounded displacement projecting to 

our chosen generator system of π 1 (M i )/N i . Moreover, there are finitely many words 

in g i1 , . . . , g il (independent of i) such that these words give a finite presentation of 

the group π 1 (M i )/N i . 

We can assume that g ij converges to g ∞j ∈ G ⊂ Iso(Y ) where (Y, ˜p ∞ ) is the 

limit of ( ˜M i , ˜p i ). Let N ∞ ⊳G be the limit of N i . By construction, G/N ∞ is discrete. 

Since π 1 (M i )/N i is boundedly presented, it follows that there is an epimorphism 

π 1 (M i )/N i → G/N ∞ induced by mapping g ij to g ∞j , for all large i. 

b). We plan to show that a subsequence satisfies b). We may assume that G⋆ ˜p ∞ 

is not connected and by the Gap Lemma (2.4) 

(23) 

2ρ 0 := min{d(˜p ∞ , g˜p ∞ ) | g˜p ∞ /∈ G 0 ⋆ ˜p ∞ } > 0. 

Let r be the maximal nonnegative integer such that the following holds. After 

passing to a subsequence there is a subgroup H i ⊆ N i of rank r satisfying 

• H i ⊳ π 1 (M i ), N i /H i is torsion free and 

• there is a chain Tor(N i ) = N i0 ⊳ · · · ⊳ N ih0 = H i such that each group 

N ih is normal in π 1 (M i ), [N i , N ih ] ⊂ N ih−1 , N ih /N ih−1 is free abelian and 

for each h = 1, . . . , h 0 the induced representations of F = π 1 (M i )/N i in 

(N ih /N ih−1 ) ⊗ Z Q and in (N jh /N jh−1 ) ⊗ Z Q are equivalent for all i, j. 

Here we used implicitly that we have a natural isomorphism between π 1 (M i )/N i 

and π 1 (M j )/N j in order to talk about equivalent representations. 

Notice that these statements hold for H i = Tor(N i ). We need to prove that H i = 

N i . Suppose on the contrary that rank(H i ) < rank(N i ). Consider ˆM i := ˜M i /H i 

endowed with the action of Γ i := π 1 (M i )/H i , and let ˆp i be a lift of p i to ˆM i . By 

construction ˆN i := N i /H i is a torsion free normal subgroup of Γ i . 

We claim that there is a central subgroup A i ⊂ ˆN i of positive rank which is 

normal in Γ i and is generated by {a ∈ A i | d(ˆp i , aˆp i ) ≤ d i } for a sequence d i → 0: 

Recall that ˆN i is generated by {a ∈ ˆN i | d(ˆp i , aˆp i ) ≤ ε i }. If ˆN i is not abelian this 

implies that [ˆN i , ˆN i ] is generated by {a ∈ [ˆN i , ˆN i ] | d(ˆp i , aˆp i ) ≤ 4nε i }. If [ˆN i , ˆN i ] is


not central in ˆN i one replaces it by [ˆN i , [ˆN i , ˆN i ]]. After finitely many similar steps 

this proves the claim. 

For each positive integer l put l · A i = {g l | g ∈ A i } ⊳ Γ i . We define l i = 2 ui as 

the maximal power of 2 such that there is an element in l i · A i which displaces ˆp i 

by at most ρ 0 . Thus, any element in L i := l i · A i displaces ˆp i by at least ρ 0 /2. 

After passing to a subsequence we may assume that ( ˆM i , Γ i , ˆp i ) converges to 

(Ŷ , Ĝ, ˆp ∞) and the action of L i converges to an action of some discrete abelian 

subgroup L ∞ ⊳ Ĝ. Finally we let ˆN ∞ ⊳ Ĝ denote the limit group of ˆN i . Let g i ∈ L i 

be an element which displaces ˆp i by at most ρ 0 . Combining d(gi k ˆp i, g k+1 

i ˆp i ) ≤ ρ 0 

with our choice of ρ 0 , see (23), gives that the sets {gi k ˆp i | k ∈ Z} converge to a 

discrete subset in the identity component of the limit orbit Ĝ0 ⋆ ˆp ∞ . Therefore, 

{g ∈ L ∞ | g ⋆ ˆp ∞ ∈ Ĝ0 ⋆ ˆp ∞ } 

is discrete and infinite. Let K ⊂ Ĝ denote the isotropy group of ˆp ∞. By Colding and 

Naber K is a Lie group and thus it only has finitely many connected components. 

Hence L ′ ∞ := L ∞ ∩Ĝ0 ⊳Ĝ is infinite as well. Since the abelian group L′ ∞ is a discrete 

subgroup of a connected Lie group, it is finitely generated. 

We choose a free abelian subgroup ˆL ⊂ L ′ ∞ of positive rank which is normalized 

by Ĝ such that the induced representation of Ĝ in ˆL ⊗ Z Q is irreducible. Notice that 

ˆL ⊂ L ∞ commutes with ˆN ∞ . Hence we can view this as a representation of Ĝ/ˆN ∞ . 

Recall that π 1 (M i )/N i 

∼ = Γi /ˆN i is boundedly represented. Let ĝ i1 , . . . , ĝ iβ ∈ Γ i 

denote the images of g i1 , . . . , g iβ . There is an epimorphism Γ i /ˆN i → Ĝ/ˆN ∞ induced 

by sending ĝ im to its limit element ĝ ∞m ∈ Ĝ for all large i. Thus, ˆL is also naturally 

endowed with a representation of Γ i /ˆN i . 

Let b 1 , . . . , b l ∈ ˆL ∼ = Z l be a basis. For large i there are unique elements h i (b j ) ∈ 

L i which are close to b j , j = 1, . . . , k. We extend h i to a Z-linear map h i : ˆL → L i . 

We plan to prove that h i : ˆL → L i is equivariant for large i. For any given 

linear combination ∑ l 

α=1 z αb α (z j ∈ Z) we know that ∑ l 

α=1 z αh i (b α ) is the unique 

element in L i which is close to ∑ lα=1 z αb α for all large i. 

For each ĝ ∞m and each b j we have ĝ ∞m b j ĝ∞m −1 = ∑ lα=1 z αb α for z α ∈ Z (we 

(∑ l 

suppress the dependence on m and j). We have just seen that h i α=1 z ) 

αb α is 

close to ĝ ∞m b j ĝ∞m −1 for all large i. On the other hand, ĝ im h i (b j )ĝ −1 

im ∈ L i is the 

unique element in L i close to ĝ ∞m b j ĝ∞m −1 for all large i. 

In summary, h i (ĝ ∞m b j ĝ∞m) −1 = ĝ im h i (b j )ĝ −1 

im for m = 1, . . . , β, j = 1, . . . , k and 

all large i. This shows that h i is equivariant with respect to the representation. 

Thus, h i (ˆL) is a normal subgroup of Γ i and the induced representation of F = 

π 1 (M i )/N i = Γ/ˆN i in h i (ˆL) ⊗ Z Q and is isomorphic to the one in h j (ˆL) ⊗ Z Q for all 

large i, j. 

There is a unique subgroup A ′ i of ˆN i such that h i (ˆL) has finite index in A ′ i and 

ˆN i /A ′ i is torsion free. Let N ih 0+1 ⊳ N i denote the inverse image of A ′ i ⊂ ˆN i = N i /H i . 

Clearly, the representation of π 1 (M i )/N i in (N ih0+1/H i ) ⊗ Z Q is isomorphic to 

the one in (N jh0+1/H j ) ⊗ Z Q for all large i, j – a contradiction to our choice of H i . 

Proof of the addendum. Thus, the sequence satisfies a) and b) but not the 

addendum. Recall that N i has a nilpotent basis of length ≤ n − dim(X). Therefore 

dim(X) = 1 and N i is a torsion free group of rank n−1. Since X is one-dimensional 

we deduce that G/N ∞ is virtually cyclic, that is, it contains a cyclic subgroup of


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.


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.


There is nothing to prove if R ≤ 2R 0 /3. Suppose the statement holds for R ′ ≤ 

R − D. We claim it holds for R. 

Let a ∈ B 3R/2 (e) \ B R0 (e). By the definition of the metric on A i , there are 

g 1 , . . . , g l ∈ A i with d(e, g j ) ≤ 10D, a = ∏ l 

j=1 g j and d(e, a) = ∑ l 

j=1 d(ˆp, g j ˆp). If 

there is any choice we assume in addition that l is minimal with these properties. By 

assumption l ≥ R0 

10D 

= k. By the choice of k, after a renumbering, we may assume 

that d(g 1 , g 2 ) ≤ D. Our assumption on l being minimal implies that d(e, g 1 g 2 ) > 

10D and we may assume 4D ≤ d(e, g 1 ) ≤ d(e, g 2 ). Thus, 

d ( e, g −2 

1 a) ≤ d ( e, (g 1 g 2 ) −1 a ) + D = d(e, a) − d(e, g 1 ) − d(e, g 2 ) + D 

≤ d(e, a) − 2d(e, g 1 ) + D ≤ d(e, a) − 7D. 

By assumption, this implies that g1 −2 a has distance ≤ R 0 to some b 2 ∈ A i with 

d(e, b) ≤ 3( 2 d(e, a)−2d(e, g1 )+D ) . Consequently, a has distance ≤ R 0 to g1b 2 2 ∈ A i 

with d(e, g 1 b) < 2 3d(e, a). This finishes the proof of the subclaim. 

Since the Ricci curvature of ˆMi is bounded below and the L-bilipschitz embedding 

ι i maps the ball B 50R0 (e) ⊂ A i to a subset of B 50R0 (ˆp i ), we can use Bishop– 

Gromov once more to see that there is a number Q > 0 (independent of i) such 

that the ball B 50R0 (e) ⊂ A i can be covered by Q balls of radius R 0 for all i. 

We now claim that the ball B 2R (e) ⊂ A i can be covered by Q balls of radius R for 

1 

all R ≥ 20R 0 and all i. This will clearly imply that 

diam(A A i) i is precompact in the 

Gromov–Hausdorff topology and that the limits have finite Hausdorff dimension. 

Consider the homomorphism 

h 8 : A i → A i , x ↦→ x 16 . 

It is obviously 16-Lipschitz since A i is abelian. Choose a maximal collection of 

points p 1 , . . . , p l ∈ B 2R (e) ⊂ A i with pairwise distances ≥ 2R 0 . 

5 

From the subclaim it easily follows that B 3R (e) ⊂ ⋃ l 

i=1 B 50R 0 

(h(p i )). Hence we 

can cover B 3R (e) by l · Q balls of radius R 0 . 

Consider now a maximal collection of points q 1 , . . . , q h ∈ B 2R (e) with pairwise 

distances ≥ R. In each of the balls B 2R (q i ) we can choose l points with pairwise 

5 

distances ≥ 2R 0 . Thus, B 3R (e) contains h · l points with pairwise distances ≥ 2R 0 . 

Since we have seen before that B 3R (e) can be covered by l · Q balls of radius R 0 , 

this implies h ≤ Q as claimed. 

Thus, 

1 

diam( ˆM i) 

ˆM i is precompact in the Gromov–Hausdorff topology. After pass- 

1 

ing to a subsequence we may assume that ˆM 

diam( ˆM i → T. Notice that T comes 

i) 

with a transitive action of an abelian group. Therefore T itself has a natural group 

structure. Moreover, T is an inner metric space and the Hausdorff dimension of T is 

finite. Like Gromov in [Gro81] we can now deduce from a theorem of Montgomery 

Zippin [MZ55, Section 6.3] that T is a Lie group and thus, a torus. 

By Lemma 10.1, this shows π 1 ( ˆM i ) is infinite for large i – a contradiction. 

□ 

Final Remarks 

We would like to mention that in [Wil11] the following partial converse of the 

Margulis Lemma (Theorem 1) is proved.


Theorem. Given C and n there exists m such that the following holds: Let ε > 0, 

and let Γ be a group containing a nilpotent subgroup N of index ≤ C which has a 

nilpotent basis of length ≤ n. Then there is a compact m-dimensional manifold M 

with sectional curvature K > −1 and a point p ∈ M such that Γ is isomorphic to 

the image of the homomorphism 

π 1 

( 

Bε (p), p ) → π 1 (M, p). 

Apart from the issue of finding the optimal dimension another difference to 

Theorem 1 is that this theorem uses the homomorphism to π 1 (M) rather than to 

π 1 (B 1 (p), p). This actually allows for more flexibility (by adding relations to the 

fundamental group at large distances to p). This is the reason why the following 

problem remains open. 

Problem. The most important problem in the context of the Margulis Lemma 

for manifolds with lower Ricci curvature bound that remains open is whether or 

whether not one can arrange in Theorem 1 for the torsion of N to be abelian. We 

refer the reader to [KPT10] for some related conjectures for manifolds with almost 

nonnegative sectional curvature. 

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Department of Mathematics, University of Toronto, Toronto, Ontario, Canada 

E-mail address: vtk@math.toronto.edu 

University of Münster, Einsteinstrasse 62, 48149 Münster, Germany 

E-mail address: wilking@math.uni-muenster.de

Margulis Lemma

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