on the fundamental group of noncompact manifolds with nonnegative ricci curvature

UNIVERSITY OF CALIFORNIASanta Barbara On the Fundamental Group of Noncompact Manifolds with Nonnegative Ricci Curvature A Dissertation submitted in partial satisfaction of the requiremen

UNIVERSITY OF CALIFORNIA

Santa Barbara

On the Fundamental Group of Noncompact Manifolds with Nonnegative

Ricci Curvature

A Dissertation submitted in partial satisfaction of the requirement for the

degree of Doctor of Philosophy in Mathematics

by

William C Wylie

Committee in charge:

Professor Guofang Wei, Chair

Professor Xianzhe Dai

Professor Daryl Cooper

June 2006

UMI Number: 3218835

3218835 2006

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All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company

300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company

The dissertation of William C Wylie is approved

Daryl Cooper

Xianzhe Dai

Guofang Wei, Committee Chairman

May 2006

DedicationDedicated to my mother and father for always supporting me.

Acknowledgments

I would like to thank my advisor Guofang Wei for all of her guidanceand encouragement I could not have hoped for a better mentor I wouldalso like to thank Lisa for keeping me sane and providing me with constantpatience and support I am eternally grateful to Liam Donohoe, JamesTattersall, and Joanna Su for spending so many hours introducing me tothe beauty of mathematics, without them I never would have come to UCSB

in the first place I am also grateful to Xianzhe Dai, Daryl Cooper, andRick Ye for many inspiring lectures and helpful discussions Finally, I mustthank all of my friends at UCSB who have made my time here so enjoyable

Vita of William C Wylie

Education

Providence College Mathematics & Comp Sci B.S 2001Univ of California at Santa Barbara Mathematics M.A 2003Univ of California at Santa Barbara Mathematics Ph.D 2006

We also derive bounds on the number of generators of the damental group for some families of complete open manifoldswith nonnegative Ricci curvature In fact we show that the fun-damental group of these manifolds behaves somewhat like thefundamental group of a compact manifold We also show there

fun-is a relationship between the volume growth of a manifold withnonnegative Ricci curvature and the length of a loop represent-ing an element of infinite order in π1(M )

2.1 The Fundamental Group of Manifolds with Nonnegative Ricci

Curvature 6

2.2 Gromov-Hausdorff Convergence 10

3 Semi-Local Fundamental Groups 16 3.1 Introduction and Statement of Results 16

3.2 Nullhomotopy Radius 23

3.3 The Halfway Lemma for G(p,r,R) 28

3.4 Localized Uniform Cut Lemma 32

3.5 Proof of Theorem 3.1.5 37

3.6 Proof of Theorems 3.1.8 and 3.1.9 39

4 The Loops to Infinity Property and Diameter Growth 45 4.1 Introduction and Statement of Results 45

4.2 The Splitting Theorem and the Loops to Infinity Property 48 4.3 The Loop Pulling Lemma 51

4.4 Manifolds with Sublinear Diameter and Large Volume Growth 554.5 α-Noncollapsing 604.6 Volume Growth and the Length of Homotopically NontrivialLoops 66

Chapter 1

Introduction

There are many interesting relationships the between the geometric andtopological structures of a smooth complete Riemannian manifold Roughlyspeaking, one can envision an n-dimensional manifold M as a subset of Rk

such that every point p has a neighborhood which is homeomorphic to anopen subset of Rn The tangent space of M at p, TpM ⊂ Rk is the space

of all vectors v such that v = c0(0) where c : (−ε, ε) → M is a curve on Mwith c(0) = p The Riemannian metric on M is the natural inner product

on TpM , that is, the restriction of the dot product on Rk to TpM TheRiemannian metric enables us consider M as a metric space by defining thedistance as the infimum of lengths of paths between two points and allows us

to define geometric concepts such as length, diameter, volume, and angle

M is a topological space so one can also study its topological propertiessuch as compactness, connectedness, and homotopy and homology groups

In studying the interactions between the geometric and topological tures of a manifold, curvature plays a pivotal role To define curvature we

struc-begin with the most simple manifolds Let S be an orientable smooth face embedded in R3 Since S is orientable we can choose a smooth normalvector field to S, n(x) The Gauss map g : S → S2 is the map which takes

sur-x to n(sur-x) The Gaussian curvature of S at sur-x, κ(sur-x), is the determinant of thedifferential of g at x The magnitude of κ(x) measures how quickly the nor-mal vector n(x) is turning at x, or how curved the surface is Moreover, ifthe surface is sphere-like around x then the gauss map preserves orientationand if S is saddle-like around x then g reverses orientation Therefore, κ(x)

is positive if S looks like a sphere around x and negative if S looks like asaddle around x Clearly curvature is not a topological quantity However,one of the most amazing results in geometry, the Gauss Bonnet theorem,shows that the integral of curvature is a topological quantity

Theorem 1.0.1 (Gauss-Bonnet) If S is a compact, orientable surfacethen

Z

S

κ = 2πχ(S)where χ(S) is the Euler characteristic of S

Since the only compact orientable surface with positive Euler teristic is the sphere, the Gauss Bonnet Theorem shows that any compactorientable surface with κ(x) > 0 for all x ∈ S is homeomorphic to thesphere As we have seen, the condition κ(x) > 0 means that S curves in onitself at all points, thus it is not surprising to find that there are not manypossibilities for the topology of these surfaces

charac-To generalize the definition of curvature to higher dimensions let v1 and

v2 be two unit length tangent vectors to M and let σ(v1, v2) be the surface

cut out of M by geodesics whose derivative is in the span of v1 and v2 Bygeodesic we mean a path that locally minimizes distance between points

in M Since a small neighborhood of x in σ(v1, v2) can be isometricallyembedded in R3, and it can be shown that curvature is invariant underisometry, we can define κx(v1, v2) as the Gaussian curvature of σ(v1, v2) at

x We call κx(v1, v2) the sectional curvature at x of the surface spanned by

v1 and v2 A Riemannian manifold has sectional curvature greater than orequal to (greater than) H if κx(v1, v2) ≥ H (κx(v1, v2) > H) for all x ∈ Mand v1, v2 ∈ TxM We denote this by secM ≥ H or secM > H

Ricci curvature is defined as the average of the sectional curvatures.Given a unit tangent vector v at a point p, complete v to an orthonormalbasis of TpM , {v, e1, , en−1} The Ricci curvature at p in the direction vis

We say M has Ricci curvature greater than or equal to (greater than) H

if all of the Ricci curvatures are greater than or equal to (greater than) H

We denote this by RicM ≥ H or RicM > H Since the Ricci curvature isthe average of the sectional curvatures, a lower bound on Ricci curvature

is a weaker condition than a lower bound on sectional curvature Still, if

M has all of its Ricci curvatures larger than a positive number then, insome sense, M curves in on itself in every direction Thus there should not

be many topological possibilities This is expressed in the famous Myers Theorem

Bonnet-Theorem 1.0.2 (Bonnet-Myers) If M is a complete Riemannian

man-ifold with RicM ≥ H > 0 then M is compact and has finite fundamentalgroup.

If we weaken the hypothesis of the Bonnet-Myers Theorem the sions no longer hold For example, the paraboloid {(x, y, z) ∈ R3 : z =

conclu-x2 + y2} has secM > 0 but is not compact And Tn × R has secM ≡ 0,

is noncompact, and has infinite fundamental group Still, the topology ofmanifolds with secM ≥ 0 is quite restrictive Cheeger and Gromoll [10] haveshown that if M is a noncompact manifold with sec ≥ 0, then M is the nor-mal bundle over a compact totally geodesic submanifold In particular, theonly noncompact manifolds with secM ≥ 0 are vector bundles over compactmanifolds with nonnegative sectional curvature Moreover, Perelman [20]has shown that any noncompact manifold with secM ≥ 0 and all positivesectional curvatures at one point is diffeomorphic to Rn, that is a vectorbundle over one point

The Ricci curvature case is much different and far less understood Forexample, Sha and Yang [23] have constructed manifolds with positive Riccicurvature and infinite second betti number These examples are not evenhomotopy equivalent to the interior of a compact manifold with boundary.However, there are a number of results concerning the fundamental group

of a manifold with RicM ≥ 0 In this dissertation we study further theinteraction between the geometry and the fundamental group of a manifoldwith nonnegative Ricci curvature There are three parts

In the first part we review results about the fundamental group of ifolds with nonnegative Ricci curvature and background involving Gromov

man-Hausdorff convergence.

In the second part we show that, for a large class of manifolds withnonnegative Ricci curvature, all of the information about the fundamentalgroup can be isolated nicely to a compact set We then use this result

to give information about a structure on M called the tangent cones atinfinity If M has nonnegative sectional curvature this structure is verywell understood, but for a manifold with nonnegative Ricci curvature verylittle is known See Section 3.1

In the third part we give bounds on the number of generators of the damental group under some natural geometric conditions In fact, we showthat the fundamental group of these manifolds behaves like the fundamentalgroup of a compact manifold See Section 4.1

fun-All manifolds in this thesis are assumed to be complete, noncompact,and without boundary For clarity we will often omit explicitly stating thishypothesis but it is always assumed

Chapter 2

Background

with Nonnegative Ricci Curvature

In this section we review some results concerning the fundamental group

of manifolds with nonnegative Ricci curvature The most basic geometrictool in studying manifolds with RicM ≥ 0 is the Bishop-Gromov relativevolume comparison theorem which bounds the volume of a metric ball interms of the corresponding ball in Rn By metric ball we mean

is nonincreasing in r In particular,

vol(B(p, r)) ≤ ωnrnwhere ωn is the volume of the ball of radius 1 in n-dimensional Euclideanspace

The Bishop-Gromov Volume Comparison Theorem motivates the lowing definition

fol-Definition 2.1.2 A manifold is said to have polynomial volume growth oforder ≤ k if

lim sup

r→∞

vol(B(p, r))

rk < ∞and has polynomial volume growth of order ≥ k if

Theorem 2.1.3 (Li, [14]) If Mn has Ric ≥ 0 and Euclidean volumegrowth then π1(M ) is finite

Of course, as we saw in the last chapter, there are manifolds with RicM ≥

0 and infinite fundamental group However, there is still a relationship

between volume growth and the structure of the fundamental group when

π1(M ) is infinite To see this we must first define the growth of a finitelygenerated group

Let Γ be a finitely generated group generated by the set {g1, , gk}.Any g ∈ Γ can be written as a word g = Y

i

gni

k i, where ki ∈ {1, , k}.Define the length of this word to beX

i

|ni|, and let |g| be the minimum ofthe lengths of all word representations of g in the generating set {g1, , gk}.Definition 2.1.4 Fix a set of generators for Γ The growth function of Γis

Γ(s) = #{g ∈ Γ : |g| ≤ s}

The function Γ(s) depends on the generating set chosen However, ifΓ(s) is a polynomial of degree k for some generating set then the growthfunction corresponding to any other set of generators of Γ must also bebounded by a (different) polynomial of degree k This motivates the fol-lowing definition

Definition 2.1.5 Γ is said to have polynomial growth of degree ≤ k if

Γ(s) ≤ askfor some a > 0

The property of having polynomial growth of degree ≤ k is independent

of the generating set and is thus a property of the group itself Applyingthe Bishop-Gromov Volume Comparison Theorem to the universal coverMilnor shows the following

Theorem 2.1.6 (Milnor, [18]) If Mn has Ric ≥ 0 then any finitelygenerated subgroup of π1(M ) has polynomial growth of order ≤ n.

Moreover, Anderson has shown that the volume growth of M controlsthe growth of π1(M )

Theorem 2.1.7 (Anderson, [2]) If Mn has Ric ≥ 0 and polynomialvolume growth of order ≥ k then any finitely generated subgroup of π1(M )has polynomial growth of order ≤ n − k

Yau [35] has shown that any noncompact manifold with Ric ≥ 0 has atleast linear volume growth, that is polynomial volume growth of order ≥ 1.Therefore, Anderson’s result shows that any finitely generated subgroup ofthe fundamental group of a noncompact manifold with nonnegative Riccicurvature has polynomial volume growth of order ≤ n − 1

Gromov [11] has shown that any finitely generated group with mial growth is almost nilpotent, that is the group has a nilpotent subgroup

polyno-of finite index Wei [32] and Wilking [33] have shown that for any finitelygenerated almost nilpotent group, G, there is a manifold with positive Riccicurvature and fundamental group G Therefore a finitely generated group

G is the fundamental group of some manifold with nonnegative Ricci vature if and only if G is almost nilpotent A major open problem is thefollowing conjecture of Milnor

cur-Conjecture 2.1.8 (Milnor, [18]) If M has Ric ≥ 0 then π1(M ) is finitelygenerated

No counterexample has been constructed to the Milnor conjecture Thereare a number of partial results For example, Wilking [33] has shown that if

one can solve the conjecture for abelian fundamental groups then the eral result follows and Sormani [27] has shown that manifolds with smalllinear diameter growth have finitely generated fundamental group We willstrengthen this result in a different direction in Chapter 3 On the otherhand, it is also an interesting problem to understand how the different ge-ometric properties of the manifold, such as volume growth, interact withproperties of the fundamental group We will prove some results in thisvein in Chapter 4.

gen-Sormani [27] has also shown that there is a relationship between thetangent cone at infinity of M and its fundamental group, in Chapter 3

we further investigate this relationship In the next section we review thedefinition of tangent cone at infinity and Gromov-Hausdorff convergence

Another tool used in studying manifolds with Ricci curvature bounded low is Gromov-Hausdorff convergence introduced by Gromov in [11] First

be-we recall the definition of Hausdorff distance

Definition 2.2.1 Given a metric space Z and A, B ⊂ Z, a ε-tubularneighborhood of A is

Tε(A) = {z ∈ Z : ∃a ∈ A, d(z, a) < ε}

and the Hausdorff distance between A and B is

dZH(A, B) = inf{ε : A ⊂ Tε(B), B ⊂ Tε(A)}

The Hausdorff distance is small if every point in A is close to a point in

B and every point in B is close to A

Definition 2.2.2 Given two compact metric spaces X and Y the Hausdorff distance between X and Y is

mathe-on M = {compact metric spaces}/isometry and this metric gives a cmathe-oncept

of convergence on the space of compact metric spaces This convergencealso has a simple geometric meaning

To see this consider the following situation You are shown two objects.You see the first object which is then removed from your view and thenyou are shown the second object You are then asked whether the twoobjects look alike To answer the question you compare the two objects

in your mind By moving images of the two objects around in your mindyou try determine whether the two objects can ever be embedded in a thirdspace so that they are very close to each other You can never determinewhether two objects are exactly the same since you can only see on a finitelength scale but you might be given a microscope which improves the lengthscale which you can observe A sequence of objects {Xi} converges to X

in the Gromov-Hausdorff topology means that no matter how powerful amicroscope you are given, you can go far enough out in the sequence so thatall the Xi will look like X.

We wish to define Gromov-Hausdorff convergence for noncompact metricspaces, one could use the same definition as in the compact case Howeverthis is not intuitively satisfactory as the following example shows

Example 2.2.3 For any θ ∈ [0, 2π) let Wθ = {z ∈ C : z = reiφ, 0 ≤

φ ≤ θ} That is, Wθ is the infinite wedge in C enclosed by an angle of

θ As θ → 0, Wθ will be a smaller and smaller wedge that geometricallylooks as though it should converge to the x-axis (see figure 2.1) However,

dGH(Wθ1, Wθ2) = ∞ if θ1 6= θ2 so the sequence does not converge

0

0000000 0000000 0000000

1111111 1111111 1111111

0

0000000 0000000 0000000 0000000

1111111 1111111 1111111 1111111

0

0000000 0000000

Figure 2.1: The Wθ should converge to the positive x-axis

Instead we define convergence of noncompact metric spaces as gence on compact sets

conver-Definition 2.2.4 Suppose Xi and Y are complete metric spaces with xi ∈

X and y ∈ Y Then the pointed spaces (Xi, xi) converge in the pointedGromov-Hausdorff topology to (Y, y) if for all R > 0 B(xi, R) converges toB(y, R) with respect to the restricted metric on the balls

Gromov-Hausdorff convergence is particularly useful in studying

man-ifolds with Ricci curvature lower bounds because of the Gromov pactness Theorem

Precom-Theorem 2.2.5 (Gromov Precompactness Precom-Theorem) The set of allpointed, complete, n-dimensional Riemannian manifolds with Ricci curva-ture bounded below by H is precompact in the pointed Gromov-Hausdorfftopology

In other words, a sequence,(Xi, xi) where all the Xihave Ricci curvaturebounded below by H will have a subsequence which converges to somemetric space (Y, y) We call Y a limit space One can show that a limit space

Y must be a complete, locally compact, length space It is an interestingproblem to study what other properties a limit space must have

The geometric properties of the limit space of a sequence of manifoldswith Ricci curvature lower bound have been studied extensively by Cheegerand Colding [5], [6], [7], [8] Sormani and Wei apply their work to study thetopology of the limit space [29], [30] They show that a limit space musthave a universal cover in the following sense

Definition 2.2.6 ([31] p 62, 82) A path connected covering space ˜Y of apath connected topological space Y is the universal cover of Y if ˜Y coversevery other path connected cover of Y and the covering projections form acommutative diagram

Theorem 2.2.7 (Sormani-Wei, [30]) If (Y, y) is the pointed Hausdorff limit of a sequence of Riemannian manifolds {(Xi, xi)} with RicXi ≥

Gromov-H then Y has a universal cover

Recall, that if a topological space Y is semi-locally simply connectedthen the universal cover is the unique simply connected cover of Y How-ever, the universal cover of a topological space may not be simply connected.

It is not known if a limit space of a sequence with bounded Ricci curvaturemust be semi-locally simply connected There are sequences of Riemannianmanifolds that converge to a metric space which does not have a universalcover When a topological space is semi-locally simply connected, the fun-damental group is exactly the deck transformation group of the universalcover This motivates the following definition

Definition 2.2.8 ([30]) Let Y be a path connected topological space with

a universal cover, the revised fundamental group π1(Y ) is the group of decktransformations of the universal cover

Sormani and Wei ([30], Corollary 4.7-4.9) go on to prove results similar

to those in the previous section of this chapter for π1(Y ) for limit spaces ofsequences of manifolds with nonnegative Ricci curvature We are concernedwith special limit spaces called tangent cones at infinity

Definition 2.2.9 ([12]) Let X be a complete length space A tangentcone at infinity of X, (Y, dY, y0), is a rescaled Gromov Hausdorff limit

(X,dX

ri , p)

GH

−−→ (Y, dY, y0)

where ri → ∞ and dX is the metric on X

Since we are rescaling X by a sequence of smaller and smaller bers, at each step the structure of a larger and larger ball is crushed into

num-a smnum-all neighborhood of the bnum-asepoint, num-and thus in the limit we see onlythe structure “at infinity” of X If X is a manifold then rescaling the met-ric also rescales curvature by a positive constant In particular, if X hasnonnegative Ricci curvature then the entire sequence of rescalings has non-negative Ricci curvature and the Gromov Precompactness Theorem impliesthat tangent cones at infinity exist.

In fact, if X has nonnegative sectional curvature then no matter whatsequence of rescalings is taken, the limit is the Euclidean metric cone overthe ideal boundary of M ([13], Lemma 3.4) In particular, the tangent cone

at infinity is contractible Cheeger and Colding have shown that if X hasnonnegative Ricci curvature and Euclidean volume growth then any tangentcone at infinity is a metric cone [5] although Perelman [21] has shown thatthis limit may depend on which sequence of rescalings is chosen In general,the tangent cone at infinity of a manifold with nonnegative Ricci curvaturemay not be a metric cone, as an example of Menguy shows [15] However,the fundamental group of this example is well behaved This leads to thefollowing question

Question 2.2.10 Is the revised fundamental group of the tangent cone atinfinity of a manifold with nonnegative Ricci curvature trivial?

In the next chapter we give a partial affirmative answer to this question

We do this by controlling the fundamental group of large metric balls in aprescribed way and applying the methods of Wei and Sormani

Chapter 3

Semi-Local Fundamental

Groups

Motivated by the work of Sormani and Wei mentioned in the last chapter,

in this chapter we study the following groups

Definition 3.1.1 For p ∈ M and 0 < r < R define G(p, r, R) to be thegroup obtained by taking all the loops based at p contained in the closedball of radius r and identifying two loops if there is a base point preservinghomotopy between them that is contained in the open ball of radius R Wecall G(p, r, R) the geometric semi-local fundamental groups of M at p (Seefigure 3.1)

It may seem strange to the reader to consider inner balls which are closedand outer ones which are open The definition is given as above because

Figure 3.1: G(p, r, R) is the group of loops contained in the inner closedball with homotopies contained in the outer open ball

we can find a nice characterization of G(p, r, R) as a subgroup of a group

of deck transformations (See Corollary 3.3.2 and Lemma 3.3.6) There is anatural map from G(p, r, R) to π1(M, p) induced by the inclusion of B(p, r)into M In this chapter π1(M ) ∼= G(p, r, R) will mean that this inducedmap is a group isomorphism We denote the image of this map by G(p, r).The geometric semi-local fundamental groups depend heavily on themetric structure of M and not just the topology Even a simply con-nected manifold may have very complicated geometric semi-local funda-mental groups as is shown in the following example

Example 3.1.2 Consider the standard, flat xy-plane sitting in R3 withstandard Euclidean coordinates For each positive integer n, remove a smalldisc in the xy-plane around each point (n, 0, 0) and glue in its place a long

capped, flat cylinder with the cap of the cylinder at the point (n, 0, 10n).See figure 3.2 Let M be the resulting simply connected metric space and let

p be the point (0, 0, 0) Then for each ball of radius n + 1/2 around p in Mthe loop that wraps around the glued in capped cylinder is nullhomotopic

in M but not inside any metric ball centered at the origin of radius lessthat n + 10n Thus G(p, n + 1/2, n + 10n) is not the trivial group for any n

Figure 3.2: The surface is simply connected but has complicated semi-localfundamental groups

The surface in example 3.1.2 can easily be smoothed and keep the samesemi-local fundamental groups The smoothed surface, however, does nothave nonnegative curvature In fact, the geometric semi-local fundamentalgroups are well behaved for a manifold with nonnegative sectional curvature.Lemma 3.1.3 If N is a complete noncompact manifold with nonnegativesectional curvature then, for any point p in the soul of S, there exists largeenough R so that π1(B(p, r), p) ∼= π1(M, p) for all r > R

Proof By a theorem of Sharafutdinov [24] (see also Thm 2.3 in [36]) , Ndeformation retracts onto a compact soul S and this deformation retraction

is distance nonincreasing For any p in S take a large enough r such thatB(p, r) contains S By following Sharafutdinov’s deformation retraction,any loop in B(p, r) can be homotoped into S while staying inside B(p, r)

This R may be very large and depend upon the point we choose, as thefollowing simple example shows

Example 3.1.4 Let S1

ε be the round circle of circumference ε Let N be aflat Sε1× [0, ∞) with the upper hemisphere of a round S2

ε glued to Sε1× {0}and let p ∈ N such that the distance, l, from the north pole of the S2 islarger than ε2 N is simply connected but G(p,ε2, l) is Z since the curvethat wraps around the cylinder is only nullhomotopic via homotoping itover the top of the attached sphere

tal groups To do so define the ray density function at p as

Theorem 3.1.5 There exists a constant Sn = 1

s→0gn(s) = 1 such that if Mn is acomplete open Riemannian manifold with nonnegative Ricci curvature and

Definition 3.1.7 [6] A complete manifold of nonnegative Ricci curvature

is asymptotically polar if all of its tangent cones at infinity (Y, dY, y0) have

a pole at its basepoint y0

In this case we have a similar theorem

Theorem 3.1.8 Let M be a complete open Riemannian manifold of negative Ricci curvature which is asymptotically polar Then, for every

non-ε > 0 and every point p ∈ M , there is R so that

π1(M ) ∼= G(p, r, (1 + ε)r) ∀r > R

There are many examples that satisfy the hypotheses of Theorems 3.1.5and 3.1.8 Specifically, Cheeger and Colding [5] have shown that manifoldswith nonnegative Ricci curvature and Euclidean volume growth are asymp-totically polar; and Sormani [26] has shown that manifolds with nonnegativeRicci curvature and linear volume growth satisfy the hypotheses of eithertheorem In [15] Menguy gives an example of a manifold with nonnegativeRicci curvature which is not asymptotically polar However, the conclusion

of Theorem 3.1.8 holds for this manifold The author is unaware of anyexamples of manifolds of nonnegative Ricci curvature which do not satisfythe conclusion of Theorem 3.1.8

The conclusions to Theorems 3.1.5 and 3.1.8 specifically imply that

π1(M ) is finitely generated Under the hypothesis of Theorem 3.1.5 Xu,Yang, and Wang (Theorem 3.1, [34]) have shown that π1(M ) is finitelygenerated As in this paper, their work is based on original work of Sor-mani who proved that π1(M ) is finitely generated when M satisfies the

hypothesis for a smaller constant Sn (Theorem 1, [27]) Similarly, rem 3.1.8 is a strengthening of Sormani’s Pole Group Theorem (Theorem

Theo-11, [27]) in which she proves that every asymptotically polar, complete,open manifold with nonnegative Ricci curvature has finitely generated fun-damental group Also note that π1(M ) for a manifold with nonnegativeRicci curvature curvature is finitely generated if and only if it is finitelypresented

Theorems 3.1.5 and 3.1.8 are stronger than the results above becausethey not only control the generators of π1(M ) but also where the homotopiesbetween loops sit in M This allows us to control the topology of the tangentcone at infinity

Theorem 3.1.9 Let X be a complete length space with x ∈ X If thereexists positive numbers k > 1 and R so that

π1(X) ∼= G(p, r, kr) ∀r > R,then any tangent cone at infinity of X has trivial revised fundamental group.Theorem 3.1.9 is not true if we only assume X has finitely generatedfundamental group Thus, the extra control gained by Theorem 3.1.5 doesyield additional information We say that a manifold that satisfies thehypothesis of Theorem 3.1.5 for some s ∈ (0, Sn] has small linear diametergrowth with respect to ray density Directly applying Theorems 3.1.5 and3.1.9 we obtain,

Corollary 3.1.10 If Mn is an open manifold with nonnegative Ricci vature and small linear diameter growth with respect to ray density then anytangent cone at infinity of M has trivial revised fundamental group

cur-This chapter is organized as follows In the next section we introduce thenullhomotopy radius function ρ which measures how different the geometricsemi-local fundamental groups are from π1(M ) We also give examplesand discuss some basic properties of the geometric semi-local fundamentalgroups and the nullhomotopy radius Just as in [27] and [34] the proofs

of Theorems 3.1.5 and 3.1.8 are the result of two lemmas, the HalfwayLemma and the Uniform Cut Lemma In Sections 3.3 and 3.4 we give proofs

of versions of these two lemmas for the geometric semi-local fundamentalgroups The main difficulty here is that we work on the universal cover

of an open metric ball which may not be a complete metric space Thesesections consist of a series of technical lemmas to work around this difficulty

In Section 3.5 we give the proof of Theorem 3.1.5 In the final section wediscuss Theorems 3.1.8 and 3.1.9

In this section we introduce the nullhomotopy radius and discuss its tionship with the geometric semi-local fundamental groups The definitionscan be applied to any length space X We also show that the nullhomotopyradius is finite for complete manifolds with nonnegative Ricci curvature(Corollary 3.2.8) which will be a very important fact in the proof of Theo-rems 3.1.5 and 3.1.8 To end the section we prove Lemma 3.2.9 which wewill apply in the proofs of Theorems 3.1.5 and 3.1.8

rela-To motivate the definition of nullhomotopy radius consider two loops γ1and γ2 based at a point p ∈ X with the image of both loops contained in

B(p, r) We would like to know whether the two loops are homotopic in X.This is equivalent to asking whether the loop ω = γ1∗ γ2−1 is nullhomotopic.

As we have seen in Example 3.1.2, it may be that ω is nullhomotopic butthere is no nullhomotopy of ω contained in B(p, R) even for an R muchlarger than r The nullhomotopy radius function measures how much biggerthan r we need to make R in order to check that γ1 and γ2 are homotopic.Definition 3.2.1 Let Ωp,r be the set of all R ∈ R such that all loops inB(p, r) that are nullhomotopic in X are also nullhomotopic in B(p, R)

If R ∈ Ωp,r then to check whether a loop in B(p, r) is nullhomotopic

we only need to check for homotopies that are contained in B(p, R) Thenullhomotopy radius is the smallest such R

Definition 3.2.2 The nullhomotopy radius function at p ∈ X is the tion ρp : R+→ R+∪ {∞} defined as

We say that ρp(r) is the nullhomotopy radius at p with respect to r

An equivalent definition of Ωp,r is as the set of R so that the natural mapfrom G(p, r, R) to G(p, r) is an isomorphism ρp(r) is the smallest number

so that G(p, r, R) is isomorphic to G(p, r) for all R > ρp(r) When it is clearwhich basepoint we are using we will suppress the point and write ρ(r).Let us illustrate the behavior of the nullhomotopy radius with a few basicexamples

Example 3.2.3 Consider the capped cylinder in Example 3.1.4 Let p ∈ Nsuch that the distance, l, from the north pole of the cap S2 is larger than

Example 3.2.4 Let N = Tn× R where Tn is the n-torus Tn = S1

2 ε if ε2 ≤ r <

√ n

2 ε

r if r ≥

√ n

Example 3.2.6 Bowditch and Mess [4] and Potyagailo [22] have shownthat there are examples of complete 4-dimensional hyperbolic manifoldssuch that ρ(r) = ∞ for all r ≥ 1 This is also pointed out by Sormani andWei, see Example 4.1 in [30]

The nullhomotopy radius function will be finite for manifolds with negative Ricci curvature To see this, recall that the results of Gromov andMilnor combine to show that any finitely generated subgroup of π1(M ) has

non-a nilpotent subgroup of finite index In pnon-articulnon-ar, every finitely genernon-atedsubgroup of π1(M ) is also finitely presented This combined with the fol-lowing general observation shows that the nullhomotopy radius is finite fornonnegative Ricci curvature

Lemma 3.2.7 Let X be a complete length space If G(p, r) is finitelypresented then ρ(r) is finite

Proof Suppose that G(p, r) is finitely presented G(p, r, 2r) is finitely erated so let {γ1, γ2, · · · , γk} be a finite collection of loops in B(p, r) suchthat G(p, r, 2r) = h{[γ1], [γ2], · · · , [γk]}i Then {[γ1], [γ2], · · · , [γk]} is a fi-nite set of generators for G(p, r) Since it is finitely presented, we can write

gen-a presentgen-ation for G(p, r) of the form

h[γ1], [γ2], · · · , [γk]|R1, R2, · · · Rli

Each Rj can be represented by a homotopy involving the representativeloops {γ1, γ2, · · · , γk} For each j, let Hj be this homotopy Since there areonly finitely many homotopies we know that there exists an R > 2r such

Let σ be a loop contained in B(p, r) with [σ] = 0 in π1(X) Then, since

R > 2r, [σ] = [w] in G(p, r, R), where w is some word in the γis [w] = 0

in π1(X) so it can be written as a product of the Rjs But w can behomotoped to the constant map via the homotopies Hj Since the image

of all these homotopies are contained in B(p, R) we see that [w] = 0 inG(p, r, R) Since we have started with an arbitrary nullhomotopic loop inB(p, r), R ∈ Ωp,r

Corollary 3.2.8 If Mn is a complete Riemannian manifold with ative Ricci curvature, then ρ(r) < ∞ for all r

nonneg-There is one more basic fact about the nullhomotopy radius that wewill state here The proofs of Theorems 3.1.5 and 3.1.8 are applications ofLemma 3.2.9

Lemma 3.2.9 Let X be a complete length space If there exists positivenumbers L, k, and N0 such that ρ(L) < ∞ and G(p, L, kr) ∼= G(p, r, kr)for all r > N0 Then there exists R0 such that π1(X) ∼= G(p, r, kr) for all

contained in B(p, L) But then, since kr > ρ(L), [α] = 0 in G(p, r, kr) Thusthe natural map from G(p, r, kr) to π1(M ) is one to one The hypothesisclearly implies that the map is onto.

In this section we establish the Halfway Lemma for G(p, r, R) which ismotivated by Sormani’s Halfway Lemma in [27] In this section N is acomplete Riemannian manifold without boundary We do not require acurvature bound in this section

Fix p ∈ N B(p, R) is an open subset of N and is thus semi-locallysimply connected Let ^B(p, R) be the universal cover of B(p, R) and fix

a lift of p to B(p, R), ˜^ p G(p, r, R) is a subgroup of π1(B(p, R)) thus,

we can identify G(p, r, R) with a subgroup of the deck transformations of

^

B(p, R) Note that B(p, R) may not be semi-locally simply connected, this

is the reason for using the open outer ball in the definition of the geometricsemi-local fundamental groups We also equip ^B(p, R) with the coveringmetric coming from B(p, R) ^B(p, R) is then a Riemannian manifold with-out boundary which is not complete Thus, large closed metric balls in

^

B(p, R) may not be compact and we do not know, a priori, that there areonly a finite number of deck transformations that move the basepoint agiven distance However, we can argue that this is true for small enoughdistances

Lemma 3.3.1 For any 0 < r < R the set {g ∈ π1(B(p, R), p) : d(˜p, g ˜p) ≤2r} is finite

Proof Let D be a smooth compact region in M such that B(p, r) ⊂ D ⊂B(p, R) Fix ˜p0 as a lift of p to ˜D Let i∗ : π1(D, p) −→ π1(B(p, R), p) bethe induced map coming from the inclusion Then i∗ maps the set {h ∈

π1(D, p) : d(˜p0, h˜p0) ≤ 2r} onto the set {g ∈ π1(B(p, R), p) : d(˜p, g ˜p) ≤ 2r}.Since D is a complete length space, so is ˜D and thus B(˜p0, 2r) is compactand the set {h ∈ π1(D, p) : d(˜p0, h˜p0) ≤ 2r} is finite

G(p, r, R) is exactly the subgroup of the group of deck transformations

of ^B(p, R) generated by deck transformations that move the basepoint shortdistances

Corollary 3.3.2 G(p, r, R) = h{g ∈ π1(B(p, R), p) : d(˜p, g ˜p) ≤ 2r}iProof Let g ∈ G(p, r, R), then there is a rectifiable loop γ : [0, L] −→B(p, r) with [γ] = g in π1(B(p, R)), p) Assume γ is parametrized by ar-clength Fix δ > 0 and let 0 = t0 < t1 < t2 < · · · < tk = L such that

ti+1− ti < δ Let σi be the minimal geodesic in B(p, r) from p to γ(ti)and let αi be the loop based at p which traverses σi from p to γ(ti) thenproceeds along γ to γ(ti+1) then returns to p via σi+1 (see figure 3.4).L(αi) ≤ 2r + δ where L(αi) denotes the length of αi Let hi = [αi], thend(˜p, hi(˜p)) ≤ 2r + δ But g = h1h2· · · hk So we have proven that

G(p, r, R) ⊂ h{g : d(˜p, g(˜p)) ≤ 2r + δ}i ∀δBut by Lemma 3.3.1 , the set {g : d(˜p, g(˜p)) ≤ 2r + δ0} is finite for asmall δ0 So there is a possibly smaller δ0 so that {g : d(˜p, g(˜p)) ≤ 2r +δ0} ={g : d(˜p, g(˜p)) ≤ 2r} Therefore, G(p, r, R) ⊂ h{g : d(˜p, g(˜p)) ≤ 2r}i Theother inclusion is trivial

Figure 3.4: Breaking the loop γ into a product of short loops.

Note that using the closed ball B(p, r) is convenient in the proof ofCorollary 3.3.2 To get a halfway generating set we would like to takeminimal geodesics in ^B(p, R) and project them down However, we mustprove that these minimal geodesics exist

Lemma 3.3.3 If g ∈ π1(B(p, R)) and d(˜p, g ˜p) < 2R then there is a mal geodesic in ^B(p, R) from ˜p to g ˜p

mini-Proof First observe that, although ^B(p, R) is not complete, it is clear thatfor all g ∈ π1(B(p, R), p) the exponential map at g ˜p is defined on B(0, R) ⊂

Tg ˜pM Therefore a minimal geodesic from ˜p to g(˜p) exists for all g withd(˜p, g ˜p) < R

Let R ≤ d(˜p, g ˜p) < 2R and let 2D = d(˜p, g ˜p) S(˜p, D) is compact since

D < R Let q be the point that minimizes the function x −→ d(x, g ˜p) for

x ∈ S(˜p, D) Then, for any ε > 0, there is a curve σεfrom ˜p to g ˜p with lengthless than or equal to 2D + ε and there is tε ≥ D such that σε(tε) ∈ S(˜p, D)

d(σε(tε), g ˜p) ≤ D + ε, therefore d(q, g ˜p) ≤ D Since D < R there is aminimal geodesic from q to g ˜p call this minimal geodesic γ2 Let γ1 be aminimal geodesic from ˜p to q Then the curve that transverses γ1 and then

γ2 has length less than or equal to 2D and therefore is a minimal geodesicfrom ˜p to g ˜p

We are now ready to prove the Halfway Lemma Let us first review thedefinition of a set of halfway generators

Definition 3.3.4 A set of generators {g1, g2, , gm} of a group G is anordered set of generators if each gi can not be written as a word in theprevious generators and their inverses

Definition 3.3.5 Given g ∈ G(p, r, R) we say γ is a minimal representativegeodesic loop of g if g = [γ] and L(γ) = d^

B(p,R)(˜p, g ˜p)

We can now state the Halfway Lemma for G(p, r, R)

Lemma 3.3.6 (Halfway Lemma) Let N be a complete Riemannianmanifold and p ∈ N Then for any 0 < r < R there exists an ordered set

of generators {g1, , gm} of G(p, r, R) with minimal representative geodesicloops, γi of length di such that

many elements Therefore, we can take g1 so that d(g1(˜p), ˜p) is minimalamong all g1 ∈ G(p, r, R) If {g1} is not a generating set of G(p, r, R),consider G(p, r, R) \ hg1i Take g2 to minimize d(g2(˜p), ˜p) among all g2 ∈G(p, r, R)\hg1i By the above, d(˜p, g2(˜p)) ≤ 2r Inductively we define a gen-erating sequence with the properties that for each i, gi minimizes d(gi(˜p), ˜p)among all gi ∈ G(p, r, R) \ hg1, g2, · · · , gi−1i and di ≤ 2r for all i where

di = d(˜p, gi(˜p))

By Lemma 3.3.3 there is a minimal geodesic joining ˜p and gi(˜p) in

^

B(p, R) Let ˜γi be this minimal geodesic and let γi be the projection of

˜i down to B(p, R) By our construction of the gis they have the propertythat if h ∈ G(p, r, R) and d(h(˜p), ˜p) < d(gi(˜p), ˜p) then h ∈ hg1, g2, · · · , gi−1i

To finish the lemma we need to show that d(γi(0), γi(di/2)) = di/2 To do

so, suppose that there is i so that d(p, γi(di/2)) < di/2 Let σ be the mal geodesic in M from p to γi(di/2) Let h1 be the element of G(p, r, R)represented by transversing γi from 0 to di

mini-2 then following σ back to p andlet h2 be the element of G(p, r, R) represented by transversing first σ then

γi from di

2 to di See figure 3.5

d(h1(˜p), ˜p) and d(h2(˜p), ˜p) are both less than d(gi(˜p), ˜p) which impliesthat h1 and h2 are in hg1, g2, · · · , gi−1i But h1h2 = gi so that gi ∈

hg1, g2, · · · , gi−1i which is a contradiction to our choice of gi

In this section Mn is a complete Riemannian manifold with nonnegativeRicci curvature and dimension at least 3 In dimension less than three,