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
UMI Microform
Copyright
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
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by ProQuest Information and Learning Company.
The dissertation of William C. Wylie is approved
Daryl Cooper
Xianzhe Dai
Guofang Wei, Committee Chairman
May 2006
Dedication

Dedicated to my mother and father for always supp orting me.
Acknowledgments
I would like to thank my advisor Guofang Wei for all of her guidance
and encouragement. I could not have hoped for a better mentor. I would
also like to thank Lisa for keeping me sane and providing me with constant
patience and support. I am eternally grateful to Liam Donohoe, James
Tattersall, and Joanna Su for spending so many hours introducing me to
the 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, and
Rick Ye for many inspiring lectures and helpful discussions. Finally, I must
thank all of my friends at UCSB who have made my time here so enjoyable.
iii
Vita of William C. Wylie
Education
Providence College Mathematics & Comp. Sci. B.S. 2001
Univ. of California at Santa Barbara Mathematics M.A. 2003
Univ. of California at Santa Barbara Mathematics Ph.D. 2006
Fields of Study
Riemannian Geometry and Global Geometric Analysis.
Publications
William C. Wylie, Noncompact Manifolds with Nonnegative Ricci Curva-
ture, To appear in Journal of Geometric Analysis.
Appointments
2005-2006 Graduate Council Departmental Mentorship Award Fellow, Uni-
versity of California at Santa Barbara.
2003-2005 (3 quarters) Teaching Associate, University of California at
Santa Barbara.
2001-2005 (10 quarters) Teaching Assistant, University of California at
Santa Barbara.
2004-2005 (3 quarters) Research Assistant, University of California at

Santa Barbara.
iv
Abstract
On the Fundamental Group of Noncompact Manifolds with Nonnegative
Ricci Curvature
by William C. Wylie
We study the fundamental group of noncompact Riemannian
manifolds with nonnegative Ricci curvature. We show that
the fundamental group of a noncompact, complete, Rieman-
nian manifold with nonnegative Ricci curvature and small lin-
ear diameter growth is almost the fundamental group of a large
ball. We make this precise by studying semi-local fundamen-
tal groups. We also find relationships between the semi-local
fundamental groups and special Gromov-Hausdorff limits of a
manifold called tangent cones at infinity. As an application we
show that any tangent cone at infinity of a complete open mani-
fold with nonnegative Ricci curvature and small linear diameter
growth is its own universal cover.
We also derive bounds on the number of generators of the fun-
damental group for some families of complete open manifolds
with nonnegative Ricci curvature. In fact we show that the fun-
damental group of these manifolds behaves somewhat like the
fundamental group of a compact manifold. We also show there
is a relationship between the volume growth of a manifold with
nonnegative Ricci curvature and the length of a loop represent-
ing an element of infinite order in π
1
(M).
v
Contents

1 Introduction 1
2 Background 6
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 Lo ops 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
vi
4.4 Manifolds with Sublinear Diameter and Large Volume Growth 55
4.5 α-Noncollapsing . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 Volume Growth and the Length of Homotopically Nontrivial
Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
vii
Chapter 1
Introduction
There are many interesting relationships the b e tween the geometric and
topological structures of a smooth complete Riemannian manifold. Roughly
speaking, one can envision an n-dimensional manifold M as a subset of R
k
such that every point p has a neighborhood which is homeomorphic to an
open subset of R

n
. The tangent space of M at p, T
p
M ⊂ R
k
is the space
of all vectors v such that v = c

(0) where c : (−ε, ε) → M is a curve on M
with c(0) = p. The Riemannian metric on M is the natural inner product
on T
p
M, that is, the restriction of the dot product on R
k
to T
p
M. The
Riemannian metric enables us consider M as a metric space by defining the
distance 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 properties
such as compactness, connectedness, and homotopy and homology groups.
In studying the interactions between the geometric and topological struc-
tures of a manifold, curvature plays a pivotal role. To define curvature we
1
begin with the most simple manifolds. Let S be an orientable smooth sur-
face embedded in R
3
. Since S is orientable we can choose a smooth normal
vector field to S, n(x). The Gauss map g : S → S

2
is the map which takes
x to n(x). The Gaussian curvature of S at x, κ(x), is the determinant of the
differential 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, if
the surface is sphere-like around x then the gauss map preserves orientation
and 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 a
saddle 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 surface
then

S
κ = 2πχ(S)
where χ(S) is the Euler characteristic of S.
Since the only compact orientable surface with positive Euler charac-
teristic is the sphere, the Gauss Bonnet Theorem shows that any compact
orientable surface with κ(x) > 0 for all x ∈ S is homeomorphic to the
sphere. As we have seen, the condition κ(x) > 0 me ans that S curves in on
itself at all points, thus it is not surprising to find that there are not many
possibilities for the topology of these surfaces.
To generalize the definition of curvature to higher dimensions let v
1
and
v
2
be two unit length tangent vectors to M and let σ(v
1

, v
2
) be the surface
2
cut out of M by geodesics whose derivative is in the span of v
1
and v
2
. By
geodesic we mean a path that locally minimizes distance between points
in M. Since a small neighborhood of x in σ(v
1
, v
2
) can be isometrically
embedded in R
3
, and it can be shown that curvature is invariant under
isometry, we can define κ
x
(v
1
, v
2
) as the Gaussian curvature of σ(v
1
, v
2
) at
x. We call κ

x
(v
1
, v
2
) the sectional curvature at x of the surface spanned by
v
1
and v
2
. A Riemannian manifold has sectional curvature greater than or
equal to (greater than) H if κ
x
(v
1
, v
2
) ≥ H (κ
x
(v
1
, v
2
) > H) for all x ∈ M
and v
1
, v
2
∈ T
x

M. We denote this by sec
M
≥ H or sec
M
> H.
Ricci curvature is defined as the average of the sectional curvatures.
Given a unit tangent vec tor v at a point p, complete v to an orthonormal
basis of T
p
M, {v, e
1
, . . . , e
n−1
}. The Ricci curvature at p in the direction v
is
Ric
p
(v) =
1
n − 1
n−1

i=1
κ
p
(v, e
i
).
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 Ric
M
≥ H or Ric
M
> H. Since the Ricci curvature is
the 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, in
some sens e, M curves in on itself in every direction. Thus there should not
be many topological possibilities. This is expressed in the famous Bonnet-
Myers Theorem.
Theorem 1.0.2 (Bonnet-Myers). If M is a complete Riemannian man-
3
ifold with Ric
M
≥ H > 0 then M is compact and has finite fundamental
group.
If we weaken the hypothesis of the Bonnet-Myers Theorem the conclu-
sions no longer hold. For example, the paraboloid {(x, y, z) ∈ R
3
: z =
x
2
+ y
2
} has sec
M
> 0 but is not compact. And T
n
× R has sec

M
≡ 0,
is noncompact, and has infinite fundamental group. Still, the topology of
manifolds with sec
M
≥ 0 is quite restrictive. Cheeger and Gromoll [10] have
shown 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, the
only noncompact manifolds with sec
M
≥ 0 are vector bundles over compact
manifolds with nonnegative sectional curvature. Moreover, Perelman [20]
has shown that any noncompact manifold with sec
M
≥ 0 and all positive
sectional curvatures at one point is diffeomorphic to R
n
, that is a vector
bundle over one point.
The Ricci curvature case is much different and far less understood. For
example, Sha and Yang [23] have constructed manifolds with positive Ricci
curvature and infinite second betti number. These examples are not even
homotopy 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 Ric
M
≥ 0. In this dissertation we study further the
interaction between the geometry and the fundamental group of a manifold
with nonnegative Ricci curvature. There are three parts.
In the first part we review results about the fundamental group of man-

ifolds with nonnegative Ricci curvature and background involving Gromov
4
Hausdorff convergence.
In the second part we show that, for a large class of manifolds with
nonnegative Ricci curvature, all of the information about the fundamental
group 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 at
infinity. If M has nonnegative sectional curvature this structure is very
well understood, but for a manifold with nonnegative Ricci curvature very
little is known. See Section 3.1.
In the third part we give bounds on the number of generators of the fun-
damental group under some natural geometric conditions. In fact, we show
that the fundamental group of these manifolds behaves like the fundamental
group of a compact manifold. See Section 4.1.
All manifolds in this thesis are assumed to be complete, noncompact,
and without boundary. For clarity we will often omit explicitly stating this
hypothesis but it is always assumed.
5
Chapter 2
Background
2.1 The Fundamental Group of Manifolds
with Nonnegative Ricci Curvature
In this section we review some results concerning the fundamental group
of manifolds with nonnegative Ricci curvature. The most basic geometric
tool in studying manifolds with Ric
M
≥ 0 is the Bishop-Gromov relative
volume comparison theorem which bounds the volume of a metric ball in
terms of the corresponding ball in R
n

. By metric ball we mean
B(p, r) = {x ∈ M : d(p, x) < r}.
Theorem 2.1.1 (Bishop-Gromov). If M
n
is a complete n-dimensional
Riemannian manifold with Ric ≥ 0 then the quantity
vol(B(p, r))
r
n
6
is nonincreasing in r. In particular,
vol(B(p, r)) ≤ ω
n
r
n
where ω
n
is the volume of the ball of radius 1 in n-dimensional Euclidean
space.
The Bishop-Gromov Volume Comparison Theorem motivates the fol-
lowing definition.
Definition 2.1.2. A manifold is said to have polynomial volume growth of
order ≤ k if
lim sup
r→∞
vol(B(p, r))
r
k
< ∞
and has polynomial volume growth of order ≥ k if

lim inf
r→∞
vol(B(p, r))
r
k
> 0
By Bishop-Gromov volume comparison, if M
n
has nonnegative Ricci
curvature, then M has polynomial volume growth of order ≤ n. If, in
addition, M has polynomial volume growth of order ≥ n, then M
n
is said to
have Euclidean volume growth. In this maximal case we have the following
result of Li which he proves using a heat kernel estimate. Anderson [2] has
also given another proof using the volume comparison theorem.
Theorem 2.1.3 (Li, [14]). If M
n
has Ric ≥ 0 and Euclidean volume
growth then π
1
(M) is finite.
Of course, as we saw in the last chapter, there are manifolds with Ric
M
≥
0 and infinite fundamental group. However, there is still a relationship
7
between volume growth and the structure of the fundamental group when
π
1

(M) is infinite. To see this we must first define the growth of a finitely
generated group.
Let Γ be a finitely generated group generated by the set {g
1
, . . . , g
k
}.
Any g ∈ Γ can be written as a word g =

i
g
n
i
k
i
, where k
i
∈ {1, . . . , k}.
Define the length of this word to be

i
|n
i
|, and let |g| be the minimum of
the lengths of all word representations of g in the generating set {g
1
, . . . , g
k
}.
Definition 2.1.4. Fix a set of generators for Γ. The growth function of Γ

is
Γ(s) = #{g ∈ Γ : |g| ≤ s}.
The function Γ(s) depends on the generating set chosen. However, if
Γ(s) is a polynomial of degree k for some generating set then the growth
function corresp onding to any other set of generators of Γ must also be
bounded 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) ≤ as
k
for 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. Applying
the Bishop-Gromov Volume Comparison Theorem to the universal cover
Milnor shows the following.
8
Theorem 2.1.6 (Milnor, [18]). If M
n
has Ric ≥ 0 then any finitely
generated subgroup of π
1
(M) has polynomial growth of order ≤ n.
Moreover, Anderson has shown that the volume growth of M controls
the growth of π
1
(M).
Theorem 2.1.7 (Anderson, [2]). If M
n
has Ric ≥ 0 and polynomial
volume growth of order ≥ k then any finitely generated subgroup of π

1
(M)
has polynomial growth of order ≤ n − k.
Yau [35] has shown that any noncompact manifold with Ric ≥ 0 has at
least linear volume growth, that is polynomial volume growth of order ≥ 1.
Therefore, Anderson’s result shows that any finitely generated subgroup of
the fundamental group of a noncompact manifold with nonnegative Ricci
curvature has polynomial volume growth of order ≤ n − 1.
Gromov [11] has shown that any finitely generated group with polyno-
mial growth is almost nilpotent, that is the group has a nilpotent subgroup
of finite index. Wei [32] and Wilking [33] have shown that for any finitely
generated almost nilpotent group, G, there is a manifold with positive Ricci
curvature and fundamental group G. Therefore a finitely generated group
G is the fundamental group of some manifold with nonnegative Ricci cur-
vature if and only if G is almost nilpotent. A major open problem is the
following conjecture of Milnor.
Conjecture 2.1.8 (Milnor, [18]). If M has Ric ≥ 0 then π
1
(M) is finitely
generated.
No counterexample has been constructed to the Milnor conjecture. There
are a number of partial results. For example, Wilking [33] has shown that if
9
one can solve the conjecture f or abelian fundamental groups then the gen-
eral result follows and Sormani [27] has shown that manifolds with small
linear diameter growth have finitely generated fundamental group. We will
strengthen this result in a different direction in Chapter 3. On the other
hand, it is also an interesting problem to understand how the different ge-
ometric properties of the manifold, such as volume growth, interact with
properties of the fundamental group. We will prove some results in this

vein in Chapter 4.
Sormani [27] has also shown that there is a relationship between the
tangent cone at infinity of M and its fundamental group, in Chapter 3
we further investigate this relationship. In the next section we review the
definition of tangent cone at infinity and Gromov-Hausdorff convergence.
2.2 Gromov-Hausdorff Conve rgence
Another tool used in studying manifolds with Ricci curvature bounded be-
low is Gromov-Hausdorff convergence introduced by Gromov in [11]. First
we recall the definition of Hausdorff distance.
Definition 2.2.1. Given a metric space Z and A, B ⊂ Z, a ε-tubular
neighborhood of A is
T
ε
(A) = {z ∈ Z : ∃a ∈ A, d(z, a) < ε}
and the Hausdorff distance between A and B is
d
Z
H
(A, B) = inf{ε : A ⊂ T
ε
(B), B ⊂ T
ε
(A)}.
10
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 Gromov-
Hausdorff distance between X and Y is
d
GH

(X, Y ) = inf{d
Z
H
(f(X), g(Y )) : f : X → Z , g : Y → Z},
where the infimum is taken over all f and g, isometric embeddings of X
and Y into a metric space Z.
Gromov-Hausdorff distance at first may seem like a convoluted mathe-
matical object. Indeed, given two metric spaces it is very difficult to calcu-
late their Gromov-Hausdorff distance. However, d
GH
is a complete metric
on M = {compact metric spaces}/isometry and this metric gives a concept
of convergence on the space of compact metric spaces. This convergence
also has a simple geometric meaning.
To see this consider the following situation. You are shown two objects.
You see the first objec t which is then removed from your view and then
you are shown the second object. You are then asked whether the two
objects look alike. To answer the question you compare the two objects
in your mind. By moving images of the two objects around in your mind
you try determine whether the two objects can ever be embedded in a third
space so that they are very close to each other. You can never determine
whether two objects are exactly the same since you can only see on a finite
length scale but you might be given a microscope which improves the length
scale which you can observe. A sequence of objects {X
i
} converges to X
11
in the Gromov-Hausdorff topology means that no matter how powerful a
microscope you are given, you can go far enough out in the sequence so that
all the X

i
will look like X.
We wish to define Gromov-Hausdorff convergence for noncompact metric
spaces, one could use the same definition as in the compact case. However
this is not intuitively satisfactory as the following example shows.
Example 2.2.3. For any θ ∈ [0, 2π) let W
θ
= {z ∈ C : z = re
iφ
, 0 ≤
φ ≤ θ}. That is, W
θ
is the infinite wedge in C enclosed by an angle of
θ. As θ → 0, W
θ
will be a smaller and smaller wedge that geometrically
looks as though it should converge to the x-axis. (see figure 2.1). However,
d
GH
(W
θ
1
, W
θ
2
) = ∞ if θ
1
= θ
2
so the sequence does not converge.

Figure 2.1: The W
θ
should converge to the positive x-axis
Instead we define convergence of noncompact metric spaces as conver-
gence on compact sets
Definition 2.2.4. Suppose X
i
and Y are complete metric spaces with x
i
∈
X and y ∈ Y . Then the pointed spaces (X
i
, x
i
) converge in the pointed
Gromov-Hausdorff topology to (Y, y) if for all R > 0 B(x
i
, R) converges to
B(y, R) with respect to the restricted metric on the balls.
Gromov-Hausdorff convergence is particularly useful in studying man-
12
ifolds with Ricci curvature lower bounds because of the Gromov Precom-
pactness Theorem
Theorem 2.2.5 (Gromov Precompactness Theorem). The set of all
pointed, complete, n-dimensional Riemannian manifolds with Ricci curva-
ture bounded below by H is precompact in the pointed Gromov-Hausdorff
topology.
In other words, a sequence,(X
i
, x

i
) where all the X
i
have Ricci curvature
bounded below by H will have a subsequence which converges to some
metric 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 interesting
problem to study what other prop erties a limit space must have.
The geometric properties of the limit space of a sequence of manifolds
with Ricci curvature lower bound have been studied extensively by Cheeger
and Colding [5], [6], [7], [8]. Sormani and Wei apply their work to study the
topology of the limit space [29], [30]. They show that a limit space must
have a universal cover in the following sense.
Definition 2.2.6. ([31] p 62, 82) A path connected covering space
˜
Y of a
path connected topological space Y is the universal cover of Y if
˜
Y covers
every other path connected cover of Y and the covering projections form a
commutative diagram.
Theorem 2.2.7 (Sormani-Wei, [30]). If (Y, y) is the pointed Gromov-
Hausdorff limit of a sequence of Riemannian manifolds {(X
i
, x
i
)} with Ric
X
i
≥

H then Y has a universal cover.
13
Recall, that if a top ological space Y is semi-locally simply connected
then 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 curvature
must be semi-locally simply connected. There are sequences of Riemannian
manifolds that converge to a metric space which does not have a universal
cover. When a topological space is semi- locally simply connected, the fun-
damental group is exactly the deck transformation group of the universal
cover. 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 π
1
(Y ) is the group of deck
transformations 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 π
1
(Y ) for limit spaces of
sequences of manifolds with nonnegative Ricci curvature. We are concerned
with special limit spaces called tangent cones at infinity.
Definition 2.2.9 ([12]). Let X be a complete length space. A tangent
cone at infinity of X, (Y, d
Y
, y
0
), is a rescaled Gromov Hausdorff limit
(X,
d

X
r
i
, p)
GH
−−→ (Y, d
Y
, y
0
)
where r
i
→ ∞ and d
X
is the metric on X.
Since we are rescaling X by a sequence of smaller and smaller num-
bers, at each step the structure of a larger and larger ball is crushed into
14
a small neighborhood of the basepoint, and thus in the limit we see only
the 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 has
nonnegative Ricci curvature then the entire sequence of rescalings has non-
negative Ricci curvature and the Gromov Precompactness Theorem implies
that tangent cones at infinity exist.
In fact, if X has nonnegative sectional curvature then no matter what
sequence of rescalings is taken, the limit is the Euclidean metric cone over
the 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 has
nonnegative Ricci curvature and Euclidean volume growth then any tangent
cone at infinity is a metric cone [5] although Perelman [21] has shown that

this limit may depend on which sequence of rescalings is chosen. In general,
the tangent cone at infinity of a manifold with nonnegative Ricci curvature
may 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 the
following question.
Question 2.2.10. Is the revised fundamental group of the tangent cone at
infinity 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 a
prescribed way and applying the methods of Wei and Sormani.
15
Chapter 3
Semi-Local Fundamental
Groups
3.1 Introduction and Statement of Results
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 the
group obtained by taking all the loops based at p contained in the closed
ball of radius r and identifying two loops if there is a base point preserving
homotopy between them that is contained in the open ball of radius R. We
call G(p, r, R) the geometric semi-local fundamental groups of M at p. (See
figure 3.1).
It may seem strange to the reader to consider inner balls which are closed
and outer ones which are open. The definition is given as above be cause
16
p
Figure 3.1: G (p, r, R) is the group of loops contained in the inner closed
ball 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 a
natural map from G(p, r, R) to π
1
(M, p) induced by the inclusion of B(p, r)
into M. In this chapter π
1
(M)
∼
=
G(p, r, R) will mean that this induced
map is a group isomorphism. We denote the image of this map by G(p, r).
The geometric semi-local fundamental groups depend heavily on the
metric 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 R
3
with
standard Euclidean coordinates. For each positive integer n, remove a small
disc in the xy-plane around each point (n, 0, 0) and glue in its place a long
17