Isometries and CAT(O) Metric Spaces
by
Naomi Lynne Wolfson, B.Math
A thesis submitted to
the Faculty of Graduate Studies and Research
in partial fulfillment of
the requirements for the degree of
Master of Science
School of Mathematics and Statistics
Ottawa-Carleton Institute for Mathematics and Statistics
Carleton University
Ottawa, Ontario, Canada
April 27, 2006
© copyright
2006, Naomi Lynne Wolfson
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Using the CAT(O) inequality, non-positively curvature in the Riemanian sense
can be generalized to the much broader setting of geodesic metric spaces. By
studying CAT(O) spaces and groups which act on them properly and cocom-
pactly by isometries, a great deal can be learned about the structure of both.
This thesis presents an introduction to CAT(O) spaces and explores the conse
quences of proper cocompact group actions on these spaces, including the Flat
Torus Theorem, the Splitting Theorem, and a solution to the word problem
for such groups.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii
Acknowledgments
I would like to thank my supervisor Dr. Ben Steinberg as well as Dr. Inna
Bumagin, Dr. Mike Moore and Valerie Daley from the School of Mathematics
and Statistics at Carleton University. Above all else, I would like to thank my
parents Eleanor Bennett and Michael Wolfson, who have always supported
me throughout all my crazy endeavors and without whom I would never have
gotten this far.
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A b stract......
Acknowledgments ...... ii
List of Figures...... iv
0 Introduction 1
1 Geodesics and the Model Spaces 4
1.1 What are Geodesics? ...... 4
1.2 Geodesic Metric Spaces and Convexity ...... 9
1.3 Geodesic Triangles and Angles Between Geodesics ...... 14
1.4 The Euclidean, Spherical, and Hyperbolic Metric Spaces . . . 22
1.5 The Model S p a c e s ...... 27
2 Introduction to CAT(k) Spaces 36
2.1 The CAT( k) Inequality ...... 36
2.2 Insights into Convexity ...... 44
2.3 Projection onto a Convex Subspace ...... 50
iii
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2.4 The Centre of a Bounded Set ...... 55
2.5 Flat Subspaces and the Product Decomposition Theorem . . . 57
3 Group Actions 75
3.1 Basics of Group A c tio n s ...... 75
3.2 Group Presentations and A ctions ...... 83
3.3 Bruhat-Tits Fixed Point Theorem ...... 93
4 Decision Problems 96
4.1 The Word Problem ...... 96
5 Investigations of Isometries 110
5.1 Displacement Functions and Translation Length ...... 110
5.2 Three Classes of Isometries...... 117
6 Groups of Isometries 127
6.1 More on the Structure of Isometries ...... 127
6.2 Splitting CAT(O) Spaces...... 132
6.3 The Flat Torus Theorem ...... 156
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1.1 Section of the Cayley graph of F(a, b) ...... 13
1.2 Geodesic Triangle ...... 14
1.3 Comparison Triangle ...... 15
1.4 Comparison Triangle ...... 17
1.5 Split a into the sum of a' and a " ...... 20
1.6 Split the geodesic segment [a, 6] by a and a ' ...... 21
1.7 Euclidean Law of Cosines ...... 23
1.8 Spherical Distances ...... 24
1.9 Spherical Law of Cosines ...... 25
1.10 Hyperbolic Law of Cosines ...... 26
1.11 ^-comparison Triangle ...... 30
1.12 Quadrilateral with vertices A,B, B and C ...... 32
1.13 Construction of B' ...... 33
1.14 Construction of Triangle A(H, B, B ' ) ...... 35
2.1 Comparison Point ...... 37
v
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2.2 Comparison Triangle for A (\p,q]',\p,x\,[x,q]) 40
2.3 Comparison Triangle For c(0), c(t0), c(t0 + e ) ...... 42
2.4 Graph of a Convex F u n ctio n ...... 45
2.5 Comparison Triangle for (c(0), c(l), c '( l ) ) ...... 47
2.6 Geodesic paths c, d and c " ...... 48
2.7 Comparison Points Relative to the Annular R egion ...... 51
2.8 Location of p e [tt(x ),I7]...... 53
2.9 Arrangement of A' and A" in E 2 ...... 59
2.10 Location of x and x' in A(p,q,q') ...... 60
2.11 Placement of Ai and A 2 ...... 63
2.12 Calculating the distance between cXl(ti) and cX2(£2) ...... 73
3.1 Constructing the path cs ...... 86
3.2 Relationship between D, CU3(r), and X ...... 89
3.3 Selection of 5i and 5 [ ...... 90
4.1 Dividing the geodesic segment [x 0,7 • xo] into f°ur parts .... 99
4.2 A(xo, 7 • xo, 7 ' • Xo) with a and b illustrated ...... 103
4.3 Comparing isosceles triangles ...... 104
4.4 7 and 7 ' in the metric Cayley graph of T ...... 106
5.1 Constructing a quadrilateral in E 2 for com parison ...... 115
6.1 Commutative Diagram for 72 , p, and p1 ...... 146
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Introduction
The objective of this thesis is to supply a straightforward introduction to
CAT(O) spaces and the groups which act on them properly and cocompactly
by isometries. The classical theory for groups acting on hyperbolic manifolds
was very successful and in the latter part of the 20th century attempts were
made to encapsulate the properties of non-positively curved manifolds in the
setting of geodesic metric spaces. Two of the schemes used to accomplish
this were Gromov’s Thyperbolicity and the CAT(O) inequality [Gromov], Al
though both approaches have been tremendously profitable, this thesis focuses
on the latter. Named for E. Cartan, A.D. Alexandrov, and A. Topogonov, the
CAT(O) inequality for geodesic metric spaces successfully generalizes many of
the properties of hyperbolic manifolds. Moreover, the Flat Torus Theorem, the
Splitting Theorem, and the Solvable Subgroup Theorem, which were proved
for the fundamental groups of compact non-positively curved Riemanian man
1
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ifolds [Gromoll & Wolf], [Lawson k, Yau] have analogous theorems for groups
which act properly and cocompactly by isometries on a CAT(O) metric space.
The principle resource available on the topic is the book Metric Spaces of
Non-positive Curvature written in 1999 by Martin Bridson and Andre Haefliger
[Bridson & Haefliger]. This 700-page text is comprehensive and encyclopedic in
nature, containing reference to more or less everything that was known about
CAT(O) spaces at the time it was written. This book is not necessarily suitable
as an introductory text for graduate students who wish to learn about CAT(O)
spaces however, as it omits numerous details in its proofs while covering more
material than appropriate for an introduction. In order to provide a more
effortless introduction, we have chosen to cover selected chapters and sections
from this book, presenting all the omitted details in the relevant proofs. By
choosing an appropriate route through this material, we build progressively
from the basic definitions of geodesic and CAT(O) metric spaces to attain the
ultimate goal of proving the Splitting Theorem and Flat Torus Theorem in
the context of CAT(O) spaces. This thesis provides a snap-shot of the theory
of CAT(O) spaces while still providing sufficient detail to be rigorous.
The thesis can be logically divided into two parts where the first part
consists of Chapters 1 and 2 and second part contains the remaining chapters.
In Chapter 1 we will introduce geodesic metric spaces, triangles and angles
as well as a collection of uniformly curved metric spaces which will serve as
a basis upon which to define CAT(O) or CAT (A) spaces. Then in Chapter 2
we define CAT(O) spaces and begin examining their global structure. We pay
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special attention to the consequences of convexity in these spaces and define
the centre and radius of a bounded set, which will be used later to prove the
Bruhat-Tits Fixed Point Theorem for complete CAT(O) spaces. The chapter
will finish with a section which examines the presence of flat subspaces in a
CAT(O) space and a proof of the Product Decomposition Theorem which will
be used to study axes of hyperbolic isometries in Chapters 5 and 6 .
In Chapter 3 we change direction and introduce group actions on topolog
ical spaces. We also prove a useful result due to Macbeath that gives a finite
presentation for a group acting properly and cocompactly by isometries on a
simply-connected geodesic metric space. The subsequent chapter elaborates
on this, showing that any group acting properly and cocompactly by isome
tries on a CAT(O) space has a decidable word problem and in fact a quadratic
isoperimetric function.
The final two chapters are dedicated to the isometries of CAT(O) spaces.
In Chapter 5 we look at the properties of individual isometries via their dis
placement functions and translation lengths. In Chapter 6 , the final chapter,
we will examine the structural consequences of groups which act properly (not
necessarily cocompactly) by isometries on a CAT(O) space. We finish our dis
cussions by presenting proofs for the Splitting Theorem and the Flat Torus
Theorem. This thesis assumes basic knowledge of metric spaces, group theory,
and topology.
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Geodesics and the Model Spaces
In this chapter, we introduce geodesic metric spaces and explore a class
of such metric spaces called the Model Spaces. The notions of curvature we
will explore in this thesis are based on the properties of the Model Spaces
which means that a solid understanding of their properties are essential in
the chapters which follow. The material in this section is presented largely
without proof and one may refer to [Bridson & Haefliger pg. 1-26] for more
details.
1.1 What are Geodesics?
In a metric space X we define a path c : [a, b] —> X to be a continuous
function from a closed bounded interval [a, b\ C M to the metric space X. A
4
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path c is said to be rectifiable if its length lc defined by
n —1 £c = sup . ,£d(c(ti),c(ti+ 1)) (1.1) (l-tf) —6 Q
exists. We may verify that in the space R" this notion of the length of a path
is equivalent to the arc length of a curve.
In a similar manner, we define a ray in a metric space X to be a continuous
map c : [a, oo) —> X and a line to be a continuous map c : R — > X. In this
thesis however, we deal almost exclusively with paths, rays, or lines in which
the distance between any two points on the path is actually equal to the length
of the portion of the path which joins them. We call such paths geodesic; the
precise definition is as follows.
Definition 1.1.1. A geodesic path in a metric space X is a continuous function
c : [0, A] —> X satisfying:
d(c(t),c(t')) = |t - t'\ (1.2)
for every t,t' £ [0, A].
Given a geodesic path c : [0, A] —»• X we say that c originates at c(0)
and terminates at c(A); or that c is a geodesic path from c(0) to c(A). As a
consequence of how a geodesic path c is defined, its length £c is equal to the
distance between its originating and terminating points; in particular we have
lc = A. Furthermore, for any t < t' < t" £ [0, A] the triangle inequality reduces
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to an equality.
d(c(t), c(t")) = d(c(t), c(t')) + d(c(t'), c(t")) (1.3)
We also apply the concept of geodesics to lines and rays in a metric space
X where a line c : M X is said to be geodesic if for every t, t' £ 1R the
equality d(c(t), c{t')) = \t — t'\ holds. Geodesic rays are defined similarly.
Each geodesic path c : [0, A] —> X in a metric space X has as its domain
the interval [0, A] where A is equal to the length Ic of that path. In a collection
of several geodesic paths however, it is not necessary that these lengths are
all the same and so the geodesics may have different domains. To simplify
matters, it is useful to regard a geodesic path as a continuous image of the
unit interval.
Definition 1.1.2. A linearly reparameterized geodesic path in a metric space
A is a continuous function c : [0,1] —> X such that for some A > 0:
d(c(t),c(t!)) = \\t — t'\ (1.4)
for every t, t' £ [0,1]. Note that in this case, it is the reparameterization factor
A which is equal to the length £c of the path.
Let / C R be a closed interval. The path, ray, or line c : I —> X is said to
be locally geodesic if for every t £ I there is a neighbourhood J C I of t such
that c|j, after being suitably reparameterized, is a geodesic path.
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Given any choice of a geodesic path c : I —> X originating at x and
terminating at y in metric space X, we write [x, y\ to denote the image of c in
X ; this image [x,y] is called a geodesic segment. We often speak of a geodesic
segment [x, y] without mention of the specific geodesic path c, whose image is
[x,y\. Geodesics need not be unique however, so we must take care to avoid
ambiguity. For example, given antipodal points a and b on the unit sphere S2,
any great circle through a and b defines two geodesic paths from a to b.
Furthermore, given any geodesic path c which defines a geodesic segment
[x, y\ there exists a linearly reparameterized geodesic path which defines the
same geodesic segment. As such, if we have a geodesic path c and a lin
early reparameterized geodesic path c' originating and terminating at the same
points such that c and c' have the same image, the two paths are equivalent
in most of our discussions.
For any geodesic segment [a, b] defined by the geodesic path c : [0, A] —> X
we define the reverse geodesic segment [ 6 , a] to be the image of c : [0, A] —> X
where c(f) = c(A — t) for each t G [0, A].
Definition 1.1.3. Given two geodesic paths c : [0, A] —► X and d : [0, A'] —> X
such that the terminating point of c is the originating point of d we define the
concatenation cd : [0, A + A'] —> X of c and d to be given by:
c(t) t G [0, A] (1.5) d(t — A) t G [A, A + Ar]
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If we have two geodesic segments [x, y] and [y, z] we shall write [x, y\ U [y, z]
to represent the image of the concatenation of their corresponding geodesic
paths. We must be careful however, for it is not necessarily the case that this
union is itself a geodesic segment.
Proposition 1.1.4. Suppose c\ : [0, Ai] —>• X and c 2 : [0, A2] —> X are geodesic
paths such that the terminating point of C\ is the originating point of c2. Then
the concatenation of these two paths is itself a geodesic path if and only if
d(x, z) = d(x,y) + d(y,z) where x = ci(0), y = Ci(Ai) = c2(0), z = c2(A2).
Proof. (=>) Suppose that the concatenation CiC 2 is a geodesic path. Then by
(1.3) we have
d(cic 2(0), cic2(Ai + A2)) = d(cic 2(0), cic2(Ai)) + d(cic2( Ai), Cic 2(Ai + A2))
= d(ci(0), ci(Ai)) + d(c 2(0), c2(A2)) (1.6)
So we have d(x, z) = d(x, y) + d(y, z) as desired.
(^=) Suppose d(x, z) = d(x,y) + d(y, z) = Ai + A2. To show that CiC2 is a
geodesic path, we only need to consider t G [0, Ai] and t' G [Ai, Ai + A2] since
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we already know that c\ and c 2 are geodesic paths. We see that:
d(cic 2(0), ciC2(A! + A2)) < d(cic 2(0), cic2(t)) + d(cic2(t), Cic2(t'))
+d(cic 2(f'), cic2(Ai + A2))
< d(cic 2(0), cic2(t)) + d(cic 2(t), CiC2(Ai))
+d(ciC2(Ai),CiC2(t'))
+d(cic 2(t'), cic2(Ai + A2))
= d(ci(0),ci(£)) + d(ci(t),ci(Ai))
+d(c 2(0), c2(t - Ai)) + d(c2(t’ - Ax), c2(A2))
= t + (Ax — t) + (t’ — Ax) + (A2 — t' + Ax)
= Ax + A2
= d(cxc2(0), cxC2(Ax + A2))
Thus, equality holds everywhere and in particular this implies that for any
t e [0, Ax] and t' G [Ax, Ax + A2] we have:
d(cxc 2(t), Cic2{t')) = (Ax -t) + (t' - Ai) =t' - t (1.8)
We conclude that C\C2 is a geodesic path as desired. □
1.2 Geodesic Metric Spaces and Convexity
We are interested in those metric spaces for which there exists a geodesic
segment joining each pair of points. Metric spaces which satisfy this condition
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are called geodesic metric spaces. A slightly less restrictive condition for X is
that every ball in X of radius r is a geodesic metric space; such spaces are called
r-geodesic. A stronger condition for geodesic metric spaces is that for each pair
of points x, y in the space X there is a unique geodesic path originating at x
and terminating at y. Such spaces are called uniquely geodesic metric spaces ;
similarly, any r-geodesic metric space in which each ball B(x,r) C X is a
uniquely geodesic metric space is called uniquely r-geodesic.
Example 1.2.1. The 2-dimensional sphere §2 is a geodesic metric space when
endowed with the metric which assigns to each pair of points x, y G §2 a
distance d(x, y) which is equal to the length of the shortest curve in S 2 which
joins them. The space S 2 with this metric, however, is not uniquely geodesic.
Proposition 1.2.2. Given geodesic segments [a, b\ and [ 6 , c] in a uniquely
geodesic metric space X. The geodesic segment [a, c] is equal to the union
[a, b] U [b, c] if and only if b G [a, c]
Proof. (=>) This direction is clear since [a, c] = [a, b] U [b,c] implies b 6 [a, c].
(<=) If b e [a, c] then d(a,c) = d(a,b ) + d(b,c ) so by Proposition 1.1.4, the
concatenation [a, b] U[b, c] is in fact a geodesic segment, say [a, b] U [b, c] = [a, c]'.
The metric space X is uniquely geodesic however, so [a, 6 ]U[6 , c] = [a, c]' = [a, c]
as desired. □
Given a geodesic metric space X, a subset C C X is called convex if for
every x, x' € C there exists a geodesic path c : [0, A] —> R in X originating at
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x and terminating at x' and for every such geodesic path, its corresponding
geodesic segment [x, x'\ is completely contained within C.
Remark 1.2.3. A convex subspace C of any geodesic metric space X is a
geodesic metric space. This is because any two points x,y E C have a geo
desic segment [x, y] in X joining them and this segment is contained within
C. Similarly, any convex subspace of an r-geodesic space is r-geodesic and
any convex subset of a uniquely geodesic space (uniquely r-geodesic space) is
uniquely geodesic (uniquely r-geodesic).
Before continuing, we present some examples of geodesic metric spaces.
Refer to [Serre] for more details.
Example 1.2.4 (Metric Cayley Graphs). A combinatorial graph G con
sists of a set of vertices V and a set of edges E together with two maps
d0 : E —> V and d\ : E —> V (the endpoint maps). We say that the edge
e originates at do(e) and terminates at d\(e). Using the combinatorial graph
G we define the metric graph Xq by taking the set E x [0,1] modulo the
equivalence relation
(e,i) ~ (e',*') <==> di(e) = dv(e') (1.9)
where [e!C) £ E and i,i' £ {0,1}. We denote this space Xq and take
the distance d(x, y) to be the infimum of the lengths of all the piecewise linear
paths joining x to y in Xq- This defines a geodesic metric space provided G
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is connected. If the corresponding metric graph X q is connected and simply
connected, then it is called a tree.
The Cayley graph for a group F with generating set S is defined to be the
combinatorial graph consisting of the vertex set T and the edge set E which
contains an edge e(7iS) originating at 7 and terminating at 7 ■ s for each pair
7 G T, s e S. The Cayley graph is denoted C's(T) but we will also abuse
notation slightly and write Cs{F) to denote the metric Cayley graph. The
metric Cayley graph is a geodesic metric space. It will be a uniquely geodesic
metric space if and only if it is a tree, which occurs if and only if T is free and
S’ is a set of free generators.
For example, the free group on two generators F(a, h) has the Cayley graph
as illustrated in Figure 1.1
This metric graph is a tree and thus it is a uniquely geodesic metric space.
Example 1.2.5. We define the Manhattan metric on R 2 as follows::
| |l_X| V = V (1.10) I \x\ + \x'\ + \y - y'\ y ^ y'
Endowed with this metric, R 2 is a uniquely geodesic metric space.
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/\
/\ /\
/\
/\
Figure 1.1: Section of the Cayley graph of F(a, b)
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1.3 Geodesic Triangles and Angles Between
G eodesics
When we begin exploring the curvature of geodesic metric spaces, our
primary method of characterization shall be by investigating the properties of
their triangles. By comparing these triangles to triangles in spaces of known
curvature, say the Euclidean plane for instance, we can begin to understand
how geodesics in our metric space behave.
Definition 1.3.1. Given three points x, y, and z in a metric space X which
have geodesic segments [x, y\, [y, z], [z, x] E X joining them, we define the
geodesic triangle A([x, y\, [y, z\, [z, x\) to be the union of the geodesic segments
[x, y] U [y, z] U [z, x] C X. If the geodesic segments [x, y\, [y, z], and [z, x] are
unique then we may omit these choices in our notation and write A (x,y, z) in
the place of A([x, y\, [y, z], [z, x]).
y
x,u
x
Figure 1.2: Geodesic Triangle
Geodesic triangles are an powerful tool for understanding the properties
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of a metric space. We shall see later that the curvature of a metric space is
associated with whether its geodesic triangles are fat or thin. First though,
we need to find ways of describing our geodesic triangles, but of what use are
triangles if you have no notion of angle ? So, to develop a meaning for angle
in our spaces, we shall associate with each geodesic triangle in a space X a
triangle in the Euclidean plane E2, which is a metric space in which we have
a good understanding of the relationships between angles and triangles. The
idea of using comparison triangles is due to Alexandrov.
Definition 1.3.2. Let A be a metric space and x,y,z G X. A comparison
triangle in E 2 for the triple (x,y,z) is a geodesic triangle A(x,y,z) G E 2 such
that d(x,y) = d(x,y), d(y,z) = d(y,z), and d(z,x) = d(z,x).
The triangle inequality guarantees that we can always find x,y,z£ E2 and
furthermore, this choice is unique up to isometry. When a specific choice of
points x,y,z G E2 for a comparison triangle is made, it is typically denoted
A (x,y,z).
Figure 1.3: Comparison Triangle
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The comparison angle of the points x,y,z at the point x is the interior
angle of any comparison triangle A (x,y, z) = A (x,y,z) at the vertex labelled
x 6 E2. The comparison angle is denoted Z.x(y, z) and is well defined provided
x ^ y and x ^ z.
We now have a way of describing the angle between triples of points in
a metric space. Typically however, we are used to defining angles at the
intersection of two lines, not at triples of points. As such, we would like to
find a way of defining the angle between two geodesic paths which originate
at the same point. In the Euclidean plane, the angle between two geodesic
paths originating at z (which are straight lines in this case) coincides with the
comparison angle between x and y at z for any choice of x from the first line
and y from the second (excluding any choice where x = z or y = z). In an
arbitrary metric space however, it not necessarily the case that two different
choices of points would yield the same comparison angle, so we introduce the
Alexandrov angle between two geodesic segments.
Definition 1.3.3. Given two geodesic paths (or linearly reparameterized geo
desic paths) c : [0, o] —> X and d : [0, a'\ —► X in a metric space X originating
from the same point c(0) = c'(O), the Alexandrov angle between c and d is
defined to be:
Z(c,d) :=lim sup Zc(0)(c(f), c'(t')) (1.11) £-^°0 In this expression, Zc( 0)(c(£),d(t')) is the comparison angle for the triple of points (c(0), c(t), d(t')) as defined previously. Given geodesic segments [a, fe] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 17 and [a, c] we write Z([a, b\, [a, c]) to denote the Alexandrov angle Z(c, d) where c and d are geodesic paths whose corresponding geodesic segments are [a, b] and [a, c] respectively. X: c(a) E 2: 7(f) c c(0) c(t') c(0) Figure 1.4: Comparison Triangle The Alexandrov angle possesses several properties analogous to the stan dard angle in Euclidean geometry. For example, the angle between two oppo site rays in a geodesic line is it, as outlined in the following proposition. Proposition 1.3.4. Given geodesic segments [a, b\ and [a, c] in a metric space X such that [b, a] U [a, c] = [b, c] is a geodesic segment, then the following equality holds Z([a,6 ], [a,c]) = vr ( 1.12) Proof. Let c : [0,1] —»• X and d : [0,1] —> X be the linearly reparameterized geodesic paths with images [a, b] and [a, c] respectively. If we pick points t, t' G (0,1] then any comparison triangle A(c( 0), c(t), d(t')) = A(c(0), c(i), d(t')) for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 18 c(0), c(t), and c'(i') is degenerate. This is because by (1.3) we have d(c(t), c'(t')) = d(c(f), c( 0)) + d(c( 0), c'(i')) (1.13) Thus Z c(0)(c(t), c'(t')) = 7r which gives: Z([a, b], [a, c]) = Z(c, c') = lim sup n = n (1.14) £^ ° o < t,t'< e as desired □ Example 1.3.5. In the metric graph X for a combinatorial graph G, the angle between two geodesic paths is either 0 or n, depending if they coincide for non-zero time or not. In Euclidean space, angles are additive in the sense that if we cut an angle into two parts, it will be the sum of the two angles created. The Alexandrov angle is only subadditive. Proposition 1.3.6. Suppose c : [0,1] —> X, d : [0,1] —> X, and c" : [0, 1] —> X are linearly reparameterized geodesic paths in the metric space X all originat ing from the same point, c(0) = c'(0) = c"(0). Then the following inequality holds: Z(c,c") Proof. First, if Z(c, c") = 0 then then (1.15) holds trivially. Let us now suppose that Z(c, c") > 0. The proof proceeds by contradiction. Suppose that the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 19 inequality Z(c, c") > Z(c, d) + Z(c',c") holds. This implies that there exists 8 > 0 such that Z(c, c") > Z(c, d) + Z(d, c") + 35; let us assume further that 8 < Z(c,d'). From the definition of the Alexandrov angle (Definition 1.3.3), we may find eq > 0 such that for all t, t' < eq: Z c(0)(c(f), d{t')) < Z(c, c') + 5 (1.16) Similarly, there exists e2 > 0 such that for all t1, t" < £2 we have: Zc(0 )(c'(f'),c "(t"))< Z (c',C") + 5 (1.17) Furthermore, if we let e = min{ei, e2} then we may find t, t" < e such that: Z40)(c(t),d'(t"))>Z(c,d')-S (1.18) Let us fix t and t" from (1.18) and consider a triangle, A(0, x,x") £ E2 such that d(0, x) = f, d(0, x") = t" and the vertex angle a at Osatisfies: Z(c, c") - 5 < a < Zm (c{t),d'(t")) (1.19) Since Zc(0)(c(f),d'(t")) < w we have that a < tt and since Zc^(c(t),d'(t")) > Z(d, d') — 8 > 0 we can select a > 0 which guarantees that A is not degenerate. Then (1.19) implies that d(x,x") < d(c(t),d'(t")). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 20 Figure 1.5: Split a into the sum of of and a" Now, if we combine our initial assumption that Z(c, c") > Z(c, d) + Z(c', c") + 35 (1.20) with our choice of a such that a > Z(c, c") — 5 then we see that a > Z(c, d) + Z(c', c") + 25 (1.21) It is now possible to find a point x' G \x, x"\ C A such that the geodesic segment [O, x'] splits a into a' and a" satisfying a' > Z(c,d) + S and a" > Z{d, c") + S. See Figure 1.5. If we let r = max.{d(0,x),d(0,x")} = max{t, t"} then, by the convexity of balls in E2 we have [x,x"] C B(0,r) which implies that d(0,x') < r = max{t, t"}. Thus, if we let if = d(0,x') then t,t' < £i which by (1.16) im plies that Zc(0)(c(£),d(t’)) < Z{c,d) + S < of. The law of cosines now gives Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 21 d(c(t),d(t')) < d(x,x'). Similarly, d(c'(t'), c"(t")) < d(x',x"). Together, these inequalities tell us that d(c(£), c"(t")) > d(x,x") = d(x, x') + d(x', x") (1.22) > d(c(t), d(t')) + d(d(t'), d'{t")) which contradicts the triangle inequality. □ Corollary 1.3.7. Consider two geodesic segments [a, b] and [c, d] in a metric space X , such that c e [a,b\, c ^ a, c ^ b. Then if [c, a] and [c, b] are geodesic segments such that [a, c] U [c, b] = [a, b] then Z([a, c], [c, d]) + Z([c, d], [c, 6]) > 7r. b a Figure 1.6: Split the geodesic segment [a, b] by a and a' Proof. By Proposition 1.3.4 we have 7r = Z([c, a], [c, 6 ]) and by Proposition 1.3.6 we also have Z([c, a], [c, 6 ]) < Z([c, a], [c, d]) + Z([c, d], [c, 6 ]). Thus the desired inequality holds. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 22 In other words, if the angles a and a' split the geodesic segment [a, b] then a + a' > 7r. For the spaces we introduce in the following two sections, we shall find that a + a' = ir, a property which is key in proving the major result of this chapter, Alexandrov’s Lemma. 1.4 The Euclidean, Spherical, and Hyperbolic Metric Spaces In this section, we introduce the following metric spaces: Euclidean space E", spherical space §n, and hyperbolic space Hn. Each of these spaces satisfies a variation on the classical law of cosines. Furthermore, we will use these spaces to form a basis upon which to describe what it means to be a negatively or positively curved space. Definition 1.4.1. The n-dimensional Euclidean space En is the vector space Kn equipped with the metric which, given x, y £ En, assigns |\x — y\ | to be the distance between x and y where 11 • 11 denotes the f^-norm. Euclidean space is a uniquely geodesic metric space where the geodesic segment joining a pair of points x, y £ En is given by the straight line segment: [x,y\ = {ty + (l-t)x :te [ 0 ,l] } (1-23) The metric space En obeys the classical law of cosines: given any triangle A (x,y,z) in En where the side lengths are given by d(x,y) = a, d(x,z) = b, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 23 d(y, z) = c, and Z([x, y\, [x, z\) = 6, we have: c2 = a2 + b2 — 2abcos 6 (1-24) y x b Figure 1.7: Euclidean Law of Cosines Now we introduce a metric space in which, unlike in Euclidean space, geodesic paths and lines are curved. Definition 1.4.2. We define the n-dimensional spherical space §n to be the unit hypersphere, {x £ Rn+1 : ||a;|| = 1} together with the metric which, for each i, j £ §n, assigns the distance from x to y to be the length of a great arc which joins them. More precisely, the metric is given by the following equation: d(x, y) — cos_1(x • y) (1-25) where x ■ y is the standard dot product in Mn+1 A geodesic segment in §" joining any two points x and y is a great arc Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 24 Figure 1.8: Spherical Distances between them and this geodesic is unique provided d(x, y) < n. Consequently, §n is a geodesic metric space which is uniquely 7r/2-geodesic and any ball of radius less than or equal to ix/2 in Sn is convex. Remark 1.4.3. The n-dimensional sphere §" is not a geodesic metric space with the induced metric from En+1. If we define the angle between any two geodesic paths to be the Alexandrov angle from the previous section then the geodesic triangles in Sn obey a law of cosines analogous to the Euclidean case. This is called the spherical law of cosines: given a triangle A (x,y,z) 6 Sn such that d(x,y) = a, d(x,z) = b, d(y, z) = c and Z([x, y\, [x, z\) = 6, we have: cosc = cos a cos b + sin a sin b cos 6 (1-26) Furthermore, geodesic triangles in §” appear thick in the sense that their sides seem to swell outwards and their vertex angles have a sum which is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 25 y x b Figure 1.9: Spherical Law of Cosines greater than or equal to n. This inevitably leads us to the question as to the existence of a metric space in which the opposite occurs. That is, geodesic triangles appear to be thin in the sense that the sum of their angles is less than or equal to tt. In the following definition, we introduce a metric space in which this does occur; it is a subspace of En,:L, which is the vector space Mri+1 together with the symmetric bilinear form: n (u, V} = -u n+1vn+1 + ^ 2 uCi (1-27) i= 1 Definition 1.4.4. Define n-dimensional hyperbolic space Hn to be the set {u E E11'1 | (u,u) = —l,un+i > 0} (1-28) together with the metric which assigns to each a,b E Hn the distance d(a, b) satisfying: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 26 cosh (d(a,b)) = ~{a,b) (1.29) The space HP is uniquely geodesic and any ball is convex. Just as in the spherical case, HP obeys a hyperbolic law of cosines with respect to the Alexandrov angle: Given a geodesic triangle A(x, y, z) in HP such that d(x, y) = a, d(x, z) = b, d(y, z) = c and A([x, y], [x, z]) = 8, we have: cosh c = cosh a cosh b — sinh a sinh b cos 8 (1.30) V x Figure 1.10: Hyperbolic Law of Cosines For a more comprehensive approach to hyperbolic geometry, refer to [An derson] . In the preceding three metric spaces, if a and d are angles which split a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 27 geodesic segment, then the following equality holds a + ol = 7r (1-31) The Alexandrov angle between two curves originating at the same point z in one of these spaces, is equal to the Spherical, Hyperbolic, or Euclidean angle respectively [Bridson &; Haefliger pg. 173]. Thus, the Alexandrov angle is equal to the angle between the tangent vectors of these curves at the point z in the tangent plane [Bridson & Haefliger pg. 17,20]. Since the additivity of angles holds in the Euclidean plane E2, (1.31) holds in all three spaces. 1.5 The Model M ™ Spaces The spaces §n and Hn, which we defined in the previous section have geo desic triangles which compare to triangles in E2 in somewhat opposite ways. We say that the n-dimensional spherical metric space is positively curved and that the n-dimensional hyperbolic metric space is negatively curved ; the n- dimensional Euclidean plane, which seems to meet the other two halfway be tween, is said to be flat or have zero curvature. A natural question to ask is whether there are spaces posessing uniform curvature which lie somewhere in-between these others. As we will see in the following definition, the answer is to the affirmative. Definition 1.5.1. The n-dimensional model space of uniform curvature n, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 28 denoted M" r tis defined as follows: • if k = 0 then M0n = E" • if k > 0 then M” is obtained from Sn by assigning to it a new distance function dK : S'1 x §" -> 1 such that for all x, y £ §n we have: dK(x, y) = d(x, y)/y/K (1.32) where d : §" x §" -> R is the standard spherical metric. • if k < 0 then M" is obtained from Hn in a similar manner. The new distance function dK : E>n x §n —> M is given by: dK(x, y) = d(x, y ) / ^ ^ (1-33) for any x,y G M" where in this case we use the standard hyperbolic m etric d : IHP x HP —> R. For any k G R we define the diameter of M" to be := ^ for k > 0 and DK = oo otherwise. Remark 1.5.2. The metric spaces of uniform curvature M”, inherit the geo desic properties of their parent spaces En, SP, and HP. In particular, M” is a uniquely geodesic metric space for k < 0 and if k > 0 then M ” is uniquely ^-geodesic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 29 Since the model spaces are obtained by scaling the metrics for Era, and IP their corresponding cosine laws may be similarly rescaled. Thus, for each k £ R we obtain the following: Given a geodesic triangle A (x,y,z) C M” such that d(x,y) = a, d(x,z) = b, d(y, z) = c and Z([x, y\, [x, z]) = 9 then one of the following cosine laws will hold: • if k = 0: c2 = a2 + b2 — 2abcos9 (1-34) • if k < 0: cosh (•\/[k[c) = cosh ( \/|k | a) cosh (v|^l b) ~ sinh ( y\n\a) sinh y\C\b cos 9 (1.35) • if k > 0: cos (v^c) = cos (y/Ha) cos (y/Hb) + sin (y/Ka) sin {y/nb) cos 6 (1.36) Note that these cosine laws imply that if a and b are held constant then c is a strictly increasing function of 9. This is a property which is used frequently. Remark 1.5.3. The model spaces also have the property that any two angles a and a' which split a geodesic sum to n. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 30 Just as we had comparison triangles in E 2 for triples of points in any metric space X, we can often define comparison triangles in M” for these points as well. Definition 1.5.4. Given x,y, z E X, provided d(x, y)+d(y, z)+d(z, x) < 2DK, we can find x,y,z & M 2 such that d(x,y) = d(x,y), d(y,z) = d(y,z), and d(z,x) = d(z,x). The triangle A(x,y,z) = A (x,y,z) is called a K-comparison triangle for the triple (x, y, z) and is unique up to isometry. Given any geodesic triangle A(x, y, z) C X we define a K-comparison triangle for A(x, y, z) to be any ^-comparison triangle A (x,y,z) for the triple (x,y,z), provided it exists. x Figure 1.11: K-comparison Triangle If a ^-comparison triangle A(x, y, z) = A(x, y, z) exists for x,y,z G X then the n-comparison angle between y and 2 at x is defined to be the Alexandrov angle between the geodesic segments \x,y\ and [x. z]. This angle is denoted ^x\y,z). Definition 1.5.5. A metric space X has the geodesic extension property if for every local geodesic path c : [a, b] —► X in X can be extended to a local Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 . GEODESICS AND THE MODEL SPACES 31 geodesic line c : M. —> X such that c\{a,b] = c- Since the Euclidean, Hyperbolic and Spherical spaces have the geodesic extension property, the model spaces M” have it as well. Furthermore, any local geodesic path with length less than DK will be a geodesic path. We are now prepared to state and prove Alexandrov’s Lemma; for clarity we have split it into two parts. Theorem 1.5.6 (Alexandrov’s Lemma the first half). Suppose we have four distinct points A, B, B C 6 Ml such that d(C, B) + d(C, B') +d(A, B) + d(A, B') < 2DK. Suppose further that B and B' lie on opposite sides of the locally geodesic line extending [A, C]. Consider A(A, B, C) and A (A, B'. C) as shown in Figure 1.12. Then if 7 + 7 ' > 7r the following inequality will hold: d(B, C) + d(B\ C) < d(B, A) + d{B', A) (1.37) Proof. In we can extend the geodesic segment [ B , C] to contain a point B' e Ml such that d(C, B') = d(C, B'). See Figure 1.13. Since [A, C\ splits the geodesic segment [B , B'], Remark 1.5.3 implies that 7 + Zc(A, B') = 7r. Combining this with the given condition that 7 + 7 ' > 7r, we conclude that Zc(A,B') < 7 ', which implies that d(A, B') > d(A,B') by the appropriate law of cosines. We remark that by the law of cosines d(A, B') = d(A, B) if and only if 7 ' = AC(A, B') if and only if 7 + 7 ' = tt. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES A Figure 1.12: Quadrilateral with vertices A, B , B', and C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES B A Figure 1.13: Construction of & Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 34 Our work so far implies d(A,B) + d{A,B') > d(A, B) + d(A, B') > d(B, B') where the second inequality is a consequence of the triangle inequality. By (1.3) we know that d(B, B') = d(B, C) + d(C, B') because C lies on the geodesic segment [B,B']. By construction d(C,Bf) = d(C,Br), so we conclude that d(A, B) + d(A, B') > d(B, B') = d(B, C) + d{B', C) as desired. □ Theorem 1.5.7 (Alexandrov’s Lemma the second half). Let us take the four points, A, B, B' , and C G M‘l from the previous lemma together with the requirement that 7 + 7 ' > it. Consider a geodesic triangle A (A, B , B') C such that d(A, B) = d(A,B), d{A,B') = d(A,~B'), and d(B,B') = d(B,C) + d(C, B'). Such a triangle exists by the first part of Alexandrov’s Lemma. Let C G [B, B'\ be the unique point such that d(B, C) = d(B, C) and d(B', C) = d(B', C). We define a, (3, $ as in Figure 1.14. Then we have a > a + a’, (3 > (3, fd' > (3' and d(A, C) < d(A, C). If any one of these is an equality, then they will all be equalities, which occurs if and only if 7 + 7 ' = 7r. Proof. Let B' be as in the proof of the first half of Alexandrov’s Lemma. Recall that in the proof of Theorem 1.5.6 we showed that d(A, W) = d(A, B') > d(A, B') (1.39) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1. GEODESICS AND THE MODEL SPACES 35 A —! Figure 1.14: Construction of Triangle A (A, B, B ) and remarked that equality holds if and only if 7 + 7 ' = 7r. Also, d(B, &) = d(B, C) + d{C, B') > d(B, B') (1.40) with equality if and only if C G [B, B’} if and only if 7 + 7 ' = ir. The inequality (1.39), combined with the law of cosines gives (3 > (3 while the inequality (1.40) gives a > a + a'. Applying the law of cosines to [3 > (3 we obtain d(A,C) < d(A, C). Any of these inequalities are equalities if and only if the corresponding inequality (1.39), (1.40) is an equality if and only if 7 + 7 ' = 7T. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Introduction to CAT(k) Spaces In this chapter we introduce CAT( k) spaces, which are spaces whose cur vature is bounded from above, and then discuss the structural properties of CAT(O) spaces. We explore the convexity in the context of CAT(O) spaces and examine the conditions under which flat spaces occur in a CAT(O) space. The chapter concludes with the Product Decomposition Theorem which will be important in the chapters which follow. 2.1 The CAT(/c) Inequality In the previous section we examined a spectrum of spaces of uniform curva ture. It is possible however, to have a curved space of non-uniform curvature which is still bounded above. That is, its geodesic triangles are thinner than the triangles in M™ for some k, e R. With this motivation, we introduce 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 37 CAT(k) spaces via the CAT(k,) inequality. The following definition uses the notion of a comparison point. Given a geodesic triangle A[\p,q\,[q,r],[r,p]) E X which has a comparison triangle A = A (p,q,r) E M% for p,q,r ; we say that x is a comparison point for x E \p,q] if it is the unique point x E [p,q ] such that d(p,x ) = d(p,x). See Figure 2.1. V Figure 2.1: Comparison Point Definition 2 .1 .1 . A geodesic triangle A = A([p, q\,[q,r\,[r,p]) in a metric space X is said to satisfy the CAT(k) inequality for k E R if the following hold: 1. The perimeter of A is less than 2 DK, so there exists a ^-comparison triangle A(p, q,r ) = A (p,q,r) E for the triangle A. 2. For any points x,y E A and their comparison points x,y E A (p,q,r), the following inequality holds: d(x,y) < dK(x,y) (2 .1 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 38 Thus we can now define what it means for X to be a CAT (re) metric space. • if re < 0 then X is called a CAT (re) space if it is a geodesic metric space and all of its geodesic triangles satisfy the CAT (re) inequality. • if re > 0 then X is called a CAT (re) space if it is a _DK-geodesic metric space and all of its geodesic triangles of perimeter less than 2DK satisfy the CAT(re) inequality. To define a CAT(re) space, item (2) in the definition of the CAT (re) inequal ity may be equivalently replaced by any of the following: [Bridson & Haefliger pg. 161] 1. For any pair of points x E [p, q] and y E [p, r] such that x ^ p and y ^ p, the following inequality holds: Z.[K\x,y) < A K\q,r) (2.2) Note that a re-comparison triangle exists for p, x, y provided one exists for p, q, r because d(p,x) + d(x,y) + d(y,p) < d(p,x) + d(x,q) + d(q,r) + d(r,y) + d(y,p) = d(p,q) + d(q,r) + d(r,p) < 2 Dk (2.3) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 39 2. The Alexandrov angle, Z(\p,q\, \p,r ]) satisfies the inequality: A M . M ) S 4 \q ,r ) (2.4) 3. If A (p,q,r) C M 2K is a geodesic triangle such that d(p,q ) = d(p,q), d(p,r) = d(p,r) and Zp(q,r) = Zp(q,r) then the following inequality must hold: (2.5) In the proceeding sections, we will use the terms CAT( k) inequality or CAT( k) condition to refer to any one of the inequalities just mentioned. Here are a few of the properties that CAT(fc) spaces satisfy: Remark 2 .1 .2 . Note that (2.4) implies that in a CAT(O) space the angles of a geodesic triangle sum to at most 7r. Proposition 2.1.3. Suppose X is a CAT( k) space, then X satisfies the fol lowing: 1. For k < 0, the metric space X is uniquely geodesic and otherwise, X is uniquely Z)K-geodesic. (Note that in the definition of CAT (A) we only required the existence of geodesics, not the uniqueness) 2. Every locally geodesic path in X of length at most DK is a geodesic path. 3. The balls in X of radius less than DK/2 are convex. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 40 Proof. 1. Let p, g G X be such that d(p,q) < DK and let [p, g] and [p, q]' be any two geodesic segments joining p and q. Let x G [p. q] and x' £ \p n Y be such that d(p,x) = d(p,xr). Now, consider the triangle A ([p, g]', [p, x], [x, g]) where [p, x] U[x, q] = [p, q]. \p, q\' Figure 2.2: Comparison Triangle for A([p, q}'. [p, x], [x, q\) If we take a comparison triangle A(p, q, x) in for A([p, x], [x, q], [p, g]) then since d(p,q) = d(p,x) + d(x,q ) then by Proposition 1.1.4 it must be the case that [p, x] U [x, q) is a geodesic segment. Since the model spaces Mf are uniquely geodesic for segments of length less than DK, we have Dh Therefore if x! G [p, g] is the comparison point for x' G [p, q]', then we may conclude that the equality x = x' holds; by the CAT(k) inequality, this implies that x = x' and thus [p, g] = [p, g]' as desired. 2. Consider a local geodesic path c : [0, A] —> X with length A < DK and let S = {t E [0, A] : cj[o,t] is a geodesic path}. We wish to show that S = [0, A] which will imply that c is a geodesic path. This will be done by showing that S is open, closed and non-empty in the connected Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 41 subspace [0, A] of R. Since c is a local geodesic, S contains a neighbourhood of zero and is thus non-empty. We now show that S is closed. Consider a sequence {tn} C S C [0, A] such that tn —> t as n —» oo. Note that t G [0, A] because [0, A] is closed. The convergence of { tn} implies that for all t',t" G [0,t) there exists U G {tn} such that t',t" £ [0,U\. Since by hypothesis, c| [c(o),c(ti)] is a geodesic, we know that d(c(t;),c(t")) = \t' — t"\ for any t',t" £ [0, t). Furthermore, for any t' G [0, t) have d(t!\t) = lim d(c(t'),c(ti)) = lim 11' — U\ = 11' — f| (2.6 ) Z— ►OO i— >OC thus we have verified that c|[ 0,t] is a geodesic path and therefore that S is closed. Now we will show that S is open. Let to £ S be such that 0 < to < A. There is no need to consider t 0 = A because in that case, S = [0, A] which would conclude the proof. Since c is a local geodesic, there exists 0 < e < A — to such that c|[to_£;to+e] is a geodesic path. This means that if we consider the geodesic triangle A(c(0), c(t0), c(t 0 + £)), then the side [c(0),c(to)j is the geodesic segment corresponding to c|[ 0,t0] and the side [c(t0), c(to + £)] is the geodesic segment corresponding to c|[t0;to+£] by the uniqueness of geodesic paths with length less than DK. Let A(c(0), c(t0), c(t0+e)) = A(c(0), c(t0), c(t0 + e) be a ^-comparison Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 42 c(t0) c(t0 - £ c( 0) Figure 2.3: Comparison Triangle For c(0), c(to), c(to + e) triangle for A(c(0), c(t0), c(t0 + £)) as per Figure 2.3 and suppose that it is not degenerate. Then we have the following strict inequality d(c(t0 ~ e), c(t0 + e)) < d(c(t0 - e), c(t0) + d(c(t0), c(t0 + e))) (2.7) Taken with the CAT( k) inequality, this implies that d(c(t0 - e),c(t0 + e)) < d(c(t0 - s)), c(t0 + e) < d(c(t0 - e), c(t0)) + d(c(t0), c(t0 + e)) = d(c(t0 - e), c(t0)) + d(c(t0), c(t0 + s)) = d(c(t0- e),c(t0 + £)) (2.8) which is a contradiction. Therefore A is degenerate, which by the CAT(«;) inequality, implies that A must be degenerate as well. Therefore c|[0,to+£] is a geodesic path and (to — £,to + e) C S implying that S is open. We have seen that S is non-empty, open, and closed so S = [0, A] and we conclude that c is a geodesic path. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 43 3. Let B(xq, r) be a ball in X such that r < Rf. Consider x, y G B(x0,r) and the geodesic segment [x, y] G X joining them. This geodesic segment exists because d(x, y) < d{x, x0) + d(y, x0) < Dn. We want to show that \%,y\ Q B(xq,t) which will require that for any x' G [x,y\ we have x' G B(x0,r). Consider A (x0,x,y) and let A (x0,x,y) = A (xo,x,y) G M be a re- comparison triangle for A(xo, x, y). Since balls of radius r < DK/2 in the model space are convex, the comparison point x! G \x,y\ is in the ball B(x,r). Therefore, by the CAT (re) inequality, d(xQ,x') < r which implies that x' G B(x,r) and so the ball in convex as desired. □ Any metric space X which satisfies the CAT(re) condition will satisfy the CAT(re') condition for any re' > re. In particular, the negatively curved model spaces M", where n < 0 are CAT(O) and thus n-dimensional hyperbolic space HP is also CAT(O) [Bridson & Haefliger pg. 165]. The following corollary to Proposition 2.1.3 is presented without proof, for details refer to [Bridson k, Haefliger pg. 208]. Corollary 2.1.4. A sufficient condition for a CAT(O) space X to have the geodesic extension property is that for any local geodesic path c : [a, b] —> X there exists e > 0 such that we can find a local geodesic path c : [a,b + s] —> X which extends c. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 44 Remark 2.1.5. Any CAT(O) space is simply connected. The proof of this has been omitted, for details refer to [Bridson & Haefliger pg. 161] Remark 2.1.6. As a consequence to Remark 1.2.3 any convex subset of a CAT( k) space will again be a Dre-geodesic space. Furthermore, its triangles will satisfy the CAT( k) inequality, so any convex subset of a CAT (A) space will again be a CAT( k) space. Example 2.1.7. A metric graph is a CAT(«) space provided it has no essential loops with length shorter than 2 DK. In particular, if k < 0 then a metric graph is a CAT(k) space if and only if it is a tree [Bridson Sz Haefliger pg. 167]. Example 2.1.8. A uniquely geodesic metric space T is called an R -tree if given any two geodesic segments [a, b] and [ 6 , c] such that [a, b] fl [6 , c] = {6 }, their concatenation [a,b] U [b, c] is again a geodesic segment. For example the plane R 2 endowed with the Manhattan metric as described in Example 1.2.5 is an R-tree. A metric space T is an R-tree if and only if it is a CAT(k) space for every k G I [Bridson & Haefliger pg. 167]. 2.2 Insights into Convexity Recall the definition of a convex function. Definition 2.2.1. If I is any interval in R, then a function / : I —> R, is said Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 45 to be convex provided that for all t, t' G I and s G [0,1] we have: /( ( l - s)t + st') < (1 - (2.9) For a convex function, given three points x < y < z G I , the image of y in the graph of / lies below the secant line joining x and z. See Figure 2.4. f{y) Figure 2.4: Graph of a Convex Function We can extend this notion of convexity to apply to functions which have a geodesic metric space X as their domain, instead of an interval in E. We say that a function / : X —» M is convex if, for any linearly reparameterized geodesic path c : [0,1] —> X, the function is convex as in the previous definition. Remark 2.2.2. For a geodesic metric space, it is equivalent to define a func tion f : X —> R to be convex if for every linearly reparameterized geodesic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 46 path c : [0,1] —> X and every t G [0,1], the following inequality holds f{c{t)) < (1 - t)f(c(0)) + tf{c( 1)) (2.10) This is because for any t, t' G [0,1] we can reparameterize the geodesic path c| to be from the unit interval. Similarly, we say that for a geodesic metric space X, the distance function d : X x X —» R is convex if for any two linearly reparameterized geodesic paths, c : [0,1] —* M. and c! : [0,1] —> K and any t G [0,1] the following inequality holds: d(c(t), c'(t)) < (1 - t)d(c(0), c'(0)) + td(c( 1), c'(l)) (2.11) Proposition 2.2.3. If X is a CAT(0) metric space then the distance function, d : X x X —► R is convex. Proof. Let c and d be linearly reparameterized geodesic paths in X with repa rameterization factors A and A' respectively. We first handle the case where c and d originate from the same point, so c(0) = c'(O). Let A(c(0), c(l), c'(l)) = A(c(0), c(l), c'(l)) be a comparison triangle in E 2 for the triangle A(c(0), c(l), c'(l)) G X. For any t G [0,1] and comparison points c(t), d(t) G A, the triangle A(c(0), c(t), d(t)) in E 2 is similar to A. See Figure 2.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 47 c'(l) c(0) cit) c( 1) Figure 2.5: Comparison Triangle for (c(0), c(l), c'(l)) This is because: d(c(0), c'(t)) d(c(0), E(t)) __ AT _ ^ (2. 12) d(c(0). cTO) rf(c(0),c'(1)) A' and similarly d(c(0), c(t)) d(c(0),c(t)) __ Xt _ ^ (2.13) d(c(0),c(I)) d(c(0), c(l)) A Thus, d(c(t), c'(t)) = td(c(l), c'(l)). By the CAT(O) inequality, we have d(c(t),E(t)) < d(c(t),d(t)) which gives us the desired result d(c(t), c'(t)) < td(c( 1), c'(l)) = td(c( 1), c'(l)) (2-14) For the case where c(0) ^ c'(O), we define c" : [0,1] —> AC to be the linearly reparameterized geodesic path such that c"( 0) = c(0) and c"(l) = c'(l). Now, by the above method applied to the comparison triangle formed by c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT (A) SPACES 48 Figure 2.6: Geodesic paths c, d and d and c", and the comparison triangle obtained from the reverse geodesic paths of d and c", one obtains the following inequalities: d(c(t), d'(t)) < td(c(l), c"(l)) (2.15) d(d'(t),d(t)) < (1 — f)d(c"( 0), c'(0)) (2.16) Combining these results with the triangle inequality yields our desired result, d(c(t), d(t)) < (1 - t)d(c(0), c'(0)) + td(c( 1), c'(l)) (2-17) □ The following proposition is used frequently when dealing with convex func tions. Proposition 2.2.4. If a function / : X —> M is convex and bounded on a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 49 geodesic metric space X with the geodesic extension property, then it must be constant. Proof. Let / : X —> R be a convex bounded function on a geodesic metric space X, say |/(x)| < B for every x E X. For any xx,x2 E X such that X\ ^ x2 let [xi,x2\ be a geodesic segment joining these two points and c : R —> X be a geodesic line which extends this geodesic segment. Consider the function p = /oc:R —>R and suppose that it is not constant on R. Then there exist rx < r 2 E R such that g{r\) ^ g(r2). Case 1: If g(rx) < g{r2) then for each integer n > 0 we have g(r2) < ^ g{r2 + n(r 2 - n )) =» 9(r2 + n(r2 - n)) > (n + l)g(r2) - ng(rf) (2.18) => 9(r2 + n(r2 - n)) > (n + l)(p(r2) - ^(ri)) + g(ri) which means that g gets arbitrarily large asn —> oo and g is unbounded. Case 2: If g{r2) < g{r\) then for each integer n > 0 we have 9(n) < ^ g{r2) + D^g{ri - n(r 2 - n )) g(r1 - n(r2 - ri)) > (n + l)^(ri) - ng(r2) (2.19) =>> g{rx - n(r2 - n )) > (n + 1 )(g(r1) - g(r 2)) + g(r2) so, just as in Case 1, g gets arbitrarily large asn —> oo and g is unbounded. In both cases, we have a contradition. Thus, we conclude that g must be constant which implies that f(xx) = f(x2). Since xx and x2 were arbitrary, / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 50 is constant. □ 2.3 Projection onto a Convex Subspace An orthogonal projection to a convex subset C is a map ir : X —» C defined as in the following proposition. Proposition 2.3.1. Let A be a CAT(O) metric space and C C X be a convex subset which is complete in the induced metric. Then: 1. For every x G X there is a unique point y d(x, C) = inf^gc d(x, v). We define 7r : X —> C by ir(x) = y. 2. Whenever x' € [x,7 r(x)], then the equality tt(x ') = n(x) holds. 3. For points x, y G X such that x ^ C and y E C, if y ^ 7r(x) then A(®)0r>y) ^ 71 12 - Proof. 1. First we show the existence of y = 7r(x). Since we have D = d(x, C) = inf d(x, y') (2 .20) y'&C we can find a sequence {yn} C C such that d(x, yn) —► d(x, C) as n —> oo. We will show that {yn} is a Cauchy sequence: Let £ > 0 and fix N e N such that for all n > N the inequality d(x,yn) < D + e holds. Consider n,m > N and let A(x,yn,ym) = A(x,yf,tfT) C E2 be a comparison triangle for A (x,yn,ym) C X. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 51 Consider the ball B(x, D + e) and the open ball B(x, D) and let A = B(x, D + e) — B(x, D) be the annular region between these balls. See Figure 2.7. Clearly yn and yT, are contained within the annular region A and by the convexity of balls in E2, the geodesic segment [y^,yln\ C E2 is contained in the larger ball, B(x, D + e). Vm Figure 2.7: Comparison Points Relative to the Annular Region Furthermore, the geodesic segment [l/F, yT\ C E2 must be contained within the annular region A. Indeed, if it were not then there would exist z G such that d(x,z) < D. This would imply that the point z G |Un-Urn] f°r which ~z is a comparison point satisfies d(x,z) < d(x,z) < D (2-21) where the first inequality is a consequence of the CAT(O) condition. Since Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 52 z e C, this contradicts our choice of D = d(x, C) and so we conclude that [|is completely contained within the annular region A. A straightforward exercise in Euclidean geometry shows that a geo desic path completely contained in this annular region A can have length at most 2\f2eD + e2. Therefore d{yn,ym) = < 2y/2eD + e2, and thus the sequence {yn} is Cauchy. Since C is complete in the induced metric, there exists y G C such that {yn} —> y as n —> oo, giving d(x,y) = d(x,C). To show the uniqueness of y, consider any other sequence {?/,',} with the above properties, and its limit y'. The sequence {y"} obtained by alternating terms from {yn} and {y'n} would still be Cauchy by the above arguments, so y = y' and thus y = n(x) is unique and has the desired properties. 2. To show that n(x) = for any x1 G [7r(a;),x], we suppose ir(x) ^ 7t(x>) and derive a contradition. Since 7r(x') is the unique closest point to x' in C, d(x',7r(x/))< d(x',7r(x)). This implies however, that we have d(x, 7r(x)) = d(x,x') + d(x',7r(x)) > d(x, x') + d(x', 7r(x')) (2.22) > d(x,7r(x')) which contradicts our choice of tt(x). Therefore ir(x) = ir(x') and we are done. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 53 3. Suppose to the contrary that ZT(x)(x,y) < tt/2. By the definition of the Alexandrov angle, there exists points x' G \n(x),x\ and y' G [n(x),y\ such that the comparison angle Z 7r(x)(x/, y') is less than 7t / 2. Consider a comparison triangle A( 7t(x), x', y') = A(7r(x),x', y') C E2 for the triangle A(7r(x), x\ y'). We can find a point p G [ir(x),yl] sufficiently close to ir(x) to make the angle Zx/(p, n(x)) acute. See Figure 2.8. Since Z-^^(x',y>) is also acute, Zp(ir(x),x!) is obtuse. x' :/ V V Figure 2.8: Location of p G [7r(x),y'] Thus we have d(p,x') < d(ir(x),x'). Taken with the CAT(O) inequality the corresponding point p G [7v{x),y'} for which p is a comparison point has d(p,x') < d(p,x r) < d( tt(x),xi) (2.23) = d(ir(x),x') Since C is convex we know that [7r(x), y\ C C which gives p G C and thus 7r(x) 7^ 7r(x'). This contradicts (2) which asserts that 7r(x') = n(x). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 54 Therefore we have derived a contradiction and it must be the case that dir(x)(x,y) > t t / 2. □ Corollary 2.3.2. If C is a complete convex subset of a CAT(O) metric space X then dc ■ X —> E given by x i—>• d(x, C) is a convex function. Proof. Let it be the projection of X onto C. Let c : [0,1] -> X be a linearly pa rameterized geodesic segment, and d : [0, 1] —> C be the linearly reparameter ized geodesic path whose corresponding geodesic segment is [ 7r(c(0)),7r(c(l))]. Since C is convex, we have the inclusion [ 7r(c(0)), 7t(c(1))] C C. Now, for any t E [0,1] we have dc{c(t)) = d(c(t),C) < d(c(t),d(t)) since d(t) € C. Furthermore, the convexity of the distance function gives us d(c(t), d(t)) < (1 - t)d{c(0), c'(0)) + td(c( 1), c'(l)) (2.24) Since d(c(0), c'(O)) = dc(c(0)) and d(c(l), c'(l)) = dc(c( 1)) we may con- elude that dc(c(t)) < (1 - t)dc(c(0 )) +tdc(c(t)) (2.25) which implies that dc is a convex function. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 55 2.4 The Centre of a Bounded Set In this section, we introduce the centre and radius of a bounded set. The existence and uniqueness of such centres in complete CAT(O) spaces will be used later to prove the Bruhat-Tits Fixed Point Theorem for isometries of complete CAT(O) metric spaces. This approach via centres is due to Serre. Refer to [Brown], [Serre], [Bruhat & Tits], Definition 2.4.1. If Y is a bounded subset in a metric space X then the radius of Y, denoted ry, is given by rY = inf{r G R | Y C B(x, r) some x G X} (2.26) Proposition 2.4.2. Let A be a complete CAT(k) metric space. If Y C X is a bounded subset of radius rY < DK/2 then there exists a unique point cY G X such that Y C B(cY,rY). The point cY is called the centre of Y. Proof. The definition of rY implies that there exist sequences of points {xn} C X with a corresponding sequence {rn} C R satisfying rn —> rY and Y C B(xn, rn) for each n G N. Our goal is to show that {xn} is a Cauchy sequence. Let Q G M% be a fixed basepoint. For e > 0 we may choose R G (rY, DK/2) and R' < rY such that any geodesic segment contained within the annular region A = B(Q, R)\B(12, R') has length less than e/2 (cf. the proof of Propo sition 2.3.1 (1)). Since rn —> rY and rn > rY for all n, there exists N > 0 such that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 56 for all n > N we have r n E [rY,R\■ Let m , n > N. For each y E Y the geodesic triangle Ay = A { y ,x n, x m )is unique because d ( y , x n) < rn < DK/2 and d ( y , x m) < rm < DK/2, giving Ay a perimeter of less than 2DK. Consider a comparison triangle Ay = A(0,x^,x^) C M/ for the triangle A(y,xn:xm). Let m be the midpoint of the geodesic segment [xn, x.m] and suppose its comparison point m E [xT xT] lies within the ball B(Q, R') C M%. By the CAT(^) inequality this gives d(m,y) < d(Q,m) < R! < ry- Thus, if it were the case that for every y EY, the comparison triangle Ay has the midpoint m of [xy, Yv] contained within B(0, R') then this would imply that d(m,y) < R! for every y E Y so Y C B(m,R') C X which contradicts our choice of R! < ry. This implies that for some y E Y, the midpoint rn C an] E Ay is contained within the annular region A. We now show that one of [x^, m] or [m, xT] lies within the annular region A. Suppose to the contrary that there exists x' E and x" E [m,x^\ such that x',x” E B(Q,R!). Then by the convexity of balls in M% this implies that the geodesic segment [x',x") and consequently fn are contained within B{Il, R'). This contradicts our choice of having fn E A. Therefore, we conclude that either d(xn, m) = d(x^,m) < e/2 or d(m,xm) = d(rri,x^) < e/2. Since m is the mid-point of [xn,xm], we have d(xn, xm) = 2d(xn, m) = 2d(m, xm) < e and the sequence is Cauchy. Since X is complete, lim{.xn} = cy must exist and its uniqueness is proved in the same manner as the uniqueness of the projection in the proof of Propo sition 2.3.1. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 57 Example 2.4.3. The centre of a bounded subset of a CAT(O) space need not be contained in that set. For example in E 2 the set {(0, 0), (0,1)} is bounded and has centre (0, |) and the set {x G E2 | d(x, (0,0)) = 1} has centre (0,0). The following lemma will be used to prove the Bruhat-Tits Fixed Point Theorem [Bruhat & Tits]. Lemma 2.4.4. Let X be a complete CAT(0) space and suppose that Y is a bounded subset of X with centre cY. If / : X —► X is an isometry such that f(Y) = Y then f(cY) = cy. Proof. Since cY is the centre of Y we have Y C B(cY,rY) where rY is the radius of Y. Since / acts by isometries, we have f(Y) C B(f(cY),rY), but Y = f(Y) so Y C B(f(cY),rY). By the uniqueness of centres, this implies that f(cY) = cY as desired. □ 2.5 Flat Subspaces and the Product Decom position Theorem The goal of this section is to prove a theorem that will allow us to split the subspace consisting of all the geodesic lines parallel to a given geodesic line c in a CAT(0) space X into the product of a convex subspace of X and M. This will require us to examine under which conditions subspaces of a CAT(0) are isometric to flat Euclidean space. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 58 Definition 2.5.1. Let X be a geodesic metric space and let A be a subset of X. The convex hull of A, denoted C(A), is the intersection of all convex subspaces of X which contain A. This will be the smallest convex subset in X containing A. The following proposition shows how one can end up with flat triangles in a non-positively curved metric space. We will later use this to describe flat quadrilaterals in a CAT(O) space and eventually flat strips. Proposition 2.5.2 (Flat Triangles). Let X be a CAT(O) metric space and consider a geodesic triangle, A = A (p,q,qr) C X. If the (Alexandrov) angle Zp(q,q') is equal to its comparison angle Zp(q,cf) then the convex hull C'(A) of A C X is isometric to the convex hull C(A) where A is any comparison triangle in E 2 for A(p, q, q'). Proof. Let A = A (p, q, q') C X be any geodesic triangle, such that Zp(q. q') = ^piQiQ1)- Let A(p,q,q') = A (p,q,q!) be a comparison triangle for A (p,q,qr). The first step of the proof is to show that the equality d(p, r) — d(p, r) holds for any point r e [q,q']- Fix r G [q, q'] such that r q and r f q'. Considering the geodesic triangles A' = A (p,q,r) and A" = A(p, r, q'), we can find points p,q,q',r € E2 such that A' = A (p,q,r) and A" = A(p,r,q') are comparison triangles for A' and A" respectively. Furthermore, these points may be selected such that q and q' lie on opposite sides of the line which runs through p and f. See Figure 2.9. Let 7 = Zf(p,q) and f' = Z f(p,q1) as in Figure 2.9. Since 7 = Zr(p,q) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 59 Q P Figure 2.9: Arrangement of A' and A" in E 2 and 7 ' = Zr(p, q') split a geodesic segment, we have 7 + 7 ' > 7r by Proposition 1.3.6. By the CAT(O) inequality we get 7 > 7 and 7 ' > 7 ' which implies 7 + 7 ' > 7r. We can now apply the second part of Alexandrov’s Lemma 1.5.7 to p , q, ^p(q,q') < ^p(q,r) + ^p(r,q') < Zp{q,r) + Ap(q',f) (2 -27) < where the first inequality is a consequence of Proposition 1.3.6, the second inequality is a consequence of the CAT(O) condition, and the third inequality we just proved. Our initial assumption that Zp(g, q') = Zp(q,q') implies that equality holds Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 60 everywhere in (2.27); in particular Zp(q, r)+Zp(q', r) = Zp(q,q'). At this point, the equality condition in the second part of Alexandrov’s Lemma 1.5.7 gives d(p,r) = d(p,r), but d(p,r) = d(p,r) so d(p,r) = d(p,r) as desired. To complete the proof, we define the map, j : C(A) — > X as follows: For every r G [q, q'\ we isometrically send the geodesic segment \p, r] to \p, r] fixing j(p) = p and j(f) = r. This is only possible because d(p,r ) = d(p,r)- Note that for any x G C(A) — {p} there is a unique point, f G \q,q'] such that x G \p, r], so j is well-defined. To show that j is an isometry, let x, x' G C'(A) be any two points such that x, x' ^ p. Let r, r' G [q, q'] be such that x G \p,f\ and x' G [p, I1]. Without loss of generality, we may assume f G [q, f']. Q P Q Figure 2.10: Location of x and x' in A (p, q, q') Let x = j(x), x' = j(x'), r = j(r), and r' = j(rf). Since d(p,r) = d(jj,r), the triangle A(p, q, f) is a comparison triangle in E 2 for A (p,q,r). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 61 Similarly, A (p ,r,r') is a comparison triangle for A(p, r, r') and A (p,r',q') is a comparison triangle for A(p, r', q'). Thus, by the CAT(O) inequality, we obtain <5i = Zp(q,r) < Z¥(q,r) = fq. Similarly, S2 = Zp(r,r') < Z¥(r,r') = S2 and <$3 = Zp(r', q') < Zp(r',q') = S3 . Combining these results gives: Ap(q, q') — <5i + S2 + S3 > + S2 + S3 (2.28) = Ap(q,r) + Zp(r,r') + Zp(r',q') > Zp(q,q') (Proposition 1.3.6) By hypothesis, we have Zp(q,q') = Zp(q,q') = Zp(q,q'). Thus equality holds everywhere in the above expression, which implies that S 2 = <52- Let us fix a comparison triangle A(p,x,x') for A(p,x,x'). We wish to show that it is congruent to A (p,x,x'). Since d(p,x) = d(p,x ) and d(p,x') = d(p,xr) it suffices to show that Zp(x, x') = Zp(x,x'). We have the following inequalities: S2 = Zp(r, r') = Zp(x,x') (since x E [p, r] and x' E [p, r']) < Zp(x,x') (by the CAT(0) condition) (2.29) < Zp(r,r') (by the CAT(0) condition) = 52 We know that S2 = S2 so in (2.29) equality holds everywhere; in particular Zp(x,x') = Zp(r,r') which implies that the comparison triangle A (p,x,x') is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 62 congruent to A (p,x,x'). So we deduce that d(x,x') = d(x,x') = d(j(x), j(x')) and we have an isometry as desired. All that remains to be shown is that C(A) = j(C(A)). Since we know that A C j(C(A)) and j(C(A)) is convex (as the isometric image of a convex set), we have C(A) C j(C(A)). Since j is injective, we define a left inverse j -1 : j(C(A)) — >■ C'(A). By using the same argument as above, we see that ^ (C ^ A )) D C(A) so C(A) I) j(C(A)). Thus, we may conclude that C(A) is isometric to C'(A) which completes the proof. □ We have seen that one may find flat triangles in a CAT(O) space. The following proposition is similar, but addresses flat quadrilaterals. Proposition 2.5.3 (Flat Quadrilaterals). Let p, q, r , s be four points in a CAT(O) space A such that Zp(q, s) + Zs(p,r) + Zr(s,q) + Zq(r,p) > 2iv. Then equality holds in the previous inequality and the convex hull of the four points is isometric to the convex hull of a convex quadrilateral in E2. Proof. Let Ai = A(p,q,s) and A 2 = A (r,q,s) be geodesic triangles. We can pick comparison triangles Ai and A 2 for Ai and A 2 respectively, such that the comparison points for q and s in each triangle coincide and the comparison points for p and r lie on opposite sides of the segment (g, s]. See Figure 2.11. Let a, 7 , f31: /?2, 5i, and 8 % be defined as in Figure 2.11. If we let a = Zp(q,s), (3 = Zq(p,r ), 7 = Zr(q,s), and 8 = Zs(r,p ) then the CAT(O) inequality gives us a < a and 7 < 7 . Furthermore, by Proposition 1.3.6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES P •X! Q r Figure 2.11: Placement of Ai and A 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 64 we have 8 = Zs(r,p) < Zs(r,q) + Zs(q,p). By the CAT(O) inequality again, q) < §2 and As(q,p ) < <5i so 8 < 81 + <52- Similarly, we have (3 < Si P P2• This implies that ex P Pi P P2 P P 81 P S2 PcxpppSp'y > 2% (2.30) The sum of the angles of a Euclidean quadrilateral is equal to 27t however, so equality holds in the above expression. In particular, a = a, 7 = 7 , 5i + 82 = 8 , and Pl P P2 = P- By the previous proposition, this implies that C'(Ai) and C(A 2) are iso metric to C'(Ai) and C(A2) respectively. Thus, we know that there exist isometries, j\ : C^Ax) — > C^Ai) and j2 : C{A2) — > C(A2). Additionally Pi + P2 = P = ^-qiPir) Ci 7T and similarly <5i + 82 < 7r which implies that the convex hull, C({p,q,r,s}) is equal to the union of C'(Ai) and C'(A2). We must show that there is an isometry mapping C({p, q , r, s}) to C({p, q, r, s Define j : C({p,q,f,s}) — > A by j\C(Si) = h and j\C(K2) = h- Since C'(Ai) fl C(A2) = [ well-defined. All we must now show is that j is an isometry. For all Xi,X2 E C'(Ai) we have d(j(xl),j(x^)) = d(ji(xi),ji(xt)) = d(xl,xi) (2.31) The same holds for points in C(A2). Thus, in order to show that j is an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 65 isometry, we must verify that j preserves the distance between points X\ E C(Ai) and x 2 E C(A 2). Let Xi E C'(Ai) and x 2 E C(A2) with images j(x7) = Xi and j(xi) = x2 respectively, such that xi, xi 7^ d(xi,x2) = d(xi,xi) it is sufficient to show Z 5(xi,x2) = Zq(xi, xi). Indeed, suppose it is true. Then by the CAT(O) inequality Zg(xi,x2) > Zq(r1,r2) = Zq(xi,xi). So if A(g, xi,x2) is a comparison triangle for A(g,xx,x2) then d(x 1, x2) = d(x 1, x2) > d(x 1, x2) by the law of cosines. To see the opposite in equality, if we let y E [xi, xi] be the unique point which intersects the geodesic segment \q,s] and y = j(y), then d(x i , x 2) < d{xi,y) +d(y,x2) = d(xi,y)+d(y,xi) (2-32) = d(TT,xi) which gives d(xi,x2) — d{x\,xi) as desired. So let us prove that Zg(xi,x2) = Zq(xi, xi). Let ji = Zg(xi,x2) and /I = Zg(xl,xi). Note that since ji and j2 are isometries, we have P < Zq(p,Xi) + fi + Zq(x2 ,r) < Zq(p, x x) + (Zg(xi, s) + Z,(s,x 2)) + Z g (x 2, r) = z5(p, Xi) + (Zg (xi, s) + Z5(s, xi)) + Zq (xi, r ) = P i + (32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 66 But Pi + 02 = 0, so equality holds everywhere in the above expression. There fore, Ag(p,xi) + p +Z.q(x 2 ,r) = Ag(p,xl) + (Zg(xl,s) + Zg(s,x^)) + Zg(x^,r) = Eq(p, x l ) + f l + Zq(xi, r) = Zq(p,Xi) +JI+ Zq(x 2,r) (2.34) so we deduce that fi = JI as desired. By the same method as in the proof of Proposition 2.5.2 we can show that the image of j is the convex hull of the quadrilateral and formed by p, q, r, s and thus the proof is complete. □ Definition 2.5.4. Let X be a geodesic metric space, and let c : M. —> X, d : M. —> X be two geodesic lines in X. We say that the lines c and d are asymptotic geodesic lines if there exists a non-negative constant K such that the inequality d(c(t), c'(t)) < K holds for all i E R. We say that c and d are parallel if d(c(t),d(t)) is constant. Just as in Euclidean space En, in a CAT(O) space, any two asymptotic geodesic lines are in fact parallel. Furthermore, their convex hull is isometric to a strip in E 2 as outlined in the following proposition. Proposition 2.5.5 (The Flat Strip Theorem). Let c : R —>■ X, d : M —> X be two geodesic lines in a CAT(O) space X. If c and d are asymptotic, then C(c(M) U c'(M)) is isometric to a flat strip R x [0, D\ C E2 for some D > 0. Proof. If we consider the subspaces c(M) C X and c'(M) C X then they are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 67 clearly complete and convex, as the isometric images of the complete and convex metric space R. Let 7r : X —> c(R) and •k' : X —> c'(R) be the projection maps of X onto c(R) and c'(R) respectively, as defined in the previous section. First observe that if a, b E R are fixed then d(c(t + a), d(t + b)) < d(c(t + a), c(t)) + d(c(t), c'(t)) + d(c'(t), c'{t + b)) < a + K + b (2.35) since d(c(t),d(t)) < K for every t G R. Hence, d(c(t + a)),c(t + b)) is bounded for all t G R so without loss of generality we may reparameterize d so that 7r(c'(0)) =c(0). Now, by Proposition 2.2.3 the function t d(c(t),d(t)) is convex and non negative. Furthermore, it is bounded by assumption. Therefore by Proposition 2.2.4, it is constant, that is for some D £ R, we have d(c(t), d(t)) = D for all i G R. Similarly, by (2.35), we see that the function t h->• d(c(t + a),d(t)) is constant for all a G R. Because 7r(c'(0)) = c(0), we have d(d(t),c(t + a)) = d(d(0 ), c(a)) > d(c'(0),c(0)) (2.36) = d(d(t),c(t)) so 7r (d(t)) = c(t) for all t 6 R. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 68 Conversely, for any t 6 M, we have d{c{0 ),d{t)) = d{c(-t),d( 0)) > d(c(o),dm so 7r'(c(0)) = c'(O) where 7r' is the projection of X onto c'(R). By the same method as in (2.36), we can show that 7r'(c(t)) = d{t) for all t G M. Let us define a map j : R x [0, D] — > X such that j sends (t, s) to the point on [c(t),d(t)\ a distance s from c(f). We will show that j is in fact an isometry. Let t < t' G R and consider the quadrilateral in X with vertices, c(t), c(t,),c'(t'), and c'{t). By Proposition 2.3.1 (3) we have the following ZcW(c(t'),c'(£)) > tt/2 Ac(t')(c(t),c,(t')) > tt/2 ^c'(t'){c(t'),c'{t)) > tt/2 z c'(t)(c'(t'),c(t)) > vr/2 By the Flat Quadrilaterals Theorem 2.5.3, this implies that the convex hull of {c(t), c(t'), c'(t)} is isometric to the Euclidean quadrilateral, [0, D\ x [t, t']. In particular, we have d((t,s), (t’,s')) = d(j(t, s), j(t', s')) as desired. So j is an isometry and by the same argument as in the previous two theorems, its image is the convex hull C(c(M) U c'(R)) as desired. □ This property shows that asymptotic lines are parallel. The big theorem of the section now requires only the following lemma as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 69 preparation: Lemma 2.5.6.Let ci, 02,03 : R —»• X be three geodesis lines in a metric space X. Suppose that the union of the images of each pair of these lines is isometric to the union of two parallel lines in E2. Let Pij be the map which assigns to each point in Cj(R) the unique closest point in q(R). Then the equality 3 o p32 °P 2,i = Pi,i holds, where piti is the identity map on ci(R). Proof. Since each pair c%, cj of lines have images which are isometric to a pair of parallel lines in E2, the projection pl,J is an isometry. Thus the composition, Pi,3 0 P3,2 0 P2,i is an isometry. The only isometries of M are reflections and translations; we claim that p ij3 o p 3 2 o p2 ] must be a translation. To derive a contradiction, suppose pi ;3 o p3 2 o p2,1 is a reflection about a point Ci (a) E R, so we have p 1)3 o p 3 2 0 P2,i(ci(a + x)) = C\{a — x). We may reparameterize Ci so that Ci(a) becomes Ci(0) and then reparameterize c2 so that P2,i(ci(0)) = c2(0). Finally, we reparameterize c 3 so that p3,2(^ ( 0)) = c3(0) Thus, pij3 op3)2 op 2,i{ci{x)) = Ci(-x) for all x G R. Set di = d(ci(0),c2(0)) d2 = d(c 2(0),c3(0)) (2.39) d3 = d(c 3(0),ci(0)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 70 If we consider a real number i £ R such that x > l/2(di + d2 + d3), then d(c1(x),p2,i(ci{x))) + d(p 2,i(ci(x)),p3i2 op2A(c1(x))) +d(p3,2 °P2,l(ci{x)),Pit3 Op32 o p2,l(Cl(x))) > d(ci(x),pi>3 op 3>2 op 2'1(ci(x))) (2-40) = 2x > di + d2 + d3 This implies that one of the following will hold: di < d(c(x),p 2,i{c(x))) d2 < d(p2A(c(x)),p 3t2 op2A(c(x))) (2.41) d3 < d(P3,2 0P2,i(c(a;)),pij3 op3j 2 op2il(c(x))) which contradicts the assumption that each pair of geodesic lines has images which are isometric to a pair of parallel lines in E2. Thus, we conclude that Pi,3 ° P3,2 0 P2,i must be a translation by a constant fe e l. This gives p \$ o P3,2 °P2,i(ci(t)) = Ci(t + b) for all fee R. By reparameterizing c2 such that p2:i [c\ (0)) = 02(0) and then reparameter- izing c3 such that p3,2(c2(0)) = c3(0) we get that p1)3 op3>2 op2,i(ci(0)) = Ci(b). Now, let ai = d(ci(E), c2(M)), a2 = d(c2(R), c3(R)), a 3 = d(c3(K),ci(M)). Since we assumed that the union of any two of the given lines is isometric to two parallel lines in E2, we have d(ci(t), c2(t + s)) = a\ + s2 for any feel. Similarly, d(c2(t), c3(t+s)) = y/a% + s2 and d(c3(t), Ci(s+t)) = yja\ + (s — b)2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 71 holds for any t € R. Thus, for all s G R we have d(ci( 0), ci((ai + a2 + Us)s + 6 )) < d(ci( 0), c2(ais)) + d(c 2(ais), c3((ai + a2)s)) +d(cs((oi + ci2)s), Ci((ai + a2 + a3)s + b)) (2.42) = + y/af+^p| + = (<3i + a2 + a3)\/l + s2 But Ci is a geodesic path, so d(ci(0), Ci((ai+a2+ a3)s + 6)) = |(ai+ a2 + a3)s+&| which tells us that | (oq + a2 + a3)s + 6| < (ai + a2 + a3)\/l + s2 for all sGR. If we consider the case where b > 0 then for every s > 0 we have {y/ s2 + 1 — 'S)(ui + a2 + a3) > b > 0 (2.43) and as s —► 00 we have y/ s2 + 1 — s —> 0 and thus 6 = 0. If, on the other hand, b < 0 then for every s < 0 we have 0 > b > — (cii + a2 + a3)(\/l + s2 + s) (2.44) and as s —> — 00we have s + y/s2 + 1 —> 0 so b = 0 and we have the equality Pi,3 0 P3,2 0 P2,i = Pi,i as desired. □ We now proceed with the main theorem of this section. It will be used later to find axes of hyperbolic isometries. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 72 Theorem 2.5.7 (Product Decomposition Theorem). Let X be a CAT(O) space and c : R -> X be a geodesic line. Then: 1. The union of the collection of all images of geodesic lines d : R —► X which are asymptotic to c forms a convex subspace Xc of X and hence is CAT(O). 2. Let 7T : X —> c(M) be the orthogonal projection of X onto c(R) and let p : Xc —»■ c(R) be its restriction to Xc. Then X® = p_1(c(0)) is a convex subset of X and Xc is canonically isometric to the product x R. Proof. 1. We wish to show that for any two points X\, ,x2 in XC) the geo desic segment [aq, x2\ is contained in Xc. First observe that the triangle inequality implies that being asymptotic is an equivalence relation on the set of geodesic lines. Since Xc is the collection of geodesic lines asymptotic to c, there exist geodesic lines c\ and c 2 such that aq 6 cq(R) and x2 £ c2(R). Since c2 are asymptotic, the Flat Strip Theorem 2.5.5 tells us that the convex hull, C(ci(R) U c2(R)) is isometric to a flat strip, R x [0, D], For any point (a, b) £ R x [0,D], the geodesic line {(p,b) : p £ R} is asymptotic to {{p, 0) : p £ R}. Thus, any point in C(ci(R) U c 2(R)) is on a geodesic line which is asymptotic to implies that [aq,£2] C C(ci(R) U 2c(R)) C Xc as desired. Furthermore, a convex subset of a CAT(O) space is CAT(O) so Xc is a CAT(O) metric space by Remark 2.1.6. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 73 2. For every x G X® there exists a unique geodesic line cx : A X asymptotic to c such that x = cx(0) and p(cx0)) ( = c(0). Moreover, for every x' G X c there exists x G X ° such that x' G cx(R). We now define a map j : X® x R —> X c by j ( x , t) = cx (t). By definition j is surjective, so we must show that j is an isometry. Let (x i,ti) and (X2 H2 ) be points in x R. Note that d((xi,ti), (x2 ,h)) = y /d (x 1, x2 ) 2 + (t\ — t 2)2. Let us compute d(cXl( t i ) ,cX2(t2))- We know that C(cXl(R) U cX2(R)) is isometric to a flat strip, so we obtain Figure 2.12, where p X2jXl is the projection of cXl onto cX2. -X\ -X\ Figure 2.12: Calculating the distance between cXl(ii) and Cx2 {h) Thus we have d(cXl(h), cX2 (t2)) (2.45) = yjd{cXl (ti), Px2,x\ ipx\ (^1)))^ + d{px2)Xl (cXl (ti)), cX2 (^2))^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2. INTRODUCTION TO CAT(k) SPACES 74 If we can show that pX2iXl(xi) = x2 then since px 2:Xl is an isome try it will follow that pX2tXl(cXl{t)) = cX2(t ) for all t and in particular Px2,x1(cXl{ti)) = Cx2(ii). Then (2.45) will become ^d{xu x2) + (t2 - C)2. Let po,x2 be the projection of cX2(R) onto c(R) and let p X2:0 be the projection of c(R) onto cXl(R). By Lemma 2.5.6 we know that p Xl),o 0 Po,x2 °Px2,x1 is the identity map on cXl (R). Thus pXu 0 o p 0x2 o p X2tXl (xi) = X\. But pXi ,o and p 0jX2 are isometries, so they have inverses po,Xl and p X2p giving p X2,Xl(xi) = pX2,o °Po,x 1(%i)- Furthermore, since X\ G X ° we have Po,Xl(xi) = c(0); similarly, p 0,X2(x 2) = c(0) which gives p X2,o{c(0)) = x 2 and thus p X2tXl(xi) = x 2. This completes the proof that j is an isometry and that Xc is isometric to X c° x R. The convexity of now follows: if x,y G X® and [x, y] C X is a geodesic segment, then [x,y\ C Xc and so j~l{[x,y\) = [(rr, 0). (y, 0)] C X°xR is a geodesic segment. Clearly, for any (z,t) £ [0*5 0), (y, 0)] we have t — 0. This means that [(rc, 0), (y, 0)] is a geodesic segment in X° x {0} from (x, 0) to (y, 0). Hence [x,y] = j(X® x {0}) = Thus X° is convex as desired. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Group Actions In this chapter we will introduce the basic definitions of group actions as well as discuss some properties of groups which act properly and cocompactly by isometries on a metric space. The main results of this chapter include Macbeath’s theorem [Macbeath], that a group is finitely presented if and only if it acts properly and cocompactly on a simply-connected geodesic metric space and the Bruhat-Tits Fixed Point Theorem. The latter is a generalization of Cartan’s theorem from the theory of Lie groups [Brown], The former will be used in Chapter 4 to solve the word problem for such groups. 3.1 Basics of Group Actions We begin by describing actions of a group on an arbitrary metric space, paying specific attention to those actions which are proper and cocompact. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 76 First though, we need a few definitions. Definition 3.1.1. Given a topological space X and a group T we define an action of T on X to be a homomorphism $ : T —► Homeo(A). We write 7 • x in place of ($( 7 ))(a;). Definition 3.1.2. Group actions with the following properties are used ex tensively throughout the proceeding sections: 1. A group action $ : T —> Homeo(A) is called faithful if ker $ = {1}. 2. The action, $ : T —> Homeo(Af) is said to be free if 7 • x x for each 7 G T — {1} and x G X. 3. If there exists a compact subset if C l such that F ■ K = X then $ is said to be cocompact. 4. The stabilizer of a point x G X is the set = {7 G T | 7 ■ x = x}. This set is a subgroup of Y called the isotropy subgroup of x. 5. Given a point x E X the orbit of x is the set T • x = {7 • x | 7 G T}. We shall be using the following definition for a group acting on a metric space. Definition 3.1.3. An isometric action of a group T on a metric space X is a group action of $ is an isometry of X, that is $(F) C Isom(A). We say that the action is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 77 proper , or that T acts properly on X, if for every x G X there exists r > 0 such that the set {7 G T | 7 • B(x, r) fl B(x, r) 7 ^ 0} is finite. Remark 3.1.4. If X is a proper metric space and T acts on X by isometries then Definition 3.1.3 is equivalent to the more general property that for every compact set K C X the set {7 G F | 7 • K n K 7 ^ 0} is finite. Note that a metric space X is said to be proper if every closed ball B(x. r) forms a compact subspace of X. In fact, every compact set K has an open neighbourhood U such that the set {7 G T | 7 • U fl U 7 ^ 0} is finite Proof. Let us fix a compact subset K of X; by Definition 3.1.3, for every x G K we may find rx > 0 such that the following set is finite { 7 £ r I 7 • B(x,r,) n B(x,rx) ^ 0} (3,1) Since the set of balls Bir,1-), x G K forms an open cover for the com pact set K. there exists a finite subset x\. x-2 -.... xn G K such that the balls B(xi,Ti/2) cover K, where 77 = rXi. Let U = U"=1 B(xi, 77 / 2) and note that U is an open neighbourhood of K. To derive a contradiction we will suppose that the following set is infinite A = {7Gr|7-(7nc/^0} (3.2) For each 7 G A we may find Xi, Xj such that 7 • B(xi, 77 / 2) fl B(xj,rj/2) 7^ 0. Since A is infinite, this implies that we can find a pair 07 , x3 such that the set Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 78 B = {7 e A | 7 • B(xi,ri/2) fl B(xj,rj/2) 7 ^ 0} is infinite. W ithout loss of generality, we will assume that r* < 77 . Let Si, s2 £ B be two distinct elements. We know that si • B{xi, 77 / 2) fl B(xj, rj/2) = B(si • Xj, 77 / 2) fl £?(xj,77 / 2) 7^ 0 and that s 2 • B(xi, 77 / 2) fl B(xj, rj/2) = B(s2 • Xj, Xj/2) fl B(xj, rj/2) ^ 0 which tells us that d(si • Xi,Xj) < ri/2 + Vj/2 < Tj (3.3) and similarly d(s2 ■ Xi, Xj) < rt/2 + Tj/2 < rj (3.4) From this we see that both Si • X* and s 2 • x* are contained within B(xj.rj) which tells us that S2-W1 ‘ (si ' xi) = s2 • x^ must be contained in both the ball (s2s^) ■ B(xj,Vj) and the ball B(xj,rj). In particular, we have (S 2SI"1) • B(xj, rj)(lB(xj, rj) 7 ^ 0. Since we have an infinite number of distinct elements Si, Sj G B, this is implies that the set {7 e T | 'y-B(xj,rj)r\B(xj,rj)} is infinite, which is a contradiction. This completes the proof. □ Proposition 3.1.5. Let T act properly and by isometries on a metric space X. Then the following hold: 1. For every there exists e > 0 such that 7 £ r.r whenever we have 7 • B(x, e) fl B(x, e) 7 ^ 0. 2. The space X/V of T-orbits of X is a metric space with the distance function: d(T ■ x,T ■ y) = inf d{^-x,^'-y) (3.5) 7 ,7 'e r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 79 3. Let Y be a subspace of X. If Y is invariant under the action of a subgroup H C T then H acts properly on Y. 4. Suppose the action of T on X is also cocompact. Then for each x E X, the isotropy subgroup is finite and the set {T^ | x E X} is the union of finitely many conjugacy classes of finite subgroups. That is, there are only finitely many conjugacy classes of isotropy subgroups. Proof. 1. Given x E X, since T acts properly on X, we can find r > 0 such that the following set is finite S = {7 G T | 7 • B(x, r) fl B(x, r) ^ 0} (3.6) Now let T = {7 1 ,7 2 , • • •, 7n} Q S be the finite subset of S consisting precisely of those members which do not stabilize x. Suppose we let £ be given by the following equation: e = ^ min{r, d(x, 71 • x),d(x, 72 ■ x),..., d(x, • a?)} (3.7) We claim that e is the desired value. Suppose 7 • B(x, e) fl B(x, e) 0; since £ < r, we have 7 E S. Let x' E B(x, e) fl 7 _1 • x' E B(x, e).Since T acts by isometries, we have x' E B(-y ■ x, e). So, d(x, 7 • x) < d(x, x') + d(x', 7 • x) < £ + £ = 2e (3.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 80 Since d(x, 7 • x) < 2e, we may conclude that 7 cannot be in the set T so this implies 7 G Tx as desired. 2. First we show that d is a pseudometric. Clearly d is symmetric. Futhermore, since T acts by isometries, we have d(T-x, T-y) = d(x,T-y) for any x, y £ X. To establish the triangle inequality, consider x,y,z £ X. For every £ > 0 we can find x' £ T ■ x and y' £ T ■ y such that d(x', y') < d(T ■ x, P ■ y) + e/2 (3.9) We can then find z' £ T ■ z such that d(y', z') < d(T -y,T ■ z) + e/2. This gives d(T ■ x, T • z) < d(xl,z') < d(x',y') + d(y',z') (3.10) < d (r • x, r • y) + d(r • y, r • z) + e Since this holds for any e > 0 we have the desired result: for any points x,y,z £ X, we have d(r ■ x,r • z) < d(r ■ x, r • y) + d (r • y, r • z) (3.11) Since it is clear that if T • x = T -y then d(T ■ x,T • y)= 0, all Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 81 that is left to be shown is that if d(T • x,T • y) = 0 then T ■ x = T ■ y. By (1), there exists e > 0 such that if 7 • B(y,s) fl B(y,s) ^ 0 then 7 E Ty. Since d(x, T-y) = 0, there exists a sequence, {7 n} C T such that d(x, 7 n ■ y) 0- This means that we may find JVeN such that for any n > N we have d(x, r j n • y) < e /2. Now, for any n,m > N this implies that we have: dhn-y,lm-y) < d(jn -y,x) + d{jm -y,x) < e/ 2 + e/2 (3.12) £ Since T acts by isometries, d(y, 7 “ x7 m-y) < £ and thus 7 “ 17 m-B(y, e)fi B(y, e) 7^ 0 which, by our choice of e, implies that 7 “17 m E Ty. This tells us that 7 “17 m-y — y so we see that 7 n-y = -y for all n,m> N. Thus, for all n > AT we have d(x, 7 n'y) = d(x, 7 jv • y). Since d(x, ■ y) —> 0 as n^o o w e have d(x, 7 at • y) = 0 and thus x = 'Jn • y. This implies that x E T • y and T • y = T ■ x as required. 3. This is a direct consequence of the definitions. 4 . First note that for any x E X and 7 E T we have 7 T X7_1 = T7.x because a E rTX 77 cry • x = 7 • x 77 7 - 1a7 ■ x = x (3.13) 77 a E 7^ 7' 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 82 Suppose that the action of T on X is cocompact and fix a compact set K C X such that V-K = X. Since T acts properly on X, for each x £ K there exists rx > 0 such that {7 £ T : ^-B(x, rx)f\B{x, rx) ^ 0} is finite. The set of all such balls B(x, rx) is an open cover for K so by compactness we can find a finite subcover {£?(xi, 77 ), B(x2, r2),..., B(xn, rn)} of K, where 77 = rXi for each 1 < i < n. Let us define E to be: n E = ( J {7 e r I 7 • B{xi, ri) n B{xu n) ± 0} (3.14) i~ 1 As a finite union of finite sets, E is clearly finite as well. For each x E X there exists 7 E T such that 7 • x £ K. Then 7 r i 7 _1 = r ra;. Since r r:r stabilizes 7 • x £ K, for each 7 ' £ T^.x we have 7 ' • K fl K ^ 0. Thus 7 ' £ E and T7.a; C E. So T^ is conjugate to a subset of the finite set E and thus the isotropy subgroups of T are finite. Furthermore, since each conjugacy class of isotropy subgroups has a member which is contained in E, there can only be finitely many such classes. □ Remark 3.1.6. If X is a geodesic metric space such that there exists a group which acts properly and cocompactly on X, then X is complete and locally compact. Thus, by the Hopf-Rinow Theorem, it is a proper geodesic space [Bridson & Haefliger pg. 35, 132]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 83 Example 3.1.7. We can define an action of T on the metric Cayley graph Ca (T) where A is a finite generating set for T. For each 7 e T we define the action of 7 to take each edge e originating at 70 labelled by a £ A isometrically to the edge labelled by a originating at 770 . This action is proper because any ball of radius 1/2 in CU(T) intersects only its trivial translate. Conversely, the translates of any closed ball of radius 1 will cover CA(r) so the action is cocompact. 3.2 Group Presentations and Actions The aim of this section is to prove that a group acts properly and cocom pactly on a simply-connected geodesic metric space if and only if it is finitely presented. We will begin by describing group presentations and then pro ceeding to Macbeath’s theorem [Macbeath] which provides a presentation for any group acting properly and cocompactly by homeomorphisms on a simply- connected topological space. Given a set S we write F(S) to denote the free group generated by S. Definition 3.2.1. A presentation for a group G consists of a set S, an epi- morphism 7r : F(S) —► G and a subset R C F(S) such that ker7r = ((R)}, where ((R)) is the normal closure of R. A presentation is normally denoted G — (S | R) where S is called the set of generators for the presentation of G and R is the set of relators. If the sets S and R are finite then the presentation is said to be finite and if a group G admits a finite presentation, then G is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 84 called finitely presentable. The following lemma is a standard consequence of the Seifert-van Kam- pen theorem from Algebraic Topology and is stated without proof. For more details, refer to [Bridson & Haefliger pg.135] and [Massey pg. 113-116]. Lemma 3.2.2. Let T be a group with presentation (A | R) and let CA(T) be its metric Cayley graph. Let R' C ((R)) and consider the 2-complex obtained by gluing a 2-cell to each loop in whose edges are labelled by a reduced word, r G R!. Then this 2-complex is simply connected if and only if ((R')) = «*»■ The following theorem is due to Macbeath [Macbeath]. Theorem 3.2.3. Let T be a group acting by homeomorphisms on a non-empty connected topological space X. If U is an open set such that T -U = X, then: 1. The set S = {7 G T | 7 • U fl U 0} is a generating set for T 2. If we also require that U be path-connected and X be both path-connected and simply connected then T has a presentation F = {As \ R) where As is a set of symbols indexed by S and the set of relators R is defined as follows: R = {aSlaS2a“ 1 | Si G S'; U Fl si • U fl s3 • U ± 0; sis2 = s3}. (3.15) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 85 Proof. 1. Let H = (S) C T be the subgroup generated by S. We wish to show that H = T so if we let I = F — H then it is equivalent to show that I is empty. Suppose first that H ■ U fl I ■ U ^ 0. This implies that there exists elements u, it' E U and h E H, i E I such that h ■ u = i • u ' . This gives h~li • u ' = u 6 U which implies that hrli E S and thus i E HS. Unfortunately, HS = H so we conclude that i E H which is a contradiction. We now know that H ■ U fl I • U = 0. The set U is open however, so H ■ U = (J{/t • U \ h E H} and I ■ U = |J{i • U \ i E 1} are unions of open sets since T acts by homeomorphisms. Thus H ■ U and I ■ U are also open. Since T = H U I, we have X = T - U — H - U \J I - U. But X is connected, so we conclude that either H • U = 0 or / • U = 0. The set H-U is non-empty since both U and S are non-empty (1 E S). This implies that I • U = 0 and since U ^ 0, it must be the case that I = 0, as desired. 2. Let CUS(T) be the metric Cayley graph of T with generators in As- If K is the 2-complex obtained from C'as (T) by attaching a 2-cell to each of the edge loops labelled by a word in R, according to the Lemma 3.2.2, in order to show that R is a set of defining relators of T, it is sufficient to show that K is simply-connected. Note that we may extend the action of T on Chs(T) to an action on K where 7 E T sends the 2-cell glued to the edge loop labelled by r, based at the vertex x, homeomorphically to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 86 the 2-cell glued to the edge loop labelled by r based at the vertex 7 • x. In order to prove that K is simply connected, we will show that any continuous map £ : dD —>• CaS{U) from the boundary of a disk D can be extended to a continuous map £ : D —> K. We will first construct a V-equivariant map from K to X. Fix a base point Xq G U. Now, for each generator s E S, we select a point xs in the non-empty set U fl s • U. Since U is path-connected, we can now find a path c~ in U originating at :r0 and terminating at xs. Futhermore, since s • U is path-connected, we may find a path cs' from xs to s ■ xo E s ■ U. Let cs : I —> X be the concatenation of these two paths (so cs originates at Xq and terminates at s • xq). See Figure 3.1. s • x 0 Figure 3.1: Constructing the path cs We can now define a map p : 6 b s (T) —> X which sends any vertex labelled by 7 G F to the point /y-x0 and any edge labelled by as originating at 7 and terminating at 7 ■ as to the path 7 • cs in X. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 87 To construct a T-equivariant extension of p to K. we will extend p to each cell glued to edge loops which are based at 1. Let r E R and consider er : S1 —> C a (T), the loop based at 1 and labeled by r. Let cr be the cell which was glued to this loop. Since X is simply-connected, we may extend to map p o er : S1 —> X to a map p o er : S1 U c° —> X where cr is the interior of the cell cr, thus we can find a continuous extension Per '■ CU(T) U cr —>■ X of p. If we find this extension per for each r E R then we can define p : K —> X as follows: p(x) X G Cyle(r) p(x) = v ' sV ; (3.16) 7 •PeT[x) XG7-Cr This map is continuous since it is continuous when restricted to Cas (T) as well as when restricted to each open 2-cell in K. Furthermore, it is T-equivariant by definition. Let us now return to our map I : dD C7is(r). We assume without loss of generality that £ is a closed edge path labeled by a cyclically reduced word (so the first letter is not the inverse of the last). Since X is simply-connected, the map p o £ : dD —> X can be continu ously extended to a map 0 : D —> X. Our goal is to find a triangulation T for the disk D such that a suitably reparameterized version of our loop, £' : D —> Cas(T) can be continuously extended to a map 4 >: T —> Ca„(T)- Furthermore, we shall do this in such a manner that the image of each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 88 triangle in T is an edge loop in C,\s (T) which is labeled by a word in R, which will allow us to extend this map to a map from the entire disk D. Therefore 0 can be extended to D and we shall be able to conclude that I is null-homotopic. Let us now examine the image of I in Cas more carefully. Let y 0 be a base point for the loop t and let S 1S2S3 • • • sn be its edge labels. Let 7 j = 70 S1S2S3 • • • Si be the label of the ith vertex through which £ runs, % = 1, 2,..., n. Let d0, d\,..., dn- 1 be the points on dD which are mapped under i to 70 • x0, 7 i • 72 • %o, • • •, 1 n-i • respectively. Note that 7 „ • xq = xq. See Figure 3.2 Thus, if we consider the loop p o £ : dD —> X we shall see that it is the concatenation of the paths 70 • cSl, 71 • cS2... jn-i • csn-i- Moreover, we defined each path cSi to be the concatenation of a path in U from x0 to xs. and a path in s* ■ U from xSi to S{ ■ x0. For each 0 < i < n — 1 let d[ G dD be the point which gets mapped under f o £ to ■ji ■ xs. Now, this means that image of the segment of dD from d '_ 1 to d[ for 1 < i < n (as well as from d ^_1 to d'0) under p o £ is contained within 7 i+i • U (respectively 70 • U). Since D is compact, there exists et for each 0 < i < n such that any ball of radius around a point in the segment from d[ to d'i+ 1 is mapped into 7 i+i • U by p o £. Furthermore, since T - U = X, the set {0-1 (7 • U) : 7 G F} is an open cover of D. Thus by the Lebesgue Covering Lemma [Munkres], we can Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 'In— 1 jn—l 'n— 1 72 * 7i ‘^0 7o 7o * z 7n—1 * *0 Figure 3.2: Relationship between D, C,\s (T). and X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 90 find e' > 0 such that any ball of radius ef in D lies within one of the open sets of the cover. In particular, this means that the image under 0 of any such ball is completely contained within 7 ■ U for some 7 £ F. Let e = min-jV, £0, £1,... £n-i} and take a finite triangulation T of the disk D such that the diameter of every triangle is less than £ and each of d0, di,..., dn- 1, d'o, d[,... d'n_l is included in the vertex set of T. In particular, the ball of radius £ around a vertex v in the triangulation T contains all the triangles in T which are incident at v. 7i+i ' U 7i+2 ' U 7i ' zo ■ + 1 7i+l ’ Si S'i 8i+i Si+2 Figure 3.3: Selection of Si and S[ We can now define a map from the vertices of T not on dD to T by v i—► 7 u where we fix 7 „ to be an element of T such that (f> maps all of the triangles incident at v into 7 „ • C/. For each v G dD we choose 7 ,, as follows: • if v = d[ for some 0 < i < n - 1 we choose 7 „ = 7 j+i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 91 • if v lies on the segment of dD between d[ and d'i+l for 0 < % < n — 1 then we choose *yv = 7,;+1 and if it lies between d,n_l and d'() we choose = 70 Let us examine a triangle t G T with vertices vi,v2 ,v3. 0 = lv1 (UO(%11 -fV2 -U)C\{%1ljV3 -U)) (3.17) = 7«a((7^17t;i • C/) n [/ n ( 7 "17 „3 • 17)) Thus, if we let h = 7“ 17^> let b2 = %2 lv3, and let b3 = 7 “ 17 ^ then bi,b 2 ,b 3 G 5 and furthermore, cp,, cib2 (h 3 G 7?. We can now extend our map u i—> 7 ,, to a map 4> : T —>■ C'J4S(T) by sending the edge which connects 17 to to the edge b\ as defined above and similarly for the other edges. Note that 0|ao is a reparameterization of £. Now, since each triangle is mapped to a circuit in CUS(T) which is la belled by a word in R, we can extend 7 to a continuous map D —> K. Therefore K is simply connected as desired. □ Corollary 3.2.4. A group T is finitely presented if and only if it acts properly and cocompactly by isometries on a simply-connected geodesic metric space A. Proof. (7 =) Suppose T is a group which acts properly and cocompactly on a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 92 simply-connected geodesic metric space X. Let K be a compact set such that r • K = X and take an open ball B(xf). r) which contains K. Then we have T • B(xo, r) = X and can apply Theorem 3.2.3 with U = B(xo, r). By Remark 3.1.6, the space X is proper and the action of T is proper, so we know that the set {7 G T : 7 • B(x0, r) fl B(x0, r)} is finite by Remark 3.1.4. Since as defined in Theorem 3.2.3, S is contained in this set, it is finite and hence R is finite as well. (=>) Suppose T = (A\R) with A and R finite. If we construct the simply- connected geodesic metric space K by gluing an n-gon to each edge loop labeled by a word r G R of length n in the Cayley graph C'4 (F) then the action of T on CU(T) extends naturally to an action on K. In this action, if 7 G T and c is an n-gon labeled by r G R and based at 70 G T, then 7 sends c isometrically to the n-gon glued to the edge labeled by r based at 770 . The space K is simply-connected by Lemma 3.2.2. Let m be the length of the longest word in R. If we take B = B(l, m) C K then B is compact since both A and R are finite. Furthermore, the transitivity of the action of T on the vertex set of C '4 (T) guarantees that the translates of Y cover K. Thus, the action of F on K is cocompact. Furthermore, for any ball B(x, 1/2) in K, if 7 G T such that 7 • B(x, 1/2) C\B(x, 1/2) 7 ^ 0 then 7 = 1 so the action is proper and so we have defined a proper cocompact action of T on a simply-connected geodesic metric space, as desired. □ Corollary 3.2.5. A group T which acts properly and cocompactly on a proper Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 93 CAT(O) space is finitely presented. Proof. Any CAT(O) space is a simply connected geodesic space by Remark 2.1.5, so by Corollary 3.2.4, we have that F is finitely presented. □ 3.3 Bruhat-Tits Fixed Point Theorem In this section, we present a fixed point theorem for isometries of CAT(O) spaces as well as several properties regarding conjugates of finite subgroups. Theorem 3.3.1. Let T be a group acting by isometries on a CAT(O) space X. 1. [Bruhat-Tits Fixed Point Theorem] If X is a complete CAT(O) space and T has a bounded orbit then the fixed-point set of F is a non-empty convex subspace of X. 2. If T acts properly and cocompactly on X then each finite subgroup is contained in an isotropy subgroup and hence there are only finitely many conjugacy classes of finite subgroups in T. Proof. 1. Let Y be a bounded orbit of T and let 7 e P be any element. Then by Proposition 2.4.4 the centre cy of Y is fixed under the action of 7 . Thus cy is in the fixed-point set of T. To show that the fixed point set is convex, let x, y be fixed points of F and consider the (unique) geodesic segment [x, y] joining them. For any Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 94 7 G T, the set 7 -[x,y\ is a geodesic segment from 7 • x to 7 • y because 7 is an isometry. Since x and y are fixed by 7 and X is uniquely geodesic, we conclude that 7 • [x,y\ = [x, y\ and [x,y\ is in the fixed set of T as desired. 2. Suppose H is a finite subgroup of T. Let xq G X. Then H ■ Xo is a bounded orbit and thus by (1) we know that H has a non-empty fixed point set. Suppose x G X is in the fixed point set of H, then H Q Tx. Since by Proposition 3.1.5 we also know that there are only finitely many conjugacy classes of isotropy subgroups in T and hence finitely many conjugacy classes of subgroups of isotropy subgroups, we conclude that T contains only finitely many conjugacy classes of finite subgroups. □ Corollary 3.3.2. Suppose T acts properly and cocompactly by isometries on a CAT(O) space. Then T has only finitely many conjugacy classes of elements of finite order and in particular finitely many central elements of finite order. Proof. Choose a representative for each of the finitely many conjugacy classes of finite subgroups. Each 7 G T of finite order is conjugate to one of the finitely many elements of subgroups from this finite list of subgroups. The second remark follows since central elements are their own conjugacy class. □ An important application of the Bruhat-Tits Fixed Point Theorem is to the case of a group acting on a tree. Serre [Serre] uses the result to prove if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3. GROUP ACTIONS 95 Ti *h r2 is an amalgamated product, then any finite subgroup is conjugate to a subgroup of Ifi or T2. Similarly to the case of a finite subgroup, if a topological groupT acts by isometries on a CAT(O) space and K C T is compact, then K has a bounded orbit and hence a fixed point. Cartan’s fixed point theorem is a special case of this. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Decision Problems In this chapter we shall prove that any group which acts properly and cocompactly on a CAT(O) metric space has a decidable word problem. 4.1 The Word Problem We begin with a definition. Definition 4.1.1. A group T with a finite generating set A has a decidable word problem if there exists a Turing machine halting on each input that decides whether or not a given word w 6 F(A) represents the identity in T. It must be noted that we only require the existence of such a Turing ma chine, not an effective method by which to construct it. Since the aim of this section is to prove that such a Turing machine exists for groups which act properly and cocompactly by isometries on a CAT(O) 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 97 metric space, let us fix such a group T and CAT(O) metric space X for the entire section. Note that by Remark 3.1.6, X is a proper space. Since the action of V on X is cocompact, there exists a compact subset K C X such that r • K = X. Furthermore, since K is bounded, we can find a radius D/3 and basepoint xq € X such that K C B(xq, D/3). This tells us that T • B(x(h D/3) = X ; let us also fix this constant D for the remainder of the section. We begin our discussions with a lemma which identifies a generating set for our group T. Lemma 4.1.2. The set A = {o G T : d(a-x0, x0) < D +l} is a finite generating set for T. Moreover, for any 7 G T such that d(y • xq, x0) < 2D + 1 there exists a four-letter word aia 2o3a4 G F(A) which represents 7 in the group T. Proof. If we let U = B(x0 ,D/3) then we have V ■ U = X which by Theorem 3.2.3 implies that the set S = {7 G T : 7 • B(x0 ,D/3) D B(x0, D/3) ^ 0} is a generating set for T. Pick any element s G S; we wish to show that s is an element of A. There exists x G B(x0, D/3) such that x is in s ■ B(x0 , D/3) where s • B(x0, D/3) = B(s ■ xo, D/3) since T acts by isometries. From this, we can compute: d(x0 ,s-x0) < d(x0, x) + d(x, s ■ x0) < 2D/3 (4-1) < D + l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 98 Consequently, we may conclude that s G A and thus S C A. Since we know that (S) = T we see that (A) = T as desired. Now we shall show that A is finite. Consider the set: T = {7 e r : 7 • B{x0, D + 1) n B(x0, D + 1) ^ 0} (4.2) The metric space X is proper, so the closed ball B(xo, D + 1) is a compact subspace of X. Since T acts properly on X , by Remark 3.1.4 we see that the set T must be finite. Given an element a e A, by definition we have d(a • xq. xq) < D + 1 which implies that a ■ x0 £ B(x0, D + l). From this we deduce that a ■ B(xo, D + 1) D B(x0, D + 1) 7^ 0 which gives a G T. Thus, A is a subset of T and the set A is a finite generating set for T as desired. To prove the second half of the lemma, let 7 e T be an element such that d{7 • x0,x 0) < 2D + 1 and consider the geodesic path c 7 : [0, A] —> X which originates at xq and terminates at 7 • £o- Note that the length of the path A is equal to d(x0, 7 • x0) and so A < 2D + 1. We fix the following points in the image of c7: Xl = Cy( A/3) x2 = c7( A/2) (4-3) x3 = c7 (2A/3) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 99 Since T • B(x0, D/3) = X we can find elements 71,72, 73 £T such that: x1 £ B (7 i • x0,D/3) x2 £ B(72 • x0,D/3) (4.4) £3 e 5(73 •x 0,D/3) D/3 A/3 A/6 A/6 A/3 — — = » - - « = — - s * - -*s- Figure 4.1: Dividing the geodesic segment [xo,7 • ^0] into four parts For the element 71 we have: d(x0, Ji-x0) < d(x 0,xi) + d(x i, 7 i • x0) < A/3 + D/3 (4.5) < (2D + l)/3 + D/3 < D + l So we may conclude that 71 £ A. Let us assign <27 = 71 . Similarly, we find that d(x0, 7]”172 ■ x0) < D + l and d(x 0,7^’173 • x0) < D + l. See Figure 4.1. Thus we may assign a2 = 7 ^ 7 2 £ A and a 3 = 7 ^ 7 3 £ A. Finally, we note that d(xo,73"17 • £0) < D + 1 so we have a4 = 73~17 € A. We conclude by pointing out that the word aia 2a3a4 in F(A) represents the element 7 i(7 r 172 )(72 ~173 )(73~17 ) = 7 in T and is therefore our desired Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 100 representation. □ Definition 4.1.3. A function / : N —> R is an isoperimetric function for a group T given by a finite presentation (A | R) if all words w in F(A) of length at most n representing the identity in T can be written as a product of at most f{n) conjugates of relators and their inverses. The following theorem says that T has a quadratic isoperimetric function and a decidable word problem [Epstein et. al.]. Theorem 4.1.4. Let R C F(A) be the set of reduced words of length at most 10 which represent the identity in T. Then a word w G F(A) represents the identity in T if and only if the following equality in F(A) holds: N ] [ i .r.r, 1 (4.6) 1=1 where N < (D + 1 )\w\2 and \xi\ < (D + l)|iy| with each n G R. Proof. For each 7 G T we associate with it a word a7 in F(A) as follows. Let c7 : [0, A] —► X be the unique geodesic path which originates at x0 and terminates at 7 • x0. We extend c 7 to a ray by taking c7 (t) = 7 • x0 for each t > A. For each integer i > 0, since T • B(xq, D/3) = X, we can find cr 7 (z) G T such that Cy(i) G B{a7 {i) ■ x0 ,D/3). For i = 0, let us take a7 (i) = 1 and for each i > d(x0, 7 • x0) = A we take a7 (i) = 7 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 101 For each integer i > 1 we make the following observation: d(cr7(i — l) 1 a7 (i) ■ x0 ,xo) = d(a 7 (i) ■ x0, a7(i — 1) • x0) < d(a7 (i) ■ x0, Cy(i)) + d(c7 (i), c7(i — 1)) +d(c7(i - 1), a7(i - 1) • x0) < D/3 + 1 + D/3 < D + l From this we conclude that a7(i — 1) 1 cr7 (i) 6 A so set a* = o7 We can now define the word a7 to be 0 ^ 2 ■ • • an where d(7 • xo, xo) < n < d(7 • xq, Xq) + 1. It is easily verified that n V(i) (4.8) 1=1 and so ct 7 E F(A) represents 7 in T. Let us now fix an element 7 E T and a letter b E A; let 7 ' = 7 b. We shall be considering the words cr7 and ay in F(A) representing 7 and 7 ' respectively, as defined previously. By adding sufficiently many empty characters to the shorter word, we may assume that the two words have the same length, that is | show that the distance from er 7 (i)-x0 to ay(i)-x0 is bounded above by 2D + 1. Note that in adding empty letters to either the word consistent with our choice of a* = a7(i — l)-1cr7(i) and a' = cry (i — l ) _1cry(i) since a7 (i) = 7 for every * > d ( j ■ x0, xq). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 102 Our goal is to show that the distance between cr7 (i) ■ x0 and a7 >(i) • xq is bounded above by 2 D + 1. This is an analogue of the fellow traveler property [Epstein et. al.]. We begin by making the observation that d(rf' xo, i • xo) = d(j ■ x0,76 • x0) = d(x0, b ■ x 0) (4-9) < D + l The last inequality arises due to the fact that b is an element of A. Let c7 and c7> be the unique geodesic paths originating at x0 and termi nating at 7 • x0 and 7 ' ■ x0 respectively. Just as before, we define c7 (t) = 7 ■ x0 for every t > d{+ • x0, xo) and cy(t) = 7 ' • x0 for every t > d(7 ' • x0, x0). The next step is to show that the inequality: d(c7 (t), Cy(t)) < D + 1 (4-10) holds for every t > 0. Consider the geodesic triangle A(x0, 7 • x0,7 ' • x0) and its comparison triangle in E2, given by A(x0,7 • x0,7 ' • x0) = A(xo, 7 • x0, j' ■ x0). W ithout loss of generality, we assume that d(7 • xo,xo ) < d(7 ' • We further assume that the triangle A is not degenerate, for if it were then the CAT(O) inequality would guarantee that the inequality (4.9) holds for every t > 0. Let a G [To, 7 ' • x0] be the unique point such that d(xo,a) = d(x^, 7 • x0) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 103 and let b be any point on the geodesic segment [a, 7' • x0] such that b ^ 7 ' • rr0. 7 • xq Figure 4.2: A(x0,7 • x0,7 ' • x0 ) with a and b illustrated Let a, a', /?, and (3' be angles as shown in Figure 4.2. Since the triangle A(xo, a, 7 • xo) is isosceles and non-degenerate, we know that ct < 7t / 2. This implies that /? > 7t/2 and since /?' = /? + a ' we know that f3' > k/2 (note that if the points a and b coincide, then (3' = (3 > 7t / 2). Since the angle at b in the non-degenerate triangle A( 6 ,7 ' • x0,7 • x0) is obtuse, we conclude that d(b, 7 • x0) < d(7 ' • x0, 7 • x0) < D + 1 (4-11) Also, (4.11) holds trivially if b = 7 ' • x0. Taking a = b, the inequality d(7 • Xo, a) < D + 1 holds Consider any t > 0 and let c 7 (t) and cy(t) be comparison points in A for c7 (t) and c 7 /(t) respectively. We shall split our discussion into two cases: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 104 1. If t < d(+ ■ x0, x0) then we find that cy (t) lies on the geodesic segment \xo,a] in A. Furthermore, the geodesic triangle A(xo, c7 (t), cy(t)) is similar to A(xq,7 • Xq, a). 7 • x0 Cy(t) C y (t) Figure 4.3: Comparing isosceles triangles We now have the following inequalities: d(c 7 (t),c( V(i)) < d(&y(t),Oy>(t)) < d(7 ■ x0 ,a) (4-12) < D + l The first inequality is a consequence of the CAT(O) condition. 2. The other possibility is that t > d^-x0, xo) in which case c7 (t) = j-x0. This means that if we consider the comparison points c 7 (f) and cy(t) for c7 (f) and cy(f) respectively, then c 7 (t) = 7 • x0 and cy(t) e [a, 7 ' • x0]. We now have the following inequality: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 105 d(Cy(t), Cy (t)) < d(Cy(t) , Cy' (t) = d (7 • x0,c7 /(t) (4-13) < D + l The first inequality is a consequence of the CAT(O) condition and the last inequality is a consequence of (4.11) To conclude this part of the proof, for every integer i > 0 we have the following inequalities: d(a1 (i) • x0, oy (i) • x0) < d (a 7(i) • £0, (^(i)) + d(c7 (i), cy(i)) +d{cy (i),aY(i)-x0) (4.14) < D/3 + (D + 1) + D/3 < 2D + 1 This gives us the inequality d(a7(i)_1ay(«) ■ Xq,xo) < 2D + 1 which by Lemma 4.1.2 implies that we can find a four-letter word a(i) in F(A) which represents we take a (0) to be the empty word and for i = n we set a(n) = b which is possible since cr 7 (n)- 10y(n) = 7 - 17 ' = b in T. To split Py(i) = a^a'2 ■■■ a I for each integer 0 < i < n. Note that we assign p7 (0) and p7 /(0) to both be the empty word. It is now trivial to verify that the following Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 106 1 Figure 4.4: 7 and 7 ' in the metric Cayley graph of P. telescoping equality holds in the free group F(A): n —1 0 ^ 0 -} = ]^[p7/(i)[a(i)_1ai+1a(i + l)(a-+1)-1]py(i)_1 (4.15) i=0 The word a(z)_ 1ai+i ment in T: + l))(a7(i + 1 + l))(<77/(i)-1oy (z + I))-1 (4.16) which reduces to the identity. Furthermore, it has no more than 10 letters so it is a word in our set R. We can now proceed with the proof of the isoperimetric function. Let w = bib 2 ---bm e F(A) be a word which represents the identity in T and let 7 j be the element in T represented by b\b 2 • • -bi for each 0 < i < m, so 70 = 1 = 7 m- Denote by |m| the length of the word w. If \w\ < 10 then w can clearly be written in the desired form in ^(^4) since w E R; this means Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 107 we may restrict our attention to \w\ > 10. We may trivially equate w to the following telescoping product in F(A): m w = W an-ibi(J^ (417) i=i For each 1 < j < m, if we let 7 = 7,-1 and 7 ' = 7 ,• then we have by (4.15), the following equality in F(A): h3a n = n P'yjii )R ijP'rj(i )~ 1 (4-18) i= 0 where n( 7 j_i,7 j) is the length of the longer word is a word in the set R. We have now written vj as a product in F(A) of the correct form: w = U n P i A i )R ijP'rj(i )~ 1 (4-19) j=0 i= 0 This means that all that remains to be shown is that \plj (i)| < (D + l)|w| for each pair i,j in the product, and that the following bound holds: m < (D + l)\w\2 (4.20) 3 = 0 To prove that these bounds hold, we first show that the length of each word cr7. is less than 1(D + l)|iu| + 1. Consider 7 7- and note the following, recalling Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 108 that 7o • Xq = 7 m • Xq = Xo d\lj-xo,Xo) < mm E/T=j+i d(7fe-i ' ®o, 7fc • ®o) . , j{D + 1) < mm < (m — j)(D + 1) (4-21) < l{j(D + 1) + {™-j)(D + 1)) = 2 ± lm = Since d(jj • Xo,xo ) < | ^Y^-\w\ + 1 as desired. The word p7. (i) is a subword if cr73. and thus satisfies |p 7i(i)| < |w| + l < (.D + l)|iu| since \w\ > 10. Furthermore, n(jj, 7 j-i) < \{D + l)|w| + 1 < {D + l)|w| so m ^Erc(7 j_i,7 j) < (D + l)\w\m = (D + 1)M2 (4.22) 3=0 This completes the proof of Theorem 4.1.4. □ C orollary 4.1.5. If a group T acts properly and cocompactly on a CAT(O) metric X, then the group T has a decidable word problem Proof. Given the sets A, R and a freely reduced word w 6 F(A) as in Theorem 4.1.4, a Turing maching can enumerate the finite list of words of the form (4.6) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. DECISION PROBLEMS 109 within the given upper bounds and compute their freely reduced forms. The Turing machine then checks if w is in the list of reduced words. By Theorem 4.1.4, w represents the identity in T if and only if it can be written in the form of (4.6), within the given bounds. Thus, the Turing machine answers yes if w is in the list and no otherwise. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Investigations of Isometries Having presented the basic properties of CAT(O) spaces in Chapter 2, as well as introduced the fundamental concept of group actions in Chapter 3, we may begin to characterize the isometries of CAT(O) spaces. In this chapter, we shall be paying special attention to the properties of individual isometries and how they interact with the structure of our metric space. We will always present our isometries as group elements a c t i n g on our space. 5.1 Displacement Functions and Translation Length In Section 5.2 we shall divide isometries into three classes, depending on the properties of their displacement function and their translation length. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 111 Definition 5.1.1. Given a metric space X and an isometry 7 of X we define the displacement function of 7 to be d 7 : X —> R where for any x G X the displacement is given by d 7 (a:) = d(y • x, x). To get a better impression of how 7 affects the entire space X, however, we are more interested in the translation length of 7 which is given by |7 | = inf{d 7 (a:) | x G X} (5.1) The set of elements in X for which d1 actually attains this infimum is called Min(7) and if we are investigating a group T of isometries, then we set Min(T)= fl7erMin(7)- We can now separate the isometries of a CAT(O) space into two types, those for which Min(7) is empty and those for which it is not. In the case that Min(y) is non-empty we say that 7 is a semi-simple isometry of X. Later, we shall see that two of our to-be-defined types of isometries are semi-simple while the other is not. Before proceeding however, we shall collect several useful facts about the displacement function, the translation length, and the set Min(7). Proposition 5.1.2. Let X be a metric space and T a group acting by isome tries on X. If 7 is an element of T, then the following hold: 1. If a G T is another isometry of X then the translation length |y| is equal to the translation length |ct7a-1| and a-Min(7)=Min(a:7a:_1). Thus, if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 112 a and 7 commute then Min( 7 ) is an cr-invariant subspace of X and if lV < risa normal subgroup, then Min(A’) = p|ne.vMin(n) is T-invariant. 2. The sets Min(7) and Min(T) are 7-invariant. 3. If X is a CAT(O) space, then the displacement function d 7 is convex and Min(7) is a closed convex subset of X. 4. Given a non-empty, complete, convex, 7-invariant subset C of a CAT(O) space X then the translation length of 7 is equal to the translation length of 7 when restricted to C. Furthermore, 7 is semi-simple if and only if 7 1c is semi-simple. In particular, Min(7) is non-empty if and only if C'flMin(7) is non-empty. P r o o f . 1. First observe that for all x E X we have d ( a j a ~ x • (a ■ x), a ■ x) = d (7 • x , x) (5.2) For any e > 0 there exists x E X such that d(y • x,x) < |y| + e. But this implies that d{a^a~l ■ (a ■ x), a ■ x) < |y| +s by (5.2), so we see that |«7 «-1| < |t|+ £ f°r every £ > 0. This tells us that |a7 a _1| < lyj. Using a similar argument, we derive that |y| < |a 7 a _1| and thus the desired equality holds. To show that Min(a7Q;_1)=Q;-Min(7), consider x eMin(7). We know that the equality, d(y • x , x ) = |q| holds and we wish to show that a ■ x G Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 113 Min^cryar1). By (5.2) we have d ( a ~ / a 1 • {a ■ x), a ■ x ) = |y| = |crya 1 (5.3) so a • x € Mu^crpa x) and a-Min(7) C Min(a7a x). Dually, we have a T Min(o;7Q! : ) C M in(y) so Min(ci!7Q; x) = a- Min(7) as desired. The above equality clearly implies that if a and 7 commute, then Min (7) is a-invariant. Now, suppose N is a normal subgroup of T. Then 7 • Min(lV) = flneiv7-M in(n) = flneJVMin(7W7 X) (5.4) = rUrMin(n) = min(A^) and Min(iV) is T-invariant as desired. 2. Since T < T we may conclude by (1) that Min(T) is T-invariant and thus 7-invariant. Similarly, since 7 commutes with itself, we apply (1) to conclude that Min(y) is 7-invariant. 3. Fix 7 6 T; to show that d7 is convex, let c : I — > X be a linearly reparameterized geodesic path. We wish to show that for all t G I the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 114 following inequality holds: dj(c(t)) < (1 — f)d7(c(0)) + idy(c(l)) (5.5) In a CAT(O) space however, the distance function is convex by Propo sition 2.2.3. If we consider the linearly reparameterized geodesic path c along with the linearly reparameterized geodesic path 7 • c we have the following: dy(c(t)) = d(j ■ c(t), c(t)) < (1 — t)d((j • c)(0), c(0)) + td((-y ■ c)(l), c(l)) (5.6) = (1 - t)d(7 • c(0), c(0)) + td(7 • c(l), c(l)) = (1 — t)d7 (c(0 )) + td7 (c(l)) so d7 is convex as desired. To show that Min(7) is convex, consider x, y E Min(7) and z 6 [x, y\. Let c : [0, 1] —> X be a linearly reparameterized geodesic path with corresponding geodesic segment [x,y\ and suppose z = c(t). Then by (5.6) we have d1 (z) < (1 — t)d1 (x) + tdj(y) < ( l - i ) |7 |+ t |7 | (5-7) = M so z e Min(y) as desired. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTERS. INVESTIGATIONS OF ISOMETRIES 115 4. We first wish to show that the translation length of 7 when restricted to C is equal to |q|. Let 7T : X —► C be the orthogonal projection of X onto C. We begin by showing that 7r is 7-equivariant. For any x E X and any c € C, since 7-1 ■ c E C we have that d(7_1 • c,x) > d(7r(x), x). Moreover, since 7 is an isometry, this now implies that d(c, 7 • x) > d(7 • 7r(x),7 • x) for every c G C. As C is 7-invariant, we have that 7 • 7r(x) G C and hence 7r(7 ■ x) = 7 • 7r(x). Next, we shall establish that d( 7 • x, x) > d(7r( 7 • x), 7r(x)) = ^ ( 7 ■ 7r(x), tt(x)) (5.8) Consider comparison triangles A = A(x, 7r(7 • x)) for the trian gle A(x, 7r(x), 7t(7-x)) and A7 = 7r(7 • x)) for A(x, 'y-x,Tr(/y-x)) such that the segments [x, 7r(7 • x)] coincide in each triangle and 7 • x and 7r(x) lie on opposite sides of the geodesic line which runs through x and 7r ( j ■ x). See Figure 5.1. Figure 5.1: Constructing a quadrilateral in E 2 for comparison Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 116 By the CAT(O) condition, we know that 7r(7 ' x)) > A(x) (x, tt(7 • a:)) > tt/2 (5.9) where the last inequality is a consequence of Proposition 2.3.1 (3). Fur thermore, we have + Z^(x,ir{x)) A - i x ^ - x ) (7 ■ X, x) + Z .n ( < y .x ') (x, 7r(x)) / r in \ (5.10) > Z ^ rx)(-f ■ x,7v(x)) > 7r/2 again by the CAT(O) inequality and Proposition 2.3.1 (3). Thus we have d { x ^ T ~ r x ) > d { 7r(a:),7r(7 • x )) which implies that we have the inequality d ( x , 7 • x) > c?(7r ( x ) , 7r(7 ■ x )) as desired. Clearly |q| < |7|c|- To show the opposite inequality, let e > 0, then there exists x G X such that d(q • x, x) < jqj + e. Then by (5.8) we have rf(q • 7r(x), 7t(x)) = d( 77(7 • x ), 7r(x)) < d( 7 -x,x) (5-11) < M + e Thus |7 |c | < d (7 • 7r(x),7r(x)) < |7 |. Since e was arbitrary, I 7 I0 I < N and equality holds as desired. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 117 It is now immediate that Min( 7 ) DC = M in ^ c )- □ We are now prepared to introduce the three classes of isometries. 5.2 Three Classes of Isometries We now proceed with dividing the individual isometries of a CAT(O) space into three different types based upon the properties of their translation lengths. Definition 5.2.1. Consider a CAT(O) metric space X and a group T which acts on X by isometries. If 7 G T is an individual isometry, then 7 may be classified as one of the following: 1. We say that 7 is elliptic if it has a fixed point. This means that |y| = 0 and Min (7) ^ 0. 2. If Min(7) ^ 0 but |7| > 0 then we say that 7 is hyperbolic. 3. Finally, in the case that Min(7) = 0 then 7 is said to be parabolic. Note: elliptic and hyperbolic isometries are semi-simple and by the previous proposition, conjugate isometries are of the same type. A semi-simple isometry 7 such that Min (7) = X is called a Clifford Translations. Proposition 5.2.2. Let A be a complete CAT(O) metric space and let 7 be an isometry of X. Then 7 is elliptic if and only if 7 has a bounded orbit. Furthermore, if 7" is elliptic for some n > 0 then 7 is elliptic as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 118 P r o o f . Suppose that 7 is elliptic. Therefore we can find an element x E X such that 7 • x = x . This implies that the orbit of x under 7 is the singleton {x} which is clearly bounded. Suppose that 7 has a bounded orbit Y. The set Y is T-invariant so by the Bruhat-Tits Fixed Point Theorem 3.3.1 (1), 7 has a non-empty fixed point set, so 7 is elliptic. To prove the second part of the theorem let 7” be elliptic. By definition, there exists a point 1 6 I such that 7" - x = x . In particular, this implies that the orbit of x under (7) is finite and thus bounded. Applying the first half of the proposition we see that 7 is elliptic, as desired. □ Example 5.2.3. Every isometry of En is semi-simple. Before proceeding, we state a small lemma, which will used in the proof of the theorem which follows. The proof of this lemma may be found in [Bridson & Haefliger pg. 239]. Lemma 5.2.4. Suppose A is a product of the metric spaces X\ and A2. An isometry 7 E Isom(A) decomposes as a product (7 1 ,72 ) where 71 is an isometry of X i and 7 2 is an isometry of A2 if and only if, for every x \ G Ai, there exists a point 71(27) such that 71 ({2 7 } x X f ) = {71(2:1)} x A 2. A semi-simple isometry 7 induces a splitting on Min(7) as we shall see in the following theorem. Theorem 5.2.5. Let A be a complete CAT(O) metric space and consider an isometry 7 of A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTERS. INVESTIGATIONS OF ISOMETRIES 119 1. The isometry 7 is hyperbolic if and only if there exists a geodesic line c : M. —> X whose image is a 7-invariant subspace of X and the action of 7 when restricted to c(R) is a non-trivial translation c(t) c(t + I7I). Such a geodesic line is called an axis of 7. 2. If 7 is a hyperbolic isometry of X then the axes of 7 are parallel to each other and their union is equal to Min(7) 3. If 7 is hyperbolic then Min(7) splits as a product 7 xR and the restriction of 7 to Min(7) acts as (y, t) 1—»• (y, t+ I7I). Moreover, Y is a convex subset of X and hence Y xRisa CAT(O) space. 4. If a is an isometry which commutes with 7 then it leaves Min(7) invari ant. Furthermore, its restriction of to Min(7) = 7 xR splits as ( a ' , a " ) where a' is an isometry of Y and a" is a translation of M. 5. If we take X to be a complete metric space and 7™ is a hyperbolic isometry for some m > 0 then 7 is hyperbolic. P r o o f . 1. (=t>) Suppose 7 is hyperbolic. This means that we can find an element x € X such that d(7 ■ x,x) = I7I. Let c : [0, 17I] —» X be the geodesic path whose corresponding geodesic segment is [x,j ■ x\. Consider the set of paths <7 : [0, |7|] —> X defined by q(f) = 7* • c(t) for every integer i. Since 7 is an isometry, these are geodesic paths and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 120 furthermore, for every integer i we have ci+1(0) = 7 i+1c(0) = 7i(7*c(0)) (5.12) = 7i -c(|7l) = c;(l7l) This means that we can define a line c : M. —>• X to be the concate nation of the paths q. More precisely, if t E M and t = a + b\~f\ where b E N and a E [0, j"y|) then c(t) = Cft(a). Our goal is to show that this line is locally a geodesic. Since for every integer i we know c| [i|'y|,(i+i)|'y|] is a geodesic, it is sufficient to show that for every integer i we have c | [ ( i - i / 2) h i , ( i + i / 2)|7 |] is a geodesic path. In light of Proposition 1.1.4 it is sufficient to show that d{c((i — 1/ 2)I'y|), c((i + 1/ 2)|7 |)) (5.13) d{c((i - 1/ 2)|7 |), c(i|7 |)) + d(c(i\-f\), c((i + 1/ 2)|7 |)) From the triangle inequality we know that d{c((i — 1/ 2)j7 |), c((i + 1/ 2)|7 |)) (5.14) < d(c((i - 1/ 2)|7 |), c(i|7 |)) + d(c(«|7 |), c((i + 1/ 2)|7 |)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 121 but we also know that d{c{(i - 1/ 2)|7 |),c(i|7 |)) + d(c(i| 7 |), c((i + 1/ 2)|7 |)) 1/ 2 |7 | + 1/ 2|7 | It I (5.15) < d(c((i - 1/2)|71), 7 • c((i - 1/ 2)I7 I)) = d(c((i — 1/2) |7|), c((i + 1/2)|7 |)) and thus equality holds and we may conclude that c | [ ( i _ i / 2)|-y|,(i-t-i/ 2)|'y|] is a geodesic path and c is a locally geodesic line. In CAT(O) spaces however, local geodesics are geodesic which means that c is a geodesic line. Furthermore, from the way we defined c it is clear that the action of 7 on the image of c is by translations of length |y| as desired. ( invariant and such that 7 acts on c by translation. Then c(R) is com plete, convex, and 7-invariant and so by Proposition 5.1.2 (4) we have |7 |c (r )| = M and thus 7 is hyperbolic. Furthermore, this implies that the translation length of 7 is the amount by which 7 translates c(M). 2. Let c : R —» X and d : R —»• X be axes of 7. For any f 6 R we have d(c(t),d(t)) = d(j ■ c(t),7 • d(t)) (5.16) = d (c (i + |7l),c'(*+l7l) This shows that the function t 1—> d(c(t),d(t)) is periodic and thus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 122 bounded. Furthermore, it is convex so it must be constant by Proposition 2.2.4 , implying that c and d are parallel. Any axis of 7 is contained in Min(7) so to show that Min(7) is actually equal to the union of all the axes of 7 consider a point x G Min(7). Using the construction in the proof of (1) we find an axis of 7 which contains x. Thus Min(7) is equal to the union of all the axes of 7 as desired. 3. We proved in Proposition 5.1.2 (3) that Min(7) is closed and convex for any 7 G T so by Remark 2.1.6, it is a CAT(O) metric space in the induced metric. By applying the Product Decomposition Theorem 2.5.7 to Min(7) we have Min(y) = 7 x 1 where each {y} x R is an axis of 7. Furthermore, for any (j/,f) 6 7 x 1 we have 7 • (y,t) = (y,t+ I7I) as desired. 4 . Suppose that a is an isometry of X which commutes with 7. The invariance of Min(7) is a consequence of Proposition 5.1.2 (1). We now show that a splits as a product (a.1 ,0 1 2 ). For each y E Y we claim that the set a • ({y} x R) is an axis of 7. If we let c : R —>• X be the map c(t) = (y, t) then c is an axis of 7 and a ■ c(R) = a ■ ({y} x R). Thus we need to show that a ■ c is also an axis of 7. For any t E R we have 7 • (a ■ c(t )) = a • (7 • c(t)) (5.17) = a ■ c(t + |7 |) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 123 so 7 • ( a ■ c(t)) G a ■ c(R) so a ■ c(R) is 7-invariant and a • c is an axis of 7 and thus a ■ ({?/} x R) = {y'} x R for some y' G Y. Thus, if we define ati(y) for each y G Y to be given by ax(y) = y' then we can apply Lemma 5.2.4 to deduce that a splits as (07,0:2) where 07 is an isometry of Y and 07 is an isometry of R. Since 7 which splits as ( i d , 7') where 7' is a translation of R and a 2 commutes with 7' we conclude that a 2 is also a translation of R. This is because the only isometries of R which commute with a translation are translations. 5. Suppose 7m is hyperbolic for some m G N. By (3) Min(7m) splits as a product 7 x R and 7™ splits as ( i d , 7') where 7' is a translation of R. Since 7™ commutes with 7, by (4) we see that 7 splits as (71, 72) where 7™ is the identity on Y and 72 is a translation of R. Note that Min(7) is a complete CAT(O) metric space, as it is a closed and convex subspace of X and thus by the Product Decomposition The orem 2.5.7, Y is a complete CAT(O) space. Since 7™ is elliptic, by Proposition 5.2.2, we know that 71 is elliptic and hence has a fixed point c y G Y. Thus 7 acts by translations on { c y } xR by some constant k > |7|. The slice {cy} x R is an axis of 7™, however, so \'ym \ = m k > m\^\. By the triangle inequality, we have m k = |7m| < m|7| = m k and thus the equality |q| = k holds and 7 is semi-simple. Since m k = |7m| > 0 we deduce that k > 0 and so 7 is hyperbolic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 124 □ Proposition 5.2.6. Let X be a metric space which splits as the product Xi x X 2 and suppose 7 is an isometry which splits as a pair of isometries (7X, 72) preserving this splitting. Then the equality Min (7) = Min(7i) xMin(72) holds. In particular, this implies that 7 is semi-simple if and only if both 7x and 72 are semi-simple. Proof. Let x\,yi G Xi and let :r2, II2 G X 2 then the following two properties hold: d(7i(a;i),£i) < ^ (s/i),^ ) (5.18) x2), (xu x2)) < d{7 {yi, x2), {yi,x2)) and d(72(2:2), 2:2) < d{^2 {y2), y2) (5.19) d( 7 (xi, x2), (xi, x2)) < d('j(x1 ,y2), (xu y2)) Take (xi, x2) G Min(7), then for any y) G X 2 we have d(~i(xi,x2), (xi,x2)) < d{~i(xuy2 ),(xlty2)) (5.20) which using (5.19) implies that d{72(^2), x2) < d('y2 {y2 ),y2) (5.21) so x2 G Min(72). Similarly, using (5.18) one shows that x\ G Min(7i). Conversely, let us take (xi,x2) G Min(7i)xMin(72) and consider any pair Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 125 (2/i, 2/2 ) e Xi x X 2. We have d ( ' y ( x1 , x 2) , { x 1 , x 2))2 = d(7i(a:i),a:1)2 P d { ^ 2{ x 2 ) , x 2 f < rf(7 i(yi),yi )2 + ^(72(2/2), y2)2 (5-22) = d('y(yi,y2),(yi,y2))'2 Thus, (27,2:2) £ Min(j) and we conclude that Min(7)=Min(71)xMin(72) as desired. □ Lemma 5.2.7. Let X be a CAT(O) space with the geodesic extension property. Let T be a group which acts cocompactly by isometries on X and a G Isom(X) be an isometry which commutes with every isometry in T; then Min (a) = X (so a is a Clifford Translation). P r o o f . Consider the displacement function d a of a . For any x G X and any 7 G T we have d a { l - x ) = d ( a ■ (7 ■ ir),7 • x ) = d ( 7 a • x , 7 • x ) V (5.23) = d ( a ■ x , x ) = da{x) which implies that the displacement function d a is T-equivariant. Since the action of V is cocompact, there exists a compact subset K such that Y-K = X. In particular, we can find a ball B ( x 0, r ) D K which gives T • B ( x 0, r ) = X. Let d = d ( a • x 0 , x 0 ) . For each i f l there exists 7 G T such that 7 • x = x ' G Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5. INVESTIGATIONS OF ISOMETRIES 126 B(xo, r). This gives d a ( x ) = d a { 7 ' x ) d ( x ' , a • x ' ) (5.24) < d ( x ' , x 0 ) + d ( x o, a ■ x 0) + d (a ■ x 0,a ■ x') < r + d + r so d a is bounded. The function d a is convex and so, since X has the geodesic extension prop erty, it must be constant by Proposition 2.2.4. This gives us the desired result that d a ( x ) = d a ( y ) for every x , y € X , s o Min (a) = X. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Groups of Isometries In the last chapter we examined the properties of individual isometries of a CAT(O) space A and what they reveal about the structure of this space. In this chapter we will look more broadly at entire groups of isometries which act properly and cocompactly on a CAT(O) space X to find some relationships between the structure of the group and the structure of the space. 6.1 More on the Structure of Isometries We shall first collect some properties of proper and cocompact actions of a group on a metric space X. Proposition 6.1.1. Let A be a metric space and suppose that T is a group which acts properly by isometries on X. 1. If the action of T on A is cocompact then any 7 G T is a semi-simple 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 128 isometry of X 2. Assume that X splits isometrically as a product X' x X" and that each element 7 of T splits as (V, 7"). Suppose that N is a normal subgroup of T formed by elements of the form ( i d x >, n ) for which there exists a compact subset K C X" such that (J (idx,n)eNn ' ^ = X". Then the induced action of T/N on X' is proper (where in the induced action, (7', 7 ")N acts as 7' on X' ). P r o o f . 1. Let A be a compact set such that V • K = X and take a sequence of points {x n} G X such that d 1 ( x n ) —► I7I as n —> 00. Since T • K = X, for each x n we can find x n G T such that x n • x n G K: let Vn = In' xn for each n . We note that d { l n l l n l ' yn, V n ) = d( 77 ^ ' V n , I n 1 ‘ V m ) = d ( - f x n , x n ) (6.1) -»• It I as n — > oo. Furthermore, for every x G K we have d i X n l l n 1 - x , x ) < ^(7n77n 1 ' X, 7n77n 1 ’ V n ) + ^(bn77n 1 ' U rn U n ) + d ( y n , x ) (6.2) < 2diam ( K ) + d (7 • xn, xn) so we see that the sequence {d (,yn'y'yf 1 ■ x , x)} is bounded by a constant M for each independent x G K. Let x q G K such that K C B ( x q , R ) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 129 Then for every n, we have 'yn ' j ' y ~ 1B(xo, R + M ) fl B ( x o, R + M) y 0 by (6.2). By Remark 3.1.6 X is proper so by the properness of the action, this means that the set {7n77n1 I 71 > 0} is finite. By passing to a subsequence, we may assume that 7 n 77 y 1 = 7 f°r every n > 0 and some 7 G R Since K is compact, by passing to another subsequence we may assume that the sequence {7„ • x n } converges to y G K. This gives dirX1 ■ y, 77i_1 • y) = ^(7i77i_1 • y > y) = lim ^oo d (7 ■ yn,yn) = limn^oo d(77“1 ■ yn, 7 " 1 • yn) (6-3) = limn^oo ^(7 • xn, xn) = It| Thus 7 is semi-simple as desired. 2. Let x’ G X' be any point. We wish to show that there exists e > 0 such that the following set is finite: s = {(y, 7")at e r/N | y • b {x ', e) n b {x \ y y 0} (6 .4) Since the action of T is proper by Remark 3.1.4, there exists e > 0 such that the set A = {7 G T | 7 • (B(x', e) x K) n (B(x', e) x K ) y 0} is finite. Suppose (7 ', 7 ")]V G T/N is such that 7 ' • B(x', £) fl R(x',e) y 0. Let Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 130 x" G K. Since U (id n)€Nn ' K = X " , there exists (idX',n) G N such that ny" ■ x" G K. Thus we have both y1 • B(x’,e) fl B(x',e) ^ 0 and ny" ■ K fl K 7 ^ 0 which means that ( 7 ', n'y") is in the set A. Since each coset in the set S has a representative in the finite set A, we can only have finitely many elements ( 7 ', 7 ")N in S and can conclude that the action of T/N on X' is proper. □ T heorem 6 .1 .2 . Let X be a CAT(O) metric space and let T be a finitely generated group acting faithfully on X by hyperbolic isometries (aside from the identity). Suppose T has a central subgroup A = Zn. Then there exists a finite index subgroup H C T which contains A as a direct factor. Proof. We shall prove the result by induction on n, the rank of A. If the rank of A is zero then A is trivial and we may take H = T. Now assume that for any subgroup A! = T? of any group F' acting faithfully by hyperbolic isometries on a CAT(O) space, with j < n — 1, the claim holds. Suppose that A has rank n and fix a non-trivial element a G A. Since a in in the centre of T, Min(a) is T-invariant. Furthermore, Min(a) splits as a product Y x R and the restriction of each 7 G T to Y x i splits as (7', 7") where 7" is a translation of R by Theorem 5.2.5. In particular, a acts as a non-trivial translation on {y} x R for each y G h We can now map T onto a finitely generated group of translations of R via the homomorphism 7 1—> 7". Since T is finitely generated, this group Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 131 of translations is isomorphic to Zm for some integer m > 0, so we define ip : T — > Zm in the obvious manner. The image i p ( T ) is non-trivial because i p ( a ) is non-trivial. Since the image ip (a) is non-trivial, we may find a suitable direct factor in Zm such that if (p : T —> Z is the composition of tp with the projection onto that factor, then the image a e A such that 0(a) generates 0(A). Let us say 0(a) = i and (p(A) = iZ, i 7 ^ 0. Let H0 = 0_1(0 (A)) = 0~1(zZ). Then [T : H0\ = [Z : iZ] = i and so H0 has finite index in T. Let / : homomorphism as 0(a) freely generates 0 (A). We then have the following split short exact sequence where K = ker 0. 1 -> K -» H0 -► 0 (A) -> 1 (6.5) Clearly (a) Cl K is trivial. Thus we have that H0 is the semidirect product of K and (a). But as a is central, this reduces to the direct product, H0 = K x (a). Thus A = (K fl A) x (a), so if we let A' = iL fl A then since A = A' x (a) we know that A' has rank n — 1. So by the inductive hypothesis applied to A' and K there exists a finite index subgroup H' of K such that H' splits as H' = H" x A'. This gives us H = H' x (a) = H" x A' x (a) = H" x A so all that is left to show is that H has finite index in T. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 132 We know however, that H' has finite index in ker p so II' x (a) has finite index inkx (a). But H = H' x (a) and H0 = K x (a), so H has finite index in Ha. Since H0 has finite index in T we may conclude that H has finite index in T to complete the proof. □ As a corollary to the first part of the proof to Theorem 6 .1.2 we have the following: Corollary 6.1.3. If T is a finitely generated group acting by isometries on a CAT(O) space X and the centre of T contains an element acting by a hyperbolic isometry, then X contains a subspace isometric to M. and there is a non-trivial homomorphism ip : T —» R. The above corollary applies in particular when T acts properly and cocom- pactly on a CAT(O) space and T has a central element of infinite order by Proposition 6.1.1 6.2 Splitting CAT(O) Spaces The goal of this section is to prove a splitting theorem for CAT(O) metric spaces. We begin this section by presenting a lemma which is used in the proof of the splitting theorem. The reader can refer to [Bridson & Haefliger pg. 239] for the proof. Lemma 6.2.1. Suppose that X is a CAT(O) space with the geodesic exten sion property and that T is a group which acts properly and cocompactly by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 133 isometries on X. If T has a torsion-free subgroup of finite index and trivial centre, then the centralizer of F in Isom(W) is trivial. The following lemma outlines another property which will be used later. Lemma 6.2.2. Suppose X is a complete CAT(O) metric space which has the geodesic extension property. If T is a group which acts cocompactly on X by isometries then the only closed convex, T-invariant subsets of X are the empty set and the entire space X. Proof. Let C be any non-empty, closed, convex, T-invariant subset of X. We wish to show that C is equal to the entire space X. Let it : X —> C be the orthogonal projection onto C. Since the action is cocompact there exists a ball B(x o, r/2) C l such that T • B(x0 ,r/2) = X. In particular, as C is T-invariant, B(x0 ,r/2) DC^0. For every i g l w e can find an element 7 G T such that 7 • x G B(x0 ,r/ 2). Thus d(j -x,C) xeX. Now suppose that there exists x G X — C and consider the non-constant geodesic segment [x,tt(x)]. Since X has the geodesic extension property, we can extend this geodesic segment to a geodesic ray c : [0, 00] —> X w ith c(0) = 7t(x ). For any t > 0 we have it(c(t)) = tt{x) by Proposition 2.3.1(2). This implies that d(c(r + 1),C) = r + 1 contradicting our conclusion that d(z,C) < r for every z G X. This means that no such i G l - C exists and C = X as desired. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 134 Lemma 6.2.3. Let T be a group acting by isometries on a CAT(O) space X. Let C be a T-invariant, closed, convex subspace of X. Then the projection p : X —► C is T-equivariant. Proof. Let x E X, then we have d(j ■ T P(7 • z)) Similarly, d (7 • x, 7 • p(x)) = d(x,p(x)) < d(x, 7 _1 • p(y • x)) = d(y ■ x,p (7 • x)) (6.7) Thus we combine ( 6 .6 ) and (6.7) to deduce that d(^-x,p(^-x)) = d( 7 -x, 'j-p(x)) and since projections are unique, we conclude that p(j • x) = 7 • p(x). □ Before presenting the splitting theorem, we need two technical results that will be used in the proof. Proposition 6.2.4. Let X be a complete CAT(O) space with the geodesic extension property and let T be a group which acts properly and cocompactly on X. If the group T splits as T = Ti x f 2 and there exists a closed convex hull C of a Ti-orbit in X such that the action of lb on C is cocompact, then X splits as a product of metric spaces X\ x X 2 and T preserves this splitting. Furthermore, for this splitting the closed convex hulls of the L i-orbits are precisely the sets of the form X\ x {x2}. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 135 Proof. Let E be the set of all non-empty closed, convex, Ti-invariant subspaces of X and let N be the subset of E containing all those sets which are minimal in E with respect to inclusion. If C\ and C 2 are in E then they are both closed, convex, and Fi-invariant so their intersection C\ fl C2 is also closed, convex, and Ti-invariant and thus in E. This implies that since the intersection of any two elements of E is again in E, we have that the elements of N are disjoint. Additionally, for any C' G N if x G C' is an element then the closed convex hull of the IVorbit of x, C(Id • x) is contained in E and is also a subset of C' by definition. Since C' is minimal under inclusion however, we conclude that the equality C' = C{Ti- x) holds and thus all the elements in N are the closed convex hulls of some Tx-orbit in X. Claim 1. The set N is non-empty. Let us take C to be the closed, convex hull of a Tx-orbit as in the proposition and let A be a compact subset of X such that Ti ■ K = C. Let M be the set of subspaces in E which are contained in C and consider a decreasing sequence of such subspaces in M: Co 2 C1 D C2 D C3 ... (6 .8) For each Cn in the sequence we have Cn C C = Tx • K. Since Cn is non empty, we pick xn G Cn and find 7,, G Tx such that qn • xn G K. Since Cn is Tx-invariant however, we have • xn G Cn so Cn fl K ^ 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 . GROUPS OF ISOMETRIES 136 It follows that c0nKDC1nKDC2nKD... (6.9) is a decreasing chain of closed subsets of K and hence has a non-empty inter section by the compactness of K. Thus C' = Di^o Cn 7^ 0- As an intersection of closed sets, C' is itself closed. To show that C' is convex, consider x,y £ C'. We have x. y £ Cn for every n > 0 and since Cn is convex for each n, we have [x,y\ £ Cn. So [x,y\ £ C' and C' is convex. Similarly, for every x £ C' and 7 £ b we know that x £ Cn, which by the T-invariance of Cn implies that 7 • x £ Cn for each n > 0. Thus j-x £ C', so C' is Ti-invariant. This tells us that C' is an element of E but C C C so C £ M. By Zorn’s Lemma, we can now conclude that M has a minimal element. This element will also be minimal in E so we have that N is non-empty as desired. This completes the proof of Claim 1. Consider two elements C\, C2 of N and let pi : X C, be the orthogonal projection. Let d = d(Ci, C2 ) be the distance between the two sets. C laim 2 . There is a unique isometry j : Cj x [0, d] —> C(C\ U C2 ) such that j(x, 0) = x and j(x, 1) = p 2 (x). Consider the function dct : C*2 —> M given by dc^x) = d(x,C\ ) for each x £ C2 . We claim that this function is constant. If it were not constant, then there would exist elements x,y £ C2 such that dc^ix) < dc^y). Let us consider the set A = {z £ C2 : dc^z) < dCl(x)j, note that x £ A and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 137 y tfz A. Since the function dc 1 is continuous, this set is clearly closed in C2 and thus closed in X. Furthermore, the convexity of dcx implies that A is convex. Indeed, if z1 )z2 E A and c : [0,1] —> C2 is a linearly reparamaterized geodesic path originating at zi and terminating at z2, then by the convexity of dc±, for any t E [0,1] we have dcA^t)) < (1 -t)dCl(zi) + tdCl(z2) < (1 -t)dCl(x)+tdCl(x) = dCl{x) (6 .10) thus c{t) E A and [zi, z2] C A so we conclude that A is convex. Moreover, for any 7 G Fi we have dCi( l'x) = d(7-i,Ci) = d(x,Ci) (6-11) = dCl(x) so A is Fi-invariant. Thus A is a non-empty, closed, convex, Ti-invariant subspace strictly contained in C2 which contradicts the minimality of C2. Thus we may conclude that dc1 is constant on C2, so dc-L (■'?’) = d for all x G C2. We shall now define an isometry j : C\ x [0, d] —» C(C\ U C2) as follows. For any (x,t) G Ci x [0, d] we let j(x,t) = cx(t) where we define the path cx : [0, d] —> C{C\ U C2) to be the unique geodesic path which originates at x and terminates at p 2 (x). Note that we are only able to define this map because d(x,p 2 (x)) = d. To show that this map is an isometry, consider any two points (aq ,ti), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 138 (x2 * t2) G Ci x [0, d]. If we consider the quadrilateral formed by the geo desic segments [xu x2], [xi,p2{x i)], [^2(^1),^2(^2)], and [^ 2(^2), ^2] then by Proposition (2.3.1)(3) we see that each angle in this quadrilateral is greater than or equal to 7r/2 which guarantees that the sum of these angles is greater than or equal to 2-7T. We can now apply the Flat Quadrilaterals Theorem 2.5.3 to see that j , when restricted to [x\. x2] x [0, d] is in fact an isometry, so d(j((xi,ti)), j({x2 ,h)) = d((xi,ti), (^2,^2)) and C{C\ U C2) is isometric to Ci x [0, d]. This completes the proof of Claim 2. Claim 3. For every x G X there exists a unique element of N, denoted Cx which contains x. The set Cx is equal to the closed convex hull of the Tx-orbit of x, C(Ti • x). If such a Cx exists for each x G X, its uniqueness is clear as the elements of N are disjoint. To show that such a Cx exists for each x G X we let B = U{Cj : Cj G N}. It is sufficient to show that B = X; this follows from Lemma 6.2.2 once we show that B is closed, convex, non-empty and T-invariant. Clearly B is non-empty. To show that B is T-invariant, note that for each x G B the set Cx — C(Ti • x) is contained in N. Then for each 7 G T, it is sufficient to show that 7 • Cx G iV. We know that 7 = a/3 where a G Ti and f3 G T2 so 7 • Cx = P ■ Cx since Cx is Ti-invariant. Since (3 is an isometry, we know that (3 ■ Cx = C((3 ■ Ti • x) = C(Ti • (/3 ■ x)) is a closed convex Tx- invariant subspace of X. This set is also minimal in E for if it were not, then there would exist a non-empty closed convex Fx-invariant subspace C' of X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 139 such that C' C C(Ti ■ f3 ■ x). This would give j3~l ■ C' C C contradicting the minimality of C. Thus B is T-invariant. To show that B is convex, consider x,y E B, so Cx, Cy E N. By Claim 2 we know that C'(C,iUC'2) is isometric to Cy x [0, d] via an isometry j : CyX [0, d] —> C(CiUC2). Furthermore, any subspace of the form j(Cy x {t}) is Ti-invariant because each element 7 € Ti takes the geodesic segment[x,p 2 (x)\ isometrically to the geodesic segment [y-x,'y-p2{x)\ = [y-x,p2(i~f-x)\. Moreover, j(Cy x {t}) is closed, convex, and non-empty. If it were not minimal then its projection onto C\ would contradict the minimality of Cy. Thus C(CX U Cy) C B and B is convex. Now we prove that B is closed. Let {xn} be a sequence of points in B which converges to x E X. We wish to show that x E B. By passing to a subsequence, we may assume that d(xn,xn+1) < l / 2n for each integer n > 0. For each element xn in our sequence, let Cn be the element of N containing xn and let pn : X —> Cn be the orthogonal projection. Consider the composition, P n P n -i ■ ■ ■ P3P2 '■ X —> Cn and let Pn : C\ —»■ X be the restriction of this function to Cy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 140 For each z E Ci the sequence Pn{z) satisfies the following inequality: d(Pn(z),Pn+k(z)) < d(Pn(z),Pn+ 1 (z)) + d(Pn+1 (z),Pn+2 (z)) d{Pn+2 {z)i Pn+3 (z) + • • • + d(Pn+k-l(z), Pn+k(z )) = d(Cn, Cn+i) T • • • 4" d(Cn+k~i, C'n+fc) ^ d(xn, X yj-i-x) "F ■ • ■ “I- d(xn+k—i , xn+k) < 1/2"-1 (6.12) Thus {Pn} is a Cauchy sequence in the space of continuous functions from Ci to X with the uniform metric. Since X is complete, so is the function space [Munkres] and thus Pn converges to a continuous function P : Ci —> X. Moreover, P is Ti-equivariant since for every y 6 C'i and 7 G T1 we have the following: P ( i • y ) = limn^oo P n { i ■ y ) = lim n^ool-Pn{y) (6.13) = 7 • lim ^oo Pn(y) = 7 -P(y) where the second equality is because Pn is Ti-equivariant and the third equality is because 7 acts as a continuous map. Now we shall show that P is an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 141 isometry. For any y, z G C\ we have d(P{y),P(z)) = d(limn_+ 00(Pn(?/),Pn(,z)) = linin—KXj d(Pn(r/), Pn(^)) (6.14) = <%*) This implies that P(C'i) is not only F| -invariant, but also closed, convex and non-empty. Furthermore, it is minimal in E as P -1 would take any closed convex Fx-invariant subset of P(Ci) to an analogous subset of C\. Thus we conclude that P(C'i) G N. The final step is to show that x G P(C i), so we shall construct a sequence in the closed set P(C'i) converging to x. For each xn in our original sequence, let zn G C\ be the unique element such that Pn{zn) = xn. Then we find that d(P(zn),x ) < d(P(zn),xn) + d(xn,x) = d(P(zn), P, i(zn)') T d(xn, x) (6.15) < l / 2n_1 + d(xn, x) which decreases to zero as n —> oo so we conclude that x G P{C\) and thus B is closed. By Proposition 6.2.2 we now know that X = B so for all x G X we may find Cx G N such that x G Cx. Claim 4. For any three sets C\,C2 ,Cs G N if we consider the projection maps Pi : X —► Ci then the equality pi = pip 2 holds when restricted to C 3 If each Ci G N is a single point then the claim holds trivially. Otherwise, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 142 we may assume that some element of N, and hence each element of N by Claim 2, has at least two points. This means that for each x E C3 we can find a distinct element y E C3 and thus extend the geodesic segment [x, y] to a non-trivial geodesic line c : E —> X. For any C E N such that C ^ C3, wo know by Claim 2 that C{C^CC) is isometric to C3 x [0, d] where d = d(C3, C). This implies that c(R) cannot intersect C and thus the line c is completely contained within C3 as X = p| N. Claim 2 also guarantees that the lines p i c and P 2 C are asymptotic to c and thus, by Proposition 2.5.6, we know that the following equality holds P3,l|pic(R) °Pl,2|p2c(M) 0 4*2,3|c(R) — P3,3|c(R) ( 6.16) and so Pi,2 |p2c(R) 0 P2,3 |cR) — Pl,3|c(R) (6.17) Therefore p i ( x ) = PiP2{x) for every x E C 3 and thus p i = p i p2 when restricted to C 3 , as desired. This completes the proof of Claim 4. We are now prepared to define the splitting of our CAT(O) space X. Fix any X\ E N and let p : X —> X i be the orthogonal projection. Then p is Fj-equivariant by Lemma 6.2.3. Let X 2 be the metric space ( N , d ' ) where the distance function is given by d ' ( C , C ' ) = inf {d(x, x') \ xE C , x ' E C"} (6.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 143 Note that Claim 2 implies that for any x G C we have d!{C, C') = d(x, p'{x)) where p' : X C' is the orthogonal projection. To prove that X2 is a metric space: 1. For any C,C' G N the inequality d'(C,C') > 0 holds trivially, as does symmetry. 2. Clearly, for any C G N we have d'(C, C) = 0. Furthermore, if for some C,C' G N the equality d'(C,C') = 0 holds, then by Claim 2, for any x G C we have d(x,p'(x)) = 0 which implies that x G C'. Hence C = C' since elements of N are disjoint. 3. To show that the triangle inequality holds, consider C, C', C" G N and let x G C. If j/ : X C and p" : X C" are the orthogonal projections onto C' and C" respectively, then we have the following d'(C, C") = d{x,p"{x)) < d(x,pf{x))+d(p?(x),pf'(x)) (6.19) = d(x,p'(x)) + d(p'(x), p"(p 1 (x))) (Claim 4) = d'{C,C') + d\C',C") so the triangle inequality holds, as desired. We have shown that ( N , d') is a metric space. The next step is to define an isometric action o fr = r i x r 2on metric space X\ x X2. We wish for the action to preserve the splitting, so we need to do this by defining an action © Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 144 of Ti and T2 on each of the factors separately and showing that each of these actions are by isometries. Since Xi is Ti-invariant, we can define the action of Fi on Xi to be the restriction to Xi of the old isometric action, so for every 7 G r\ and every x G Xi we define the action © : Id xX \ —» Xi to be y©x = 'y-x. Furthermore, each C G IV is Fi-invariant, so we define the action of Fi on X 2 to be trivial, which is clearly isometric. The action of F2 on X 2 is defined by 7 © Cx = C-f.x for each 7 G T2 and Cx G N. Note that as sets, we have 7 • Cx = 7 © Cx for any 7 G T2. This is because 7 • Cx is minimal, closed, convex, T]-invariant (since Ti and T2 commute and Cx is Ti-invariant) and thus in N . Furthermore, 7 • x G 7 ■ Cx so by the disjointness of elements in N we have 7 G Cx = CTX = 7 ■ Cx. This implies that the action is well-defined. Let Cx,Cy be any two elements of N and consider 7 G T2. If we define p1.v : X —> C1.y to be the orthogonal projection onto Cry then we have d(CTX, CTy) = d(7 ■ x,pTy(7 • x)). Since 7 maps Cy surjectively to Cry we can Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 145 find z G C y such that 'j ■ z = Pj.yi'j ■ x). This give us d(7 © Cx, 7 © C y ) = d ( C T X , C T y ) = d{-f ■ x ,p r y (-f ■ x)) = • X j'y ■ z ) (6 .20) = d ( x , z ) > d { C x , C z ) = d ( C x , C y ) By replacing C x by 7 © C x . C y by 7 © Cj, and 7 by 7_1 we obtain the opposite inequality and can thus conclude that the action is by isometries and since the action of Id on X 2 is trivial, the actions commute. We define the action of lb on X] to be given by 7 © x = p(7 • x ) for every 7 G r 2 and x G X\. Let 71, 72 G T2. Consider Figure 6.1 where p :72 -X\ —► X i and p' : 7172 -C\ —»• 71-Ci are the restrictions to 72 - X i and 7 ^ 2 - X ± respectively of the orthogonal projections onto X\ and 71 • X\ respectively. Since 71 is an isometry, this diagram commutes. Thus, for any X\ G X\ we have 7i ‘ p {l2 ■ a:i) = p '(7i72 • aq) =► p (71 • p(72 • X i ) ) = p(p'(7i72 • 27)) (6.21) =7 p (7i ' P {72 • aq)) = p(7i72 ■ a:i) (by Claim 4) = > 7i © (72 © x i ) = (7172) © X i and the action is well-defined. Furthermore, this action clearly commutes with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 146 7 i 72 ■ X: 7 7i • Xi Figure 6.1: Commutative Diagram for 72 , p. and p' the action of Ti on X\ since p is Ti-equivariant. To show that the action is by isometries, consider x, y 6 Xi and 7 £ T2. We already established that 7 • x and 7 ■ y he in the same element 7 • X-L of N. Furthermore, by Claim 2, the projection p when restricted to 7 • Xi is an isometry, so we have d(~fQx,~f@y) = d(p('y-x),p('y-y)) = d{pf • x, 7 • t/) (6 .22) = d(x,y) so the action is by isometries, as desired. The last step of the proof is to show that there exists a T-equivariant isometry f : X —> Xi x X 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 147 We define / to be the map x i—► (p(x),Cx) which is clearly T-equivariant since for every 7 = 7172 with 71 £ Id and 72 £ T2 we have /( 7 i72 ■ x) = (p(7172 • x), Clll2.x) = 7i72 © x, 7172 © Cx (o.zoj = (7172) © (p{x),Cx) = (7172) ©/(a:) Now to show that / is an isometry, let x, x' £ X and let p' : X —>■ Cx> be the orthogonal projection of X onto Cx>. The by Claim 2 we have d{x,x')2 = d(x,p'(x ))2 + d(p'(x), x ')2 = d(C2, C ^ ) 2 + d(pp'{x),p(x '))2 = d(Cx, Cx>)2 + d(p(x),p(x '))2 (by Claim 4) (6.24) = d((p(x),Cx), (p(x'),CX’)) = d{f{x),f(x’)) which means that / is a T-equivariant isometry as desired. Finally we show that / is surjective. If (x, Cy) £ X\ x then p : —> Xi is an isometry. Thus, there exists z £ Cy such that p{z) = x. Then f(z) = (p(z), Cz) = (x, Cy) and / is surjective. This means that the splitting X = X\ x X2 has the desired properties and the proof is complete. □ We now present a second technical lemma. It will be used to satisfy the requirement in the previous proposition that the action of Tx on the closed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 148 convex hull of some IVorbit be cocompact. Lemma 6.2.5. Let X be a proper CAT(O) space and T = Ti x F 2 be a group acting properly on X by isometries. Suppose moreover that the centre of T 2 is finite. Let C C X be a closed, convex, F i -invariant subspace for which there exists a compact subset K C X such that C C T ■ K. Then the action of Ti on C is cocompact. Proof. The proof shall proceed by contradiction; suppose that no compact set K' C C such that T\ ■ K' — C exists. Since K is compact we can find x £ K and d > 0 such that K C B(x,d). Construct a sequence of compact subsets of C as follows. For each n > (J set Kn = B(x , d + n) n C (6.25) Since X is proper, the closed ball B(x, d + n ) is compact for each n > 0 which means that since Kn is the intersection of a closed set and a compact set, it is also compact. Since Kn C C, the F i -invariance of C implies that F 1 ■ Kn C C. Our initial assumption then implies that C — (Fi • Kn) ^ 0. Now we can construct a sequence {xn} such that for each n > 0 we have xn £ C and xn Ti • Kn. Since xn ^ Ti ■ Kn for each n > 0, we have 7 ■ xn Kn for any 7 G Ti. Since 7 • xn £ C this implies that 7 • xn B(x, d + n). So if k £ K and 7 € T\ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 149 we have d(7 • xn, k) + d > d(^n ■ xn, k) + d(k, x) > d(r/-xn,x) (6.26) > d + n and thus d(xnTi ■ K) > n which implies that d(xn, Ti ■ K) —> oo as n —> oo. Since C C T ■ K, for each n > 0 we can find a „ e r such that an ■ xn E K. As T = Ti x r2, we can write an = 7 n(3n where 7 „ E Ti and (3n E T2 and thus InPn • xn E K which implies that (3n ■ xn E Ti • K. We fix this sequence {/3n}- Fix xn and e > 0. We can find z E T\ ■ Kand y E j3 ~l ■ K such that d(x, y) < d(Ti ■ K , /5" 1 • K) + e. This gives: d{xn) r \ • AT) d(-yn • ■ A") < ^ ( 7 n ’ Xn, Z) < d{^n ■ xn, y) + d(y, z) (6.27) < d(pnr/n ■ Xn, Pn-y) + d(T i• K, (3~l ■ K) + e < diam (AT) + d(Pl • K, j3~l ■ K) + e Since d(f3~l ■ K,Ti • K) < d{f3~l ■ K, K) the above inequality implies that d(xn, Ti • K) — diam (K) < d((3~l • K, K ). Since d(xn,T 1 • K) —► cxd as n —»• 00, we conclude that d(f3~1 • K, K) —>• 00 as n —> 00. This means that by passing to a subsequence, we may assume that the elements of {Pn} are distinct. For each (3 E T2 consider the displacement function dp. For any 7 G IT Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 150 and y £ X we have dffii-y) = • y,i • y) = d{^(5 ■ y , ^ ■ y) (6.28) = d(P ■ y, y) = dp(y) This implies that dp is constant on any IVorbit in X. Let r = ma.x{dp(y) \ y £ K}. The compactness of K guarantees that r is defined as dp\x ■ K —> M is continuous. Let us define the set U C X as follows U = {y£ X | dp(y) < r} (6.29) Since dp is continuous, this set is closed. Furthermore, it was shown in (6.28) that dp is constant on L|-orbits, so U is Ti-invariant. Finally, we note that for any it, v £ X and any linearly reparameterized geodesic path c : [0,1] —» X originating at u and terminating at v, if z = c[t) then by the convexity of dp we have dp(z) < { t- 1 )dp(u) + tdp(v) < r (6.30) so z £ U and U is convex. By definition we have K C U so since U is closed, convex, and Fi-invariant, we have C{T\ ■ K) C U. Thus, we conclude that dp is bounded by r on C(Fi • K). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 151 Now we have for any y 6 K d{PnPPn 1 • y, y) < d(pnppn 1 • y, PnPPn 1 ■ (A»7n • Xn)) S ~ d ( p n [3 Pn • (Pn'Jn ‘ Xn ), P r d n ' X n ) + d(/?nyra • X n , y) = 2d(/5nyn • £n, y) + dpynPnP • X n , y nPn ' *^n) < 2 diam(K) + d ^ nPn ■ xn) < 2diam (K) + r (6.31) Let K" = B{x, 2 diam(iL) + r). Since X is proper, X" is compact and since r 2 acts properly on X, the following set is finite A = {7 e r2 I 7 • K" fl K" ± 0} (6.32) Since for each Pn we have that d ^ ^ - i is bounded by 2diam (K) + r on K, we have d(x, PnPP^1 • x) < 2 diam K + r so each element PnPP “ 1 is contained in A. Since A is finite, by again passing to a subsequence, we may assume that PnPPn 1 = PmPPm f°r eaC^ n,m > 0. Thus, PppPnP — PPmPn &nd we have found an infinite number of distinct elements which commute with P. We repeat this process for each member of the finite generating set for T 2 restricting to a further subsequence {Pn} each time. In the end, we are left with infinitely many distinct elements of the form PppPn £ Z(r2). This contradicts our assumption that T 2 has a finite centre. Thus our initial assumption that no such K' exists was false and the proof is complete. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 152 We have now come to the statement and proof of the major result of this section. Theorem 6.2.6 (Splitting Theorem). Let X be a complete CAT(O) space with the geodesic extension property and suppose T = Li x L 2 is a group which acts properly and cocompactly on X by isometries. If one of the following holds 1. T has a finite centre and Id is torsion-free 2. the abelianization of Id is finite then X splits as a product X = Xi x X 2 and L preserves this splitting: the action of T = Id x T 2 is the product action on X = X\ x X 2 and both the actions of Id on X 2 and T 2 onXi are trivial. Proof. (1) Note that by Remark 3.1.6, X is a proper metric space. Suppose T has a finite centre and Id is torsion-free. Since Id commutes with Id, we have Z(Id) C Z(T) and thus the centre of Id is finite. If we pick any point x G X and let C = C(Ib • x) then since T acts cocompactly there exists a compact set K C X such that T • K = X. In particular, C C T ■ K and we can apply Lemma 6.2.5 to conclude that the action of Id on C is cocompact. Since C is the closed convex hull of a Id-orbit we may now apply Proposition 6.2.4 to obtain a splitting X = X\ x X 2 which is preserved by T and such that the subspaces of the X\ x {x2} are the closed convex hulls of Lj-orbits. Thus the action of Ti on X 2 is trivial. To complete the proof, we only need to show that the action of Id on X\ is also trivial. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 153 Since the centre of is finite and is torsion-free, its centre must be trivial. By Proposition 6.2.4, the action of Ti on X\ is cocompact since Ti acts cocompactly on the closed convex hulls of Ti-orbits. We have thus satisfied the requirements of Lemma 6.2.1 and can conclude that the centralizer of Id in Isom(Xi) is trivial. Moreover, since p2 commutes with Ti we can conclude that its image in Isom(Xi) is in the centralizer of the image of Ti in Isom(Xi) and thus by Lemma 6.2.1, T2 acts trivially on X\ as desired. This completes the proof of the first part of the splitting theorem. (2) Suppose that the centre of T is infinite. Since the abelianization of Ti is finite, the centre of Ti cannot have any elements of infinite order. Indeed, by Proposition 6.1.1 such an element 7 must act as a semi-simple isometry of X. Since 7 has infinite order and the action is proper, by Proposition 3.1.5 it cannot stabilize a point. Hence 7 acts as a hyperbolic isometry so by Corollary 6.1.3 there is a non-trivial homomorphism Ti —> M, which contradicts the assumption that the abelianization of Ti is finite. By Corollary 3.3.2 Z(Y) has only finitely many torsion elements. As the centre of P is infinite, it must therefore contain an element (07, 07) with infinite order. Since a \ G Z i T ] ) has finite order, a 2 G Z ( T 2 ) has infinite order. Since a 2 has infinite order it cannot be elliptic since by Proposition 3.1.5, the isotropy subgroups of T are finite. By Proposition 6.1.1 a 2 it is semi-simple so it must be hyperbolic and thus by Theorem 5.2.5 Min(a2) splits as Y x R where Y is a convex subspace of X. The element a 2 commutes with T so by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 154 Lemma 5.2.7, a 2 is a Clifford translation and Min(a;2) = A. In particular I = F x R and since each 7 G T commutes with a2, the element 7 splits as 7 = (7', 7") where 7" is a translation of R by Theorem 5.2.5. Since the abelianization of Id is finite, the homomorphism 7 1—> 7" taking each element of Ti to a translation of R must be trivial. In particular, the closed convex hulls of IVorbits are contained in slices of the form Y x {£}. An embedded flat in a metric space A is a subset of X which is isometric to Rn for some integer n. We shall now prove by induction on the maximum dimension of embedded flats in X , that the action of Id on the closed convex hulls of r ]-orbits is cocompact. To see that there is in fact a maximum di mension of embedded flats, refer to [Bridson & Haefliger pg. 247]. If there are no embedded flats in X then the centre of T must be finite as we have just shown that is Z(T) is infinite then X = Y x i Then we may apply Lemma 6.2.5, as in the previous case, to conclude that the action of Id on any closed convex hull of a Ti-orbit is cocompact. Suppose that the assertion holds for every group T = Id x T2 such that the abelianization of L is finite, where T acts properly and cocompactly on a CAT(O) space whose maximum dimension of embedded flats is n — 1. Now suppose that the maximum dimension of embedded flats in X is n . If the centre of L is finite, we can use the same argument as the base case to show that the action of Ifi on the closed convex hull of a r,-orbit is cocompact. If the centre is infinite however, we can find an element a 2 £ Z(T2) of infinite order, as described above to get the splitting of A as a product A = Y x R . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 155 Since a 2 acts trivially on Y we can define an action of Y / { a 2 ) = x r 2/ ( a 2) on Y by 7'(0:2) • y = 7' • y for all y E Y. By Proposition 6.1.1 (2) this action is proper. Furthermore, we can show that the action is cocompact. Indeed, there exists a compact set K C X such that T • K = X . The projection K' of K to Y is compact and for every (x, t) E Y x R there exists 7 = (7', 7") E F such that 7 • (x, t ) = (7' • x, 7" -t) E K. Therefore, for every x' EY there exists (7',7")iV E T/N such that 7' • x = (7', 7 " ) N ■ x E K'. We have thus defined a proper cocomapact action of F / { a 2 ) = Ti xT2/ (012) 011 the proper and complete CAT(O) space Y and clearly the abelianization of F\ is still finite. Since X has the geodesic extension property, given any distinct elements y i , y 2 E Y the geodesic segment [(2/1, 0), (2/2, 0)] can be extended to a geodesic line and since X = Y x R , this line will be contained in the slice Y x {0}. Thus Y has the geodesic extension property and so does Y. Since X = YxR, the space Y has a maximum dimension n — 1 of embedded flats and so we have satisfied all the requirements in the inductive hypothesis. Therefore, we can apply the inductive hypothesis to deduce that the action of Ti on the closed convex hull of a Tx-orbit in Y is cocompact. Since each T -L orbit in X is contained in a slice Y x {£} ofY x E we conclude that the action of Ti on the closed convex hull of some Fj-orbit in X is cocompact. We can now apply Proposition 6.2.4 to obtain a splitting of X = X\ x X 2 such that T preserves this splitting and the action of Ti on X 2 is trivial. To complete the proof, we must show that the action of F2 on X\, as defined in the proof of Proposition 6.2.4, is trivial. Let p : X —» X\ be the orthogonal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 156 projection; under our splitting, p is the projection to the first coordinate. For each x G X we need p(j2 ■ x) = x. If we can find a closed convex hull of a F 1 -orbit, C = X 1 x {x2} such that py2 restricts to the identity on C then for every (xi, x2) G C we shall have 72 • (xi, x2) = (p ( 7 2 • X\), 72 • x2) = (xi, 72 • x2) and thus 72 © x± = x\ for every x\ G Xj. By Proposition 6.1.1 (1), 72 is a semi-simple isometry of X. If 72 is elliptic, then take x — (x i,x 2) G Min(72). Let C be the closed, convex hull of Ti • x2. Then 72 • x = x and thus 72 • C = C since 72 commutes with T 1 and hence takes closed, convex hulls of Tj-orbits to closed convex hulls of T i-orbits. Thus, PI2 = P on C. If 72 is hyperbolic then Min(y2) splits as Y' x R. As we saw earlier in the proof, Ti preserves this splitting, and its action on the second factor is trivial. Thus each Ti-orbit in Min(72) is contained in a slice Y' x {t}. If we take x G Min(72) and the set C = C (ri • x), then the action of 72 on C is trivial and thus p72 when restricted to C is p. We have shown that the action of T2 on X\ is trivial and this completes the proof of the splitting theorem. □ 6.3 The Flat Torus Theorem Theorem 6.3.1 (The Flat Torus Theorem). Let A be a free abelian group of rank n which acts properly by semi-simple isometries on a CAT(O) space X. Then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 157 1. Min (A) = f]aeA Min(a) is non-empty and splits as a product Y x En. 2. Every a £ A leaves Min(A) invariant and respects this decomposition. The action of a on the first factor Y is trivial and the action on En is a translation. 3. The quotient of each n-flat {y} x En by the action of A is an n-torus. 4. Any isometry of X which normalizes A leaves Min(A) invariant and preserves the product decomposition. 5. If a subgroup T C IsorriA normalizes A then a subgroup of finite index in T centralizes A. Moreover, if T is finitely generated and contains A, then T has a subgroup of finite index that contains A as a direct factor. Proof. The proof of (1), (2), and (3) will proceed by induction on the rank of A. If the rank of A is zero then A = {1} and the claims hold trivially. Suppose (1), (2), and (3) hold for any free abelian group of rank n — 1. Suppose a £ A is elliptic. Then a stabilizes a point and by Proposition 3.1.5 (4) the isotropy groups of points are finite and thus a has finite order. Hence a = 1 as A is torsion-free. Since a is semi-simple, we conclude that every non-trivial element in A is a hyperbolic isometry. Since A is a free abelian group of rank n we can choose a set of free generators op, a 2, • • • > Z x E1 where op acts trivially on Z and as a translation on E 1 by Theorem 5.2.5 (3). Moreover, every a £ A commutes with a.\ so each a splits as (a', a") Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 158 where a1 is an isometry of Z and a" is a translation of E 1 by Theorem 5.2.5 (3) again. Consider the subgroup N C A consisting of precisely those elements which act trivially on the first factor Z. By Proposition 6.1.1 (2), N acts properly on E 1. The subgroup IV is a free abelian group with rank less than or equal to n. Suppose N has rank of at least 2 and suppose rq and n 2 are free generators; they will be translations of E by rji and r /2 respectively. Since rji and rj2 are Z-independent, they are Q-independent so by Kronecker’s Theorem [Hardy & Wright] the set Zrji + Z772 is dense in M. This means that for every e > 0 there exist non-zero integers a, b such that n“n .2 acts on E 1 as a translation of length less than e. This contradicts the properness of the action and thus N is cyclic. Since aq is primitive in A it is a generator for N. Moreover, the action of N = (o:j) on E is cocompact with quotient a circle. The free abelian group Aq = A/N has rank n — 1. By Proposition 6.1.1 (2), its induced action on Z is proper and by Proposition 5.2.6 this action is by semi-simple isometries. Finally, as a convex subspace of X, the space Z x {0} = Z is CAT(O). By induction, Min(A0) splits as Y x En_1 where A0 acts trivially on Y and by translations on E n_1 and the quotient of E n_1 by the action of A0 is an (n — l)-torus. Thus, using Proposition 5.2.5 we have Min(A) = Y x En_1 x E = Y x En. Since for each a G A, the element aN E A/N acts trivially on Y and by translations on En_1, and N acts trivially on Y x En+1 and by translations on E 1, we see that a acts trivially on Y and by translations on En. The quotient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 159 E l/N is a circle and by the inductive hypothesis, the quotient En~1/Ao is an (n — l)-torus. Furthermore, the action of A is compatible with this splitting. Thus the quotient of En under the action of A is an n-torus as desired. (4) If an isometry 7 normalizes Min(A) then by Proposition 5.1.2 (1) Min(A) is 7 -invariant. Furthermore, for any A-orbit A • x in X we have 7 A • x = A • (7 • x), so 7 takes A-orbits to A-orbits. From (3) we have that each A-orbit, A • (y,t) is a lattice in {y} x En and so the convex hull of each A-orbit is precisely an n-flat {y} x En. Therefore 7 • ({y} x En) = {71 • y} x En and thus by Proposition 5.2.4, the isometry 7 respects this splitting. (5) For any number r there can only be finitely many a G A such that |a| = r. Indeed, the set K = 5((y,0),r) is compact so by the properness of the action, the set {7 G A | 7 • K fl K ^ 0} is finite. Thus, only finitely many elements of A have a translation length less than or equal to r. Consider the homomorphism / : T —> Aut(A) where /( 7 )(a) = qcry-1 for each a G A and each 7 G T. We have that the image of / is finite since each generator cq for A can only be conjugated to a finite number of distinct elements ycqy -1 since |7 «i7 _1| = |cq|. Thus, ker / is a finite index subgroup of T which centralizes A. The action of T on X is faithful because T C Isom A, so if T is finitely generated and ACT, then ker / is a finitely generated subgroup containing A to which Theorem 6.1.2 applies. Thus ker / and hence P contains a finite index subgroup containing A as a direct factor. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 160 We can generalize the flat torus theorem to groups which contain a finite index subgroup which is free abelian of rank n. Corollary 6.3.2. Let F be a finitely generated group which acts properly by semi-simple isometries on a CAT(O) space X. Suppose that T contains a finite index subgroup that is free abelian of rank n. Then 1. X contains a T-invariant closed convex subspace isometric to a product Y x En. 2. The action of T preserves the product structure on Y xE", acting as the identity on the first factor and cocompactly on the second. 3. Any isometry of X which normalizes T preserves Y xEn and its splitting. Proof. Let A() = Zn be a subgroup of finite index in T. Because F is finitely generated, there are only finitely many subgroups in T of index | F/'A0 j. Let A be the intersection of these finitely many subgroups; note that A has finite index in T. Moreover, A is free abelian of rank n since A has finite index in A0. As automorphisms preserve the index of subgroups, any automorphism of T will send a subgroup of index [r/A0| to a subgroup of index (T/A0| which implies that A is characteristic in T. In particular, any isometry of X which normalizes T will also normalize A. As a subgroup of T, A acts properly and by semi-simple isometries on X so by the Flat Torus Theorem 6.3.1, Min(M) splits as a product Z x En where A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6. GROUPS OF ISOMETRIES 161 acts trivially on the first factor and cocompactly on the second. Furthermore, since A is normal in F, the action of T on Min (A) preserves this splitting. Thus, we have an induced action of the finite group T/A on Z where 7 A acts as 7 on each z E Z. According to the Bruhat-Tits Fixed Point Theorem 3.3.1 (1) the fixed point set Y of T/A on Z is non-empty and by Proposition 5.1.2 (3) this set Y is a closed convex subset of X (actually, it is a closed convex subset of Z but this means that it is a closed convex subset of X as well). The set Y x E" is our desired subspace. The group F acts trivially on the first factor and since the action of A on the second is cocompact, the action of T will be cocompact as well. Any isometry 7 of A which normalizes F also normalizes A and hence leaves Z x En invariant and preserves the structure by the Flat Torus Theorem 6.3.1. Thus, in order to show that 7 preserves the splitting Y x E ", it suffices to show that if y E Y, then 7 • y E Y. We prove that 7 • y E Y by showing that it is fixed by the action of F/A. Suppose 7 'A E F/A. Since 7 normalizes F in Isom(A) there exists 7 " E F such that 7'7 • x = 77 " • x for every x E X . Thus i A • (7 -y) = 7;7 -y = i i '- y (6.33) 7 -y Hence, 7 ■ y E Y and 7 preserves the structure. □ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] J. W. Anderson. Hyperbolic Geometry. Springer-Verlag, Berlin, 1999. [2] M. R. Bridson and A. Haefliger. Metric Spaces of non-positive curvature, volume 319 of Grudlehren der Mathematischen [Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 1999. [3] K. S. Brown, Buildings. Springer-Verlag, Berlin, 1988. [4] F. Bruhat and J. Tits. Groups Reductifs sur un corps local. I. Donnees radicielles values, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5-251. [5] I. Chiswell., Introduction to A-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001 [6] N. J. Cutland. Computability: An Introduction to Recursive Function Theory, Cambridge University Press, 1980. [7] A. Dummit and T. Foote, Abstract Algebra, Third Edition, John Wiley and Sons, New York, 2004. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY 163 [8] H. B. Enderton, A Mathematical Introduction to Logic. 2nd ed., Harcourt Acedemic Press, 2001, Toronto. [9] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Pat terson, W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston MA, 1992. [10] D. Gromoll and J. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of non positive curvature , Bulletins of the American Mathematical Society 77 (1971), 545-552. [11] M. Gromov Asymptotic invariants of infinite groups , Geometric Group Theory (G. A. Niblo and M.A. Roller,ed), LMS Lecture Note Series 182 Cambridge Univ. Press, Cambridge, 1993, 75-263. [12] M. Gromov, Hyperbolic Groups, Essays in group theory (S. M. Gersten, ed), Springer Verlag, MSRI Publ. 8 (1987), 75-263. [13] G. H. Hardy and E. M. Wright. An introduction to the theory of numbers. Oxford University Press, New York, 1979. [14] H. B Lawson and S. T. Yau, Compact manifolds of non-positive curvature, J. Differential Geometry 7 (1972), 211-228. [15] A. M. Macbeath, Groups of homomorphisms of a simply connected space, Annals of Mathematics, (2) 79 (1964), 473-488. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY 164 [16] W. S. Massey Algebraic Topology: An Introduction, Graduate texts in mathematics bf 56. Springer-Verlag, Berlin, 1967. [17] J. R. Munkres Topology: a first course, Prentice-Hall, 1975, Englewood Cliffs, NJ. [18] J. P. Serre Trees, Springer-Verlag, 1980, Translation of ’’Arbres, Amal- games, SL2”, Asterisque 46, 1977. [19] Group theory from a geometrical viewpoint. Proceedings of the workshop held in Trieste, March 26-April 6, 1990. Edited by E. I. Ghys, A. Haefliger and A. Verjovsky. World Scientific Publishing Co., Inc., River Edge, NJ, 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.