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Beyond Hyperbolicity; Generalizing Classical Results from Hyperbolic groups to all Finitely Generated Groups by Abdalrazzaq Zalloum 6/01/2019 A dissertation submitted to the Faculty of the Graduate School of the University at Buffalo, The State University of New York in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics

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Page 1: Beyond Hyperbolicity; Generalizing Classical Results from ...abdul/PhDthesis.pdf · A group admitting such an action is called a hyperbolic group. Hyperbolic groups admit lots of

Beyond Hyperbolicity; Generalizing

Classical Results from Hyperbolic groups

to all Finitely Generated Groups

by

Abdalrazzaq Zalloum

6/01/2019

A dissertation submitted to the

Faculty of the Graduate School of

the University at Buffalo, The State University of New York

in partial fulfillment of the requirements for the

degree of

Doctor of Philosophy

Department of Mathematics

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Dissertation Committee:

Professor Johanna Mangahas,

Professor Bernard Badzioch

Professor William Menasco

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Abstract

Let X be a proper geodesic metric space. We give a new construction of the Morse Boundary that realizes its

points as equivalence classes of functions on X which behave similar to the “distance to a point" function.

When G = 〈S 〉 is a finitely generated group and X = Cay(G,S ), we use this construction to give a sym-

bolic presentation of the Morse boundary as a space of “derivatives" on Cay(G,S ). For a proper complete

CAT(0) space, we show that horospheres corresponding to Morse points behave similar to horospheres in

the hyperbolic space Hn.

Let G be a finitely generated group. We show that for any generating set A, the language consisting of

all geodesics in Cay(G, A) with a contracting property is a regular language. As an application, we show

that any finitely generated group containing an infinite contracting geodesic must be either virtually Z or

acylindrically hyperbolic.

iii

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Acknowledgements

First and foremost, an enormous gratitude goes to my main adviser Johanna Mangahas and to her never-

ending support, guidance and insight. Her advice was essential to the completion of this dissertation and

has taught me innumerable lessons and insights on the workings of academic research in general.

Secondly, it is my absolute pleasure to acknowledge and thank my coadvisor, Ruth Charney for her ex-

ceptional support and guidance. Her comments and feedback were crucial in every step of writing this

dissertation, I would be nowhere without her advice and great insights.

I would like to thank all of my fellow graduate students, for providing a great atmosphere to complete

my degree. An extra special thanks goes out to Joshua Eike, all the results of Chapter 4 were obtained in

collaboration with him.

I would also like to thank Denis Osin for pointing out an interesting application of one of the theorems in

this thesis.

Special thanks goes to my committee members, Bernard Badzioch and William Menasco for an endless

number of things, to say the least, for their bearing with my questions during my five years at UB.

On a personal note, and aside from my friends and fellow graduate students from the mathematics depart-

ment, I would like to acknowledge my close friends Hani, Ozgur and Muath for their support throughout

my time as a graduate student.

Finally, on a more personal note, I want to acknowledge the endless support and enormous love of my

parents. They have always been a constant source of inspiration, and this dissertation is dedicated to them

both, Esmat Zalloum and Raed Zalloum.

iv

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Contents

Abstract iii

Acknowledgements iv

1 Introduction 1

2 Preliminaries 5

2.1 Quasi-Isometries and Geometric Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Hyperbolic Spaces and CAT(0) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Boundaries of Hyperbolic Spaces, Two Constructions . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 The Boundary as Geodesic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 The Boundary as a Quotient of Quasi-Horofunctions . . . . . . . . . . . . . . . . . 8

2.4 Boundaries of CAT(0) Spaces, Two Constructions . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 The Boundary of a CAT(0) Space as Geodesic Rays . . . . . . . . . . . . . . . . . . 10

2.4.2 The Boundary of a CAT(0) Space as a Space of Horofunctions . . . . . . . . . . . . 11

2.5 The Morse Boundary: One Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Combinatorial Aspects of Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 The Global Geometry of a Hyperbolic Group is Determined Locally . . . . . . . . 16

2.6.2 Consequences of Finiteness of Cone Types . . . . . . . . . . . . . . . . . . . . . . 17

2.6.3 The Geometry of CAT(0)-Groups is Dependent on the Presentation . . . . . . . . . 20

3 The Morse Boundary as a Quotient of Generalized horofunctions 25

3.1 Constructing Gradient Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Continuous Surjection onto the Morse Boundary . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Symbolic Coding of the Morse Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 The Behaviour of Horofunctions in CAT(0) Spaces . . . . . . . . . . . . . . . . . . . . . . 45

v

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4 Regular Languages for Contracting geodesics 52

4.1 Prefix-Contracting Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 The Prefix-Contracting Language is Regular . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Contracting is Equivalent to Prefix-Contracting . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Generating Functions for Regular Languages are Rational . . . . . . . . . . . . . . . . . . . 65

Bibliography 67

vi

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Chapter 1

Introduction

Geometric group theory studies the interplay between the algebraic structure of a group G and the geometry

of the spaces G acts on. A naive example of this phenomena is the following theorem. A group G is

free if and only if it admits a free action on some tree [17]. A fundamental observation due to Schwartz

and Milnor states that if two groups G and H act geometrically (properly and cocompactly) on the same

geodesic metric space, then they must be quasi-isometric, meaning that their Cayley graphs have the same

large scale geometry.

Often, quasi-isometry invariants control the algebraic and combinatorial properties of groups. This is most

common in groups that admit a geometric action on δ-hyperbolic spaces, which are spaces whose large scale

geometry is similar to that of Hn. A group admitting such an action is called a hyperbolic group. Hyperbolic

groups admit lots of quasi-isometry invariants, and those invariants inform almost all the algebraic and

combinatorial structures hyperbolic groups have.

Therefore, it is natural to try to extend tools from the class of hyperbolic groups to more general classes

of groups. In other words, we want to study groups exhibiting geometric actions, not necessarily on δ-

hyperbolic spaces, but on spaces that have certain directions (geodesics) similar to the ones in a δ-hyperbolic

space; we will refer to those directions as hyperbolic-like directions. An example of such a space is the

space we obtain by gluing a half Euclidean plane to a geodesic line β in H2. The hyperbolic-like rays in

this example are the ones that are contained in the interior of the H2 part of the space along with the ones

spending a finite time in the half Euclidean plane and staying in H2 forever after [7]. The collection of all

such geodesic rays is called the Morse boundary, which was introduced and shown to be a quasi-isometry

invariant in [11]. We give a different construction for the Morse boundary that describes its points in terms

1

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2 Chapter 1. Introduction

of equivalence classes of horofunctions, which are maps from X to R that behave similar to the “distance to

a point" function:

Theorem 1.0.1. (Zalloum) Let X be a proper geodesic metric space. There exists a surjection from a certain

space of horofunctions H? onto the Morse boundary ∂?X, recovering the More boundary as a quotient of

H?.

Symbolic dynamics studies the action of a group G on AG, the set of maps from this group to a finite set

A, which is called the set of symbols. Such an action is called a Bernoulli shift. In his famous paper [16],

Gromov shows that, when G is a hyperbolic group, there is a finite set of symbolsA , a subset Y ⊆AG along

with a Bernoulli shift action of G on Y such that if ∂G is the Gromov boundary of G, then there exists a

G-equivariant continuous surjection ϕ : Y � ∂G. This is called a symbolic coding or symbolic presentation

of ∂G. In other words, if a group G acts on a topological space X, we say that this action admits a symbolic

coding if there exists a finite set A, some G-equivariant subset Y of AG along with a continuous surjection

of Y onto X which is G-equivariant.

It is known that an action of a discrete group on a Polish topological space, which is a certain kind of

metrizable space, admits a symbolic coding (Theorem 1.4 of [21]). However, the Morse boundary is not

metrizable or even second countable [20]. We use the new construction of the Morse boundary in the

previous theorem to show that:

Theorem 1.0.2 (Zalloum). Let G be a finitely generated group. Then, the action of G on its Morse boundary

admits a symbolic coding. In other words, there exists a finite set of symbolsA, a G-invariant subset Y? of

AG along with a Bernoulli shift action of G on Y? such that if ∂?G is the Morse boundary of G, then there

exists a G-equivariant continuous surjection φ : Y?→ ∂?G.

This construction opens the door to using symbolic dynamical methods to understand the Morse boundary.

For example, when G is a hyperbolic group, this construction has been studied further by Coornaert and

Papadopoulos [10] where they show that Y ⊆AG is a subshift of finite type. Also Cohen, Goodman-Strauss

and Rieck [14] used a similar coding to the one in the above theorem to show that a hyperbolic group admits

a strongly aperiodic subshift of finite type if and only if it has at most one end.

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3

Since every group acts geometrically on its Cayley graph, a natural place to look for such hyperbolic-like

geodesics is the groups Cayley graph. Now, if every geodesic in the Cayley graph is hyperbolic-like, then we

will be in the extreme case where G is hyperbolic. Otherwise, we will be looking in groups whose Cayley

graphs have some hyperbolic-like geodesics along with some other different geodesics. A first natural

question to ask is, how many of those hyperbolic-like geodesics does a Cayley graph for a given group have.

In other words, does there exist a counting formula for hyperbolic-like geodesics. We answer this question

affirmatively:

Theorem 1.0.3. (Eike-Zalloum) For any finitely generated group G = 〈A〉, the growth function of hyperbolic

like-geodesics in the Cayley graph Cay(G, A) is a rational analytic function. In particular, there is a counting

formula for hyperbolic-like geodesics.

Among the several competing notions of “hyperbolic-like directions" in a space, two are particularly impor-

tant. The first is the notion of a Morse geodesic, and the second is that of a contracting geodesic. A (quasi)

geodesic γ is said to be D-contracting if projections to γ of any ball disjoint from γ must have diameter at

most D. A (quasi) geodesic γ is said to be N-Morse if quasi geodesics based on γ live in the N-neighborhood

of γ, where N depends on the quasi geodesics constants. An element g in a finitely generated group G = 〈S 〉

is said to be (contracting) Morse if its axis 〈g〉 is (contracting) Morse in Cay(G,S ). We remark that the above

two notions coincide in many settings such as CAT(0) spaces [22], but they are different in general [1]. Now

we can state Theorem 1.0.3 more formally:

Theorem 1.0.4. (Eike-Zalloum) If G = 〈A〉 is a finitely generated group, and D≥ 0, then the growth function

of D-contracting geodesics in Cay(G, A) is a rational analytic function. In particular, there is a counting

formula for D-contracting geodesics.

In fact, we prove the more general theorem:

Theorem 1.0.5. (Eike-Zalloum) For any finitely generated group G = 〈A〉 and for a given D ≥ 0, the lan-

guage consisting of all D-contracting geodesics in Cay(G, A) is a regular language.

Given a finite set A, a language L is a subset of the free monoid generated by A, and a language is said to be

regular if it is of low enough complexity that it can be produced using a finite graph.

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4 Chapter 1. Introduction

We remark that the above theorem recovers a classical result due to James Cannon stating that the language

consisting of all geodesics in a hyperbolic group is a regular language [6].

As an interesting application of Theorem 1.0.5 above, we show the following:

Theorem 1.0.6. (Eike-Zalloum) Let G be a finitely generated group with a generating set A such that

Cay(G, A) contains an infinite contracting geodesic, then G must be acylindrically hyperbolic.

We remark that Chapter 2 of this thesis contains all the necessary definitions and background.

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Chapter 2

Preliminaries

2.1 Quasi-Isometries and Geometric Actions

Definition 2.1.1. Let X be a metric space. A geodesic in X is an isometric embedding of a finite or an

infinite interval in X. We say X is a geodesic metric space if any two points in X can be connected with a

geodesic.

A geodesic metric space X is said to be proper if closed balls in X are compact.

Definition 2.1.2 (quasi-isometry). A map between two metric spaces f : X→ Y is said to be a (K,C)-quasi-

isometric embedding if there exists K ≥ 1 and C ≥ 0 such that for any x,y ∈ X, we have

1K dX(x,y)−C ≤ dY( f (x), f (y)) ≤ KdX(x,y)+C.

We say f is a quasi-isometry if there exists some A ≥ 0 such that for any y ∈ Y there exists some a ∈ X such

that d( f (a),y) ≤ A. If X is a segment of R and f is a (K,C)-quasi-isometric embedding, then f is said to be

a (K,C)-quasi-geodesic.

Definition 2.1.3 (Proper Action). Let G be a group acting by isometries on a metric space X. The action is

said to be proper if for each x ∈ X, there exists some r > 0 such that the set {g | gBr(x)∩Br(x)} , ∅ is finite.

Definition 2.1.4 (Cocompact Action). Let G be a group acting by isometries on a metric space X. The

action is said to be cocompact if there exists some compact set K ⊆ X such that GK = X.

An action by isometries is said to be geometric if it is both proper and cocompact.

5

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6 Chapter 2. Preliminaries

Theorem 2.1.5 (Schwartz –Milnor). Let G be a group that acts geometrically on two geodesic metric spaces

X and Y , then X and Y must be quasi-isometric to each others.

The following corollary is an easy consequence of Theorem 47.1 in [19]:

Corollary 2.1.6. Let X be a proper geodesic metric space and let p ∈ X. Then, any sequence of geodesics

gn : [0, Ln)→ X with g(0) = p and Ln→∞ has a subsequence that converges uniformly on compact sets to

a geodesic g : [0,∞)→ X.

2.2 Hyperbolic Spaces and CAT(0) Spaces

Definition 2.2.1 (δ-hyperbolic spaces). A geodesic metric space X is said to be δ-hyperbolic if there exists

δ≥ 0 such that for any geodesic triangle [x,y]∪ [y,z]∪ [x,z], we have [x,z] ⊆ Nδ([x,y])∪Nδ([y,z]). We say X

is hyperbolic if it is δ-hyperbolic for some δ. (Here, Nδ([x,y]) is the δ-neighborhood of the geodesic [x,y]).

Definition 2.2.2. Let G be a finitely generated group with a generating set S . We say G is a hyperbolic

group if the Cayley graph Cay(G,S ) is hyperbolic.

It is straight forward to show that the above definition doesn’t depend on the generating set.

The following lemma states that hyperbolicity is a quasi-isometry invariant:

Lemma 2.2.3. If X and Y are geodesic metric spaces and f : X→ Y is a quasi-isometry, then X is hyperbolic

if and only if Y is.

Using Definition 2.2.2, Theorem 2.1.5 and the previous lemma, we have:

Corollary 2.2.4. A group G is hyperbolic if and only if it admits a geometric action on some hyperbolic

space.

Let X be a geodesic metric space and let ∆ be a triangle in X with vertices p,q and r. A comparison triangle

∆ is defined to be a triangle in R2 with vertices p,q and r satisfying d(p,q) = d(p,q), d(p,r) = d(p,r)

and d(q,r) = d(q,r). A point x on a side of ∆, say [p,q], is said to be a comparison point for x ∈ [p,q] if

d(p, x) = d(p, x).

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2.3. Boundaries of Hyperbolic Spaces, Two Constructions 7

Definition 2.2.5. A geodesic metric space X is said to be CAT(0) if every triangle in X is at most as fat as

any of its comparison triangles in R2. That is to say, for any geodesic triangle ∆ in X and any points x,y ∈ ∆,

if ∆ is a comparison triangle and x,y are comparison points in ∆, we must have

d(x,y) ≤ d(x,y).

Definition 2.2.6. A group G is said to be a CAT(0) group if it admits a proper cocompact action on a CAT(0)

space.

Definition 2.2.7 (distance-like function). Let X be a proper geodesic metric space. A continuous map

h : X→R is said to be distance-like if whenever h(x) ≥ λ, for some λ ∈R, then the set h−1(λ) is nonempty

and h(x) = λ+ d(x,h−1(λ)).

We remark that it is implied by the definition that if a ∈ Im(h) then so is b for any b ≤ a. Denote the space

of all real valued 1-Lipschitz functions on X by Lip(X,R).

The proof for the next proposition follows easily.

Proposition 2.2.8. Any distance-like function is 1-Lipschitz.

2.3 Boundaries of Hyperbolic Spaces, Two Constructions

In this section, we recall two descriptions of the boundary of a hyperbolic space, both due to Gromov.

2.3.1 The Boundary as Geodesic Rays

Definition 2.3.1. A geodesic ray in a geodesic metric space X is an isometric embedding c : [0,∞)→ X.

Definition 2.3.2. Let X be a δ-hyperbolic space and fix a base point p ∈ X. As a set, the Gromov’s boundary

of X, denoted by by ∂Xp is defined to be the collection of equivalence classes of geodesic rays starting at p,

where α ∼ β if and only if there exists some C ≥ 0 such that d(α(t),β(t)) ≤C for all t ∈ [0,∞).

The above boundary is topologized as follows. Convergence in ∂Xp is defined as: xn→ x as n→∞ if there

exists geodesic rays αn starting at p with [αn] = xn such that every subsequence of αn has a subsequence

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8 Chapter 2. Preliminaries

converging uniformly on compact sets to some α, with [α] = x. A set B ⊆ ∂Xp is closed if and only if

whenever xn ∈ B with xn→ x, we must have x ∈ B.

The following two statements are Proposition 3.7 and Theorem 3.9 in the Gromov’s boundary chapter of [5]:

Proposition 2.3.3. If X be a proper geodesic space that is δ-hyperbolic, then the Gromov’s boundary defined

above is independent of the base point. In other words, if p,q ∈ X, the spaces ∂Xp and ∂Xq are homeomor-

phic.

In light of the above theorem, we may occasionally omit reference the base point and simply refer to the

boundary as ∂X.

Theorem 2.3.4. If X and Y are hyperbolic spaces and f : X→ Y is a quasi-isometry, then the spaces ∂X and

∂Y are homeomorphic.

Notice that using the above theorem and Theorem 2.1.5, we have a well-defined notion for the boundary of

a hyperbolic group: If G acts geometrically on a hyperbolic space X, we can define ∂G := ∂X where this

definition is independent of the space X being acted on since any other space Y with a geometric action of G,

must be quasi-isometric to X (Theorem 2.1.5) and hence by the above theorem ∂X � ∂Y . Since every group

G acts geometrically on its Cayley graph, Cay(G,S ), we can simply define the boundary of a hyperbolic

group to be the boundary of its Cayley graph.

2.3.2 The Boundary as a Quotient of Quasi-Horofunctions

In this subsection, we provide a different description of the Gromov’s boundary as a space of distance-like

functions. This description is also due to Gromov, but it has been studied in further details in [10] and [9].

Definition 2.3.5. Let X be a proper geodesic metric space. A map h : X → R is said to be quasi-convex

if there exists K ≥ 0 such that for any x0, x1 ∈ X, if xt satisfies d(x0, xt) = td(x0, x1), then: h(xt) ≤ (1−

t)h(x0)+ th(x1)+K, for all t ∈ [0,1], (see Figure 2.1).

Definition 2.3.6. (quasi-horofunctions) Let X be a hyperbolic space. A function h : X→R is said to be a

quasi-horofunction if it is distance-like and quasi-convex.

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2.3. Boundaries of Hyperbolic Spaces, Two Constructions 9

(0,0)

f (x0)

f (x1)

K

x0

x1

Figure 2.1: A Quasi-convex function.

Let H denote the space of all quasi-horofunctions given the topology of uniform convergence on compact

sets. Fix p ∈ X. To each h ∈ H, there is a natural way of assigning a geodesic ray g starting at p called

h-gradient ray. This can be done by taking successive projections on descending level sets of h. In other

words, if x is in the level set of λ1 and λ2 < λ1, then we can take a projection of x onto the level sets of

λ2. Now, we can iterate this process to obtain a gradient ray g starting at x. We remark that this gradi-

ent ray is “perpendicular" to all of the level sets it crosses in the sense that for any s, t ∈ [0,∞), we have

d(h(g(s),h(g(t)) = t− s, which justifies the use of the word “gradient" for the geodesic ray g (see Figure

2.2). This suggests the possibility of a map from H to ∂X, the Gromov boundary of X. Notice that we

don’t have such a map yet, since we might have two h-gradient rays, α and β for the same h ∈ H, such

that d(α(t),β(t)) is unbounded. However, the geometry of δ-hyperbolic spaces ensures this can’t happen

(see [10] for example), and therefore, one gets a well defined map from H to ∂X. See [16], [9] and [10] for

details. Let H denote the quotient of H taken with respect to the equivalence relation that identifies functions

which differ by a constant.

Theorem 2.3.7. There exists a well-defined continuous surjection ϕ : H� ∂X, recovering ∂X as a quotient

of H.

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10 Chapter 2. Preliminaries

λ1

λ2

ζ

Figure 2.2: Assigning gradient rays to a quasi-horofunction.

Let G be a group acting on a topological space Y . This action is said to admit a symbolic coding if there

exists a finite setA and a G-invarient subset Z ofAG (which is the space of all maps σ : G→A) along with

a continuous equivariant surjection of Z onto Y . The above theorem is used in [16] to show the following:

Corollary 2.3.8. The action of a hyperbolic group on its Gromov’s boundary admits a symbolic coding.

Chapter 3 of this thesis is devoted to generalizing Theorem 2.3.7 and Corollary 2.3.8.

2.4 Boundaries of CAT(0) Spaces, Two Constructions

For the rest of this section, X is assumed to be a proper CAT(0) space. We recall two descriptions of the

boundary of a CAT(0) space.

2.4.1 The Boundary of a CAT(0) Space as Geodesic Rays

We will define an analogous notion of the Gromov’s boundary given in the previous section for X, called the

visual boundary, but first, we state the following lemma (see [5]):

Lemma 2.4.1. Let X be a CAT(0) space and let c be a geodesic ray. For any p ∈ X, there exists a unique

geodesic ray c′ starting at p such that d(c(t),c′(t)) is bounded for all t ∈ [0,∞).

Definition 2.4.2. As a set, the visual boundary of X, denoted by ∂X is defined to be the collection of equiva-

lence classes of geodesic rays, where α ∼ β if and only if there exists some C ≥ 0 such that d(α(t),β(t)) ≤C

for all t ∈ [0,∞).

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2.4. Boundaries of CAT(0) Spaces, Two Constructions 11

Notice that by the Lemma 2.4.1, for each c representing an element of ∂X, there is a unique geodesic ray c′

starting at p with [c] = [c′]. Now we describe the topology of the visual boundary: Fix a base point p and

let α be a geodesic ray starting at p. A neighborhood basis for [α] is given by sets of the form

U([α],r,ε) := {[β] ∈ ∂X|β(0) = p and d(α(t),β(t)) < ε for all t < r}.

In other words, two geodesic rays are close together in this topology if they have representatives starting

at the same point which stay close (are at most ε apart) for a long time (at least r). Notice that the above

definition of the topology on ∂X made a reference to a base point p. Define the topological space ∂Xp to be

the set ∂X with the above topology. We have the following (Proposition 8.8 in [5]):

Lemma 2.4.3. For any p, q in X the topological spaces ∂Xp and ∂Xq are homeomorphic.

Notice that Proposition 2.3.3 states that a quasi-isometry between two hyperbolic spaces induces a homeo-

morphism on their boundaries. However, this is no longer the case for CAT(0) spaces [13]. Hence, when we

talk about "the boundary" of a CAT(0) group, we must specify the CAT(0) group G along with the CAT(0)

space that G admits a geometric action on.

2.4.2 The Boundary of a CAT(0) Space as a Space of Horofunctions

In this section we give a description of the CAT(0) boundary as a space of distance-like functions. Again, X

is assumed to be proper and CAT(0). For more details on this section see [5].

Definition 2.4.4. A convex distance-like function on X is called a horofunction.

Let H be the collection of all horofunctions (taken with the topology of uniform convergence on compact

sets) and let H denote the quotient of H taken with respect to the equivalence relation that identifies functions

which differ by a constant. Just as in the previous section, to each horofunction, and each p ∈ X, one can

associate a gradient ray starting at p. In fact, we have the following theorem (see the horofunctions section

of [5]):

Theorem 2.4.5. There exists a continuous bijection ϕ : H→ ∂X.

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12 Chapter 2. Preliminaries

Unlike the case for hyperbolic groups (Corollary 2.3.8), the above theorem cannot be used to obtain a

symbolic coding for the action of a group on a CAT(0) space. The reason is that boundaries of CAT(0) groups

depend on the CAT(0) space X being acted on, and hence they are not intrinsic to the acting group itself.

The situation is even worse, as there exists two homeomorphic CAT(0) spaces with non homeomorphic

boundaries [13].

2.5 The Morse Boundary: One Construction

As mentioned in the previous section, unlike hyperbolic spaces, boundaries of CAT(0) space are not invariant

under quasi-isometries. A natural attempt to circumvent this problem would be to come up with a sensible

way of defining a boundary for a CAT(0) space that contains only the hyperbolic-like geodesic rays, and

since boundaries of hyperbolic spaces are in fact invariant under quasi-isometries, one would hope that this

restricted notion of a boundary will be a quasi-isometry invariant. This approach in fact works; Charney and

Sultan [7] introduced the notion of the Morse boundary of a CAT(0) space and they were able to show that

this boundary is a quasi-isometry invariant for CAT(0) spaces. Their construction was later generalized by

Cordes [11] to any proper geodesic metric space. We define the Morse boundary, for more details, see [11].

The following definition is meant to capture what is meant by a hyperbolic-like geodesic:

Definition 2.5.1 (Morse). A geodesic γ in a metric space is called N-Morse, where N is a function N :

[1,∞)× [0,∞)→ [0,∞), if for any (λ,ε)-quasi-geodesic σ with endpoints on γ, we have σ ⊆ NN(λ,ε)(γ).

The function N(λ,ε) is called a Morse gauge. A geodesic γ is said to be Morse if it is N-Morse for some N.

The following lemma is an evidence that the above definition does in fact capture hyperbolic-like geodesics

[11]:

Lemma 2.5.2. A proper geodesic metric space is hyperbolic if and only if there exists a Morse gauge N

such that every geodesic is N-Morse.

Now we are ready to define the Morse boundary: As a set, the Morse boundary ∂?X is the set of all Morse

geodesic rays in X where two geodesic rays γ,γ′ : [0,∞)→ X are identified if there exists a constant K such

that d(γ(t),γ′(t)) ≤ K for all t ∈ [0,∞). In order to topologize the entire boundary, we fix a base point p ∈ X,

topologize pieces of the boundary and take a direct limit. More precisely:

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2.5. The Morse Boundary: One Construction 13

Definition 2.5.3. Let p ∈ X and let N be a Morse gauge. The N-component of the Morse boundary, denoted

by ∂N Xp, is defined by:

∂N Xp := {[α] | ∃β ∈ [α] that is an N-Morse ray with β(0) = p}

This set is given the following topology: convergence in ∂N Xp is defined as: xn→ x as n→∞ if there exists

N-Morse geodesic rays αn starting at p with [αn] = xn such that every subsequene of αn has a subsequence

converging unifromaly on compact sets to some α, with [α] = x. Closed sets of ∂N Xp are then defined as

follows: B ⊆ ∂N Xp is closed if and only if whenever xn ∈ B with xn → x, we must have x ∈ B. The Morse

boundary is defined as:

∂?Xp := lim−−→

∂N Xp

where the direct limit is taken over the collectionM of all Morse gauges. We remind the reader that in the

direct limit topology a set U is open (or closed) if and only if U ∩∂N Xp is open (or closed) for all N ∈M.

Notice that the topology used to define the Morse boundary made reference to the base point p. However,

similar to the hyperbolic and the CAT(0) boundaries, the Morse boundary is independent of the base point

[11]:

Lemma 2.5.4. Let X be a proper geodesic metric space and let p,q ∈ X. Then ∂?Xp and ∂?Xq are homeo-

morphic.

In light of the above, we may occasionally omit reference to the base point p and simply refer to the Morse

boundary by ∂?X.

Unlike the CAT(0) boundary, the Morse boundary is a quasi-isometry invariant [11]:

Lemma 2.5.5. If f : X→ Y is a quasi-isometry, then ∂?X and ∂?Y are homeomorphic.

The goal of chapter 3 is to give a different construction for the Morse boundary as a quotient of distance-like

functions, and similar to the case of hyperbolic groups, to use this construction to obtain a symbolic coding

for the action of a group on its Morse boundary.

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14 Chapter 2. Preliminaries

The following lemma will be referred to frequently in this thesis (this is Lemma 2.8 in [11]):

Lemma 2.5.6. Let X be a proper geodesic metric space X and let p, p′ ∈ X. If α is an N-Morse geodesic ray

starting at p, then there exists an N′-Morse geodesic ray β starting at p′ such that N′ depends only on N and

d(p, p′). Furthermore, N′ ≥ N.

Proof. For the full proof see Lemma 2.8 [11]. The idea is to take a sequence of geodesics βn := [p′,α(n)],

show that each βn is N′-Morse, and to use Arzela’-Ascoli to get a convergent subsequence βni → β. �

2.6 Combinatorial Aspects of Hyperbolic Groups

In Section 2.3, we saw that its possible to assign a topological Invariant for hyperbolic groups, the Gromov’s

boundary. It is an invariant which is independent of the generating set of the group or the space being acted

on, something that is not true for CAT(0) groups.

In this section, we explore a combinatorial invariant that can be assigned to hyperbolic groups. We show

that this invariant is independent of the generating set of the group, something that will also be shown to not

hold for CAT(0) groups. Intuitively, the combinatorial invariant we present means that a hyperbolic group

is highly symmetric/recursive; that any of its Cayley graphs have only finitely many local patterns; and, that

the full global picture of its Cayley graphs is determined by those finitely many local patterns.

Think of the group of integers Z: this group is infinite; however, when we draw its Cayley graph — which

is just a line connecting integer points that differ by one — we typically draw the first three edges and we put

few dots in both directions to demonstrate that the pattern repeats itself. So in this sense, we can actually

"see" the Cayley graph, there are exactly two pattern which are defined by sending n 7→ n+1, n 7→ n−1 and

the full Cayley graph is then obtained by simply iterating those two patterns. So even though the Cayley

graph of Z =< a > is infinite, its infiniteness comes precisely from repeating those two patterns. To make

this even more precise, the Cayley graph of Z can be obtained by three vertices v, v1 and v2 (see Figure 2.3,

A := a−1). We can see everything about the Cayley graph from this finite graph. So again, infiniteness of the

Cayley graph of Z, comes precisely from iterating the loops labeled a and A in this finite graph infinitely

many times, so in this sense, it is not really infinite, is it?

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2.6. Combinatorial Aspects of Hyperbolic Groups 15

v v1v2

aA

aA

Figure 2.3: The group Z is infinite, but is it really?

i

b

b

B

B

ab

B B

aA

aA

bA

A a

Figure 2.4: Similarly to how the Cayley graph of Z is determined by the finite graph in Figure 2.3, theCayley graph of F2 is determined by this finite graph.

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16 Chapter 2. Preliminaries

Now, a hyperbolic group has a very similar behaviour in the sense that there are only finitely many patterns

in its Cayley graph, and hence, all of the information about its Cayley graph can be encoded in a finite graph

(see Figure 2.4 for the hyperbolic group F2 case). This observation will be extremely useful and it lets one

solve most of the combinatorial questions geometric group theorists care about such as the word problem,

the conjugacy problem and understanding the growth function. The fact that a Cayley graph of a hyperbolic

group contains only finitely many patterns means that one can actually "see" the Cayley graph (in the sense

that you write down those finitely many patterns and then you just have to repeat, which is something we

know how to do!). Being able to see the Cayley graph immediately implies two things. First, we can solve

the word problem: two words are equal if and only if they end in the same vertex of the Cayley graph, and

since you can see the Cayley graph, you can tell whether they end at the same vertex or not. Secondly, we

can count elements in a ball of radius n in the Cayley graph, again, because we can see it, and if you can

see it, you can count it! This implies a closed form for the growth function. An additional strength to the

above is independence of the generating set. In other words, if G is a hyperbolic group, and S is any finite

generating set, then Cay(G,S ) has only finitely many patterns and hence we can solve the word problem,

the conjugacy problem and understand the growth. Typically, for a finitely generated group, one looks for

a preferred set of generators that would lead a canonical way of writing group elements. Once a canonical

form is established, the word problem is easily solvable, two words are the same if and only if they have

the same canonical forms. But again, the key for this is to find "the right" generating set that would yield

a canonical form for group elements, whereas in hyperbolic groups every generating set yields a canonical

form.

2.6.1 The Global Geometry of a Hyperbolic Group is Determined Locally

Before we state the main result of this section, I would like to quote a paragraph from one of my favorite

papers due to James Cannon [6], which will motivate the definitions and the theorems of this section:

"Hyperbolic groups can be understood and seen: feeling, as we do, that we understand the simple linear

recursion n 7→ n+ 1 in the group of integers Z, we extend our local picture of Z, recursively, in our mind’s

eye toward infinity. One obtains a global picture of the arbitrary hyperbolic group G in the same way:

first, one discovers the local picture of G, then the recursive structure of G by means of which copies of

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2.6. Combinatorial Aspects of Hyperbolic Groups 17

the local structure are integrated. Since our personal experience does not seem adequate to suggest the

best formulation, we restrict ourselves to the following result: The growth function of a hyperbolic group

satisfies a linear recursion, and hence, is a rational analytic function."

Notation: Let G be a finitely generated group and let A be a finite generating set. If v ∈ F(A), the image of

v in G under the natural map is denoted by v.

The following material can be found in [5] and [6]:

Definition 2.6.1 (Cone Types). Let G be a group with a finite generating set A. Let d denote the metric

induced by Cay(G, A). The cone type of an element g ∈G is the set of words v ∈ F(A) such that d(1,gv) =

d(1,g)+ |v|. In other words, if g is represented by a geodesic word u, then the cone type of g is the set of

words v such that uv is also a geodesic.

Example 2.6.2. The group F2 =< a,b > has 5 cone types. More precisely, two group elements x,y ∈ F2

have the same cone type if and only if they both end with the same letter, and since there are 4 possible

letters to end with {a, a−1, b, b−1}, we get 4 cone types. But the identity element e is always its own cone

type, and hence we get 5 cone types in total. Notice that vertices in Figure 2.4 are the cone types of F2.

Theorem 2.6.3. If G is a hyperbolic groups and A is any finite generating set. Then G has only finitely

many cone types.

The proof is given in [6] and [5]. The idea is that since X = Cay(G,S ) is δ-hyperbolic, one can show that

the 2δ+ 3 neighborhood of each vertex in X determines its cone type, and since X is locally finite, we get

the stated result. Chapter 4 of the dissertation is dedicated to generalizing the above result.

2.6.2 Consequences of Finiteness of Cone Types

In this subsection we give many interesting consequences of Theorem 2.6.3 above, but first, we state some

definitions.

A regular language is simply a (typically infinite) set of words of low enough complexity that it can be

produced by a finite graph. Now we give the formal definition:

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18 Chapter 2. Preliminaries

Definition 2.6.4. A finite state automaton (FSA) over a finite set A is a finite graph whose edges are directed

and labeled by elements of A; the vertices of the graph are divided into two sets—“accept” and “reject”—and

there is a distinguished vertex s0 called the initial vertex. The accepted language of the automaton is the set

of words which occur as labels on a directed edge path beginning at s0 and ending at an accept vertex.

Definition 2.6.5 (Regular langauge). Let A be a finite set and A∗ be the set of all words with letters in A.

A language over A is a subset L ⊆ A∗. A language over A is regular if it is the accepted language of some

finite state automaton over A.

The following is an easy corollary of finiteness of cone types in hyperbolic groups:

Corollary 2.6.6. Let G be a hyperbolic group and let A be any finite generating set. Then, the language L

consisting of all geodesic words in Cay(G, A) is a regular language.

Proof. Consider the finite graph whose vertices are the cone types of G and which has a directed edge

connecting the cone type of g ∈G to the cone type of ga if and only if a ∈ A∪A−1 and a belongs to the cone

type of g. Consider the set of edge-paths in this graph that begin at the cone type of the identity and follow

only positively directed edges; such paths have a natural labelling by words in the letters A∪A−1. It follows

immediately from the definition of cone type that the set of words that occur as such labels is precisely the

set of words that label the geodesic edge-paths in Cay(G, A). �

Remark 2.6.7. It is worth mentioning that existence of a regular language for all geodesics is in fact equiv-

alent to finiteness of the set of cone types. Though described in slightly different terms, this statement is

essentially due to John Myhill.

Regular languages has been shown to be useful in solving algorithmic problems of groups. For example, the

regular language above can be used to give a quadratic solution to the word problem for hyperbolic groups.

Now we give another consequence of finiteness of cone types, but first we introduce the following definition:

Definition 2.6.8. Let G be a finitely generated group with a generating set A. The growth function of G

with respect to this generating set is defined to be the map f : N→N given by f (n) = |Bn(e)|, where Bn(e)

denotes a ball of radius n around the identity in Cay(G, A).

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2.6. Combinatorial Aspects of Hyperbolic Groups 19

Even though the growth function depends on the generating set, the rate equivalence of growth doesn’t.

Since the definition of the rate equivalence of growth is technical, I will not define it here but I will refer

the reader to [15] for more details. The idea is that rate equivalence picks out polynomiality (being bounded

above by a polynomial) and even degree (though it identifies linear/sublinear), and picks out exponentiality

(being bounded below by et). One good reason to care about the growth is the following beautiful theorem

due to Gromov:

Theorem 2.6.9. A finitely generated group G has a polynomial growth if and only if it is it is nilpotent or

contains a nilpotent subgroup of a finite index.

Having given a good reason to care about the growth function, now we describe how finiteness of cone types

in hyperbolic groups implies a closed a form for their growth functions.

Theorem 2.6.10. If G is a hyperbolic group with a finite generating set A, there exists a closed form for the

growth function of G.

For a proof see [5]. The idea is that using Corollary 2.6.6, we get a finite graph whose set of accepted

paths surjects onto group elements of G. But then this graph can actually be modified so that the set of

accepted paths is in a in a 1-1 correspondence with group elements of G. Hence, counting group elements

of G is the same as counting accepted paths in this finite graph. But the later is an easy combinatorial

problem that can be solved using the adjacency matrix of the finite graph. More precisely, to each finite

graph one can associate the adjacency matrix (its the matrix whose n-th power’s (i, j)-th entry is the number

of paths of length at most n from state i to state j). But since each matrix must satisfy a linear recursion

(by Cayley-Hamilton’s Theorem), we get a linear recursion for the growth function of the group. Now, it is

an elementary fact from combinatorics that a linear recursion for a sequence implies a closed form for that

sequence and hence the statement above.

Here is one more interesting consequence to finiteness of cone types:

Proposition 2.6.11. If a hyperbolic group is infinite then it contains an element of infinite order.

The proof is left as an exercise for the reader.

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20 Chapter 2. Preliminaries

In fact, a much stronger statement holds here but since we lack some of the background needed to state

it formally and since the statement is not necessary for the development of this thesis, I will just state it

informally and refer the reader to [4] for the precise statement. In Section 2.3, we defined the Gromov’s

boundary for a hyperbolic group. This boundary (with the first description using geodesic rays) has some

special types of elements described below.

Let g ∈G be an element with an unbounded orbit in the Cayley graph (we call such an element hyperbolic)

and consider the orbit of g given by {g, g2, ...}. Let γg(n) be defined to be a geodesic starting at e and ending

at gn in the Cayley graph. The sequence γg(n) can be shown to converge to a unique element in the boundary

represented by a geodesic ray γg. Such geodesic rays γg are called special. In other words, special geodesic

rays are ones that happen to coincide with the axis of a hyperbolic isometry.

Theorem 2.6.12. Let G be a hyperbolic group, then the subset Y of all boundary points defined by special

geodesic rays is dense in ∂G. In other words, the set {[γg]| g is hyperbolic} is dense in ∂G, the Gromov’s

boundary of G.

2.6.3 The Geometry of CAT(0)-Groups is Dependent on the Presentation

We remarked in Subsection 2.6.1 that the language of geodesics in a hyperbolic group is a regular language

independently of the generating set of the group.

In this subsection, we present an example due to James Cannon that demonstrates that the above statement

is not true for CAT(0) groups. In other words, we describe a CAT(0) group G with two different generating

sets S and S ′ such that Cay(G,S ) has finitely many cone types (and therefore the corresponding language

of geodesics is regular), whereas Cay(G,S ′) has infinitely many cone types (and hence the language of

geodesics is not regular).

Before we outline the example, notice that the group Z⊕Z with the standard presentation 〈x,y | [x,y]〉 has

finitely many cone types. (see Figure 2.5). Namely, the cone of any element g in the group must coincide

with the cone of one of the following 9 elements: e, x, x−1, y, y−1, xy, x−1y, x−1y−1, xy−1.

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2.6. Combinatorial Aspects of Hyperbolic Groups 21

x

xyx−1y

x−1

x−1y−1 xy−1

y

y−1

e

Figure 2.5: Cone types of Z⊕Z

Now, consider the CAT(0) group G = (Z⊕Z)oZ2 given with the following presentation G =< x,y,α |

α2 = 1, xy = yx, xα = αy, yα = αx >, see Figure 2.6. Let S = {x,y,α}. With the given presentation, we

obtain the relations xnα = αyn and ynα = αxn. The claim is that Cay(G,S ) has finitely many cone types.

Notice that every element in G can be written in the form αix jyk where i ∈ {0,1} and j,k ∈Z.

Since each element g ∈ G can be represented in this form, we can think of this form as the “address" or

the “coordinates" of the element, first, second and third (such a form is typically referred to as a normal or

canonical form). Notice that there are 2 · 3 · 3 = 18 different types of elements, as there are two different

choices for the first coordinate (α is either there or it isn’t), three different choices for the second coordinate

(x, x−1 or nothing) and three different choices for the third coordinate. It is an easy exercise to verify that

these 18 different elements are representatives for all possible cone types. One way to see this is to notice

that the only difference between the normal form for elements in this example and the normal forms for

Z⊕Z is the possible presence of an element α in the front. Since the Z⊕Z had 9 cone types, this example

must have 2×9 = 18 cone types (depending on whether α is present in the normal form or not).

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22 Chapter 2. Preliminaries

x

y

α α

y

x

Figure 2.6: Highly symmetric, we can "see" the Cayley graph, and hence, finitely many cone types.

Now notice that S ′ = {x,d,z,α}, with d = xy, z = x2, is also a generating set as y = x−1d, see Figure 2.7.

We claim that the language L := {All geodesics in Cay(G,S ′)} is not a regular language. This should be

apparent assuming the following claim holds:

Claim 2.6.13. Let m,n be two positive integers. The edge path αzmαzn is a geodesic in Cay(G,S ′) if and

only if m > n.

We will prove this claim, but for the moment, let’s see how it implies that L is not regular. For the sake

of contradiction, suppose that L is regular. Then it has finitely many, say k, states. Consider the element

x = αzk+2αzk+1. First notice that by the previous lemma, x ∈ L since k+ 1 < k+ 2. Also, since the number

of states is k and k+1 > k, then one must go through the same state twice while reading zk+1 which in turns

produces a loop of length i where 1 ≤ i ≤ k. Therefore, all elements of the form αzk+2α(zi)tzk+1−i must be

also in L for all t ∈N which contradicts the previous lemma. The only part left now is to prove the above

claim:

Proof. By playing with the given relations, you can see that αzmα = d2mz−m. Therefore, αzmαzn =

d2mz−mzn = d2mzn−m. For the sake of contradiction, suppose that αzmαzn is a geodesic where m ≤ n. Since

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2.6. Combinatorial Aspects of Hyperbolic Groups 23

x

y

α α

y

x

Figure 2.7: Not as symmetric, infinitely many cone types.

αzmαzn is a geodesic, we have m+ n+ 2 ≤ 2m+ n−m = m+ n which is absurd.

We prove the converse by contradiction. Suppose that m > n yet the word w= αzmαzn is not a geodesic, i.e,

there exist a geodesic word v such that |v| < |w| where v= w. We will show that this implies n ≥m. Consider

the homomorphism Φ : G→ Z defined by sending x to 1, y to 1 and α to 0. This homomorphism is well

defined since the relations α2, xyx−1y−1, xαy−1α and yαx−1α all map zero. Intuitively, the map Φ counts

the number of the x and the y exponents appearing in a group element g. Again, since the relations are all

mapped to zero, this map is well defined and independent of the presentation of an element. Notice that

Φ(w) = 2m+ 2n = Φ(v). First, we show that the word v can’t contain the letter α.

Claim 2.6.14. The geodesic word v can’t contain the letter α.

Proof.

We argue by contradiction, we first show that α can’t appear more than once. Notice that, for any given word

u, Φ(u) is less than or equal to twice the number of non α letters appearing in u. Let k be the number of non

α’s appearing in v. Since Φ(v) = 2(m+ n), then by the previous observation, we get that 2(m+ n) ≤ 2k or

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24 Chapter 2. Preliminaries

m+ n ≤ k. However, by assumption, v is a geodesic word whose length is at most m+ n+ 1, therefore, if α

appears more than once, then k < m+ n which is a contradiction. This shows that α can’t appear more than

once in v. Now we show that α can’t appear in v. Define a homomorphism g : G→Z2 that sends α to 1 and

sends x,y to 0 and since g(w) = 0 we must have g(v) = 0 and therefore, if α appears in v, it has to appear

an even number of times but we have shown that is not possible. �

Now we know the word v can’t contain the letter α, and the remaining three generators all commute. Let

i, j,k be the exponents (positive and negative) of x,z and d in the word v respectively. This implies that

Φ(v) = i+2 j+2k = 2(m+n) = Φ(w). Note that by assumption, since v is assumed to be a geodesic word

whose length |v| < |w| = m+ n+ 2, we get that i+ j+ k ≤ |v| < m+ n+ 2. This gives us the following two

equations:

i+ 2 j+ 2k = 2(m+ n)

i+ j+ k < m+ n+ 2

Combining these two equations, we get j+ k ≥ (m+ n)−1. Since |v| ≥ |i|+ | j|+ |k|, we get |v| ≥ |i|+(m+

n− 1). Notice that |i| can only be 0,1, or 2. Since j+ k ≥ m+ n− 1, then in particular, we get that j+ k is

positive, in fact, one can use the above two equations along with the fact that |i| can only take the values 0,1,

or 2 to show that both of j and k have to be positive. If one writes v in the normal form in terms of α, x and

y we get that v= xi+2 j+kyk. Similarly, write w= x2ny2m. Since such a representation is unique, we get that

2n = i+2 j+ k and that k = 2m. This implies that 2n = i+2 j+2m which implies that i has to be even (and

thus zero because v is a geodesic and replacing an even number of x with z shortens the word). From this,

we have n = j+m. Since j is non-negative, this implies that n ≥ m, a contradiction. �

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Chapter 3

The Morse Boundary as a Quotient of

Generalized horofunctions

In this chapter, we describe a new construction of the Morse boundary as a quotient of a space of distance-

like functions satisfying a certain convexity property (we remark that Sections 2.3 and 2.5 are prerequisites

to full understanding of this chapter). This construction generalizes (and recovers) the one described in

subsection 2.3.2. We then use use this construction to obtain a symbolic coding of the action of a finitely

generated group on its Morse boundary. In short, we prove the following two theorems:

Theorem 3.0.1. Let X be a proper geodesic metric space. Then there exists a subset of distance-like func-

tions H? along with a continuous surjection ϕ : H? � ∂?X, the Morse boundary of X. In particular, we

recover the Morse boundary as a quotient of H?.

Theorem 3.0.2. Let G be a finitely generated group. Then, the action of G on its Morse boundary admits a

symbolic coding. In other words, there exists a finite set of symbolsA, a G-invariant subset Y? ofAG along

with a G-equivariant continuous surjection of Y? onto ∂?G, the Morse boundary of G.

It is known that an action of a discrete group on a Polish topological space (a space homeomorphic to a

complete metric space that has a countable dense subset), admits a symbolic coding (Theorem 1.4 of [21]).

However, the Morse boundary is not metrizable or even second countable [20]. We use the new construction

of the Morse boundary in Theorem 3.0.1 to show Theorem 3.0.2.

This construction opens the door to using symbolic dynamical methods to understand the Morse boundary.

For example, when G is a hyperbolic group, this construction has been studied further by Coornaert and

25

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26 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

ζ

Figure 3.1: The set of blue and red gradient rays are both h-gradient rays. The red ones are Morse andtherefore they all converge to the same point ζ in the Morse boundary.

Papadopoulos [10] where they show that Y ⊆AG is a subshift of finite type. Also Cohen, Goodman-Strauss

and Rieck [14] used a similar coding to the one in the above theorem to show that a hyperbolic group admits

a strongly aperiodic subshift of finite type if and only if it has at most one end.

We now give an informal sketch to the proofs of Theorems 3.0.1 and 3.0.2. Just as in the hyperbolic spaces

case (described in subsection 2.3.2), we start with a collection of 1-Lipschitz distance-like maps h : X→R.

To each such map h, and for any x ∈ X, we can still define a gradient ray starting at x by taking successive

projections on descending level sets. We then define H? to be the collection of all such maps having some

Morse gradient ray at some point. A key result of this chapter is to show that if α and β are two Morse

rays (with possibly different starting points and different Morse gauges) associated to such a map h, then

they must define the same point in the Morse boundary. In other words, we show that if h ∈ H? and α and

β are two Morse gradient rays associated to h, then there exists a C ≥ 0 such that d(α(t),β(t)) ≤ C for at

t ∈ [0,∞). This can be interpreted as follows. To each distance-like map h ∈ H?, one can assign an infinite

collection of gradient rays, some of which are Morse and some are not. Whereas those gradient rays do not

all converge to the same point in general, the Morse ones do, and hence define a unique point in the Morse

boundary ∂?X (see the Figure 3.1). So although in general we can’t get a well defined map from the space

of such functions to whatever notion of a boundary a space might have, if we restrict our attention to the

Morse boundary, such a map exists.

The above yields a well defined map ϕ : H? → ∂?X. This map is then shown to be surjective using Buse-

mann functions. More precisely, if c is a Morse ray, then the corresponding Buseman function bc(x) =

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3.1. Constructing Gradient Rays 27

limt→∞[d(x,c(t))− t] is in H? having c as its Morse gradient ray. This proves the first main theorem.

For the second theorem, we do the following. Let G = 〈S 〉 be a finitely generated group and let X =

Cay(G,S ). We first consider a certain G-invariant subspace B of H? whose elements h take integer values

on the vertices of X. We then show that the subspace B is still big enough to surject on the Morse boundary by

showing that for any Morse geodesic c : [0,∞)→ X, starting at a vertex of X, the corresponding Busemann

function bc belongs to B. Since maps h ∈ B take integer values on vertices, the subspace B can be thought of

as a subspace of the Lipschitz functions Lip(G,Z) rather than Lip(X,R). But then we show that each such

map h ∈ B ⊆ Lip(G,Z) is completely determined by a map dh : G→A for some finite set A. Therefore,

the space B, which surjects on the Morse boundary, has a natural identification with a subspace Y ⊆ AG

giving the second theorem. The only part left to explain is, how given h ∈ B ⊆ Lip(G,Z), we can we get a

map from G to a finite set A that completely determines h. This is done as follows. First remember that G

is generated by S . For any h ∈ B ⊆ Lip(G,Z), we define dh : G→A by g 7→ (s 7→ h(gs)− h(g)), where

A= {−1,0,1}S .

Consider Figure 3.2. To the left, we have a 1-Lipschitz map defined on h : Z⊕Z→Z, and on the right we

have its derivative dh : Z⊕Z→A. Notice that h is completely determined by dh and by the value of h at

one point p.

More generally, if G and H are finitely generated groups and S is a generating set for G, then if f ∈

Lipn(G, H), the space of all n-Lipschitz map from G to H, one can define a map d f : G → A given by

g 7→ (s 7→ f −1(g) f (gs)), whereA= (B(1H ,n))S . Notice that B(1H ,n) denotes the ball of radius n centered

at the identity in some Cayley graph of H.

This notion of a derivative was introduced by Cohen [8]. In fact, he proved that {d f | f ∈ Lipn(G, H)} ⊆ AG

is a subshift of finite type. The goal of the rest of this chapter is to work out the construction described above

in details.

3.1 Constructing Gradient Rays

The main goal of this section is to define distance-like functions and to show that there is a natural way of

associating a gradient ray to each such function.

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28 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

0-1

0

0-1

0

0-1

0-1

01 -1

0

01 -1

0

01 -1

01 -1

01 -1

0

01 -1

0

01 -1

01 -1

01

0

01

0

01

01

Figure 3.2: To the left, a 1-Lipschitz map h : Z⊕Z→ Z, to the right, its derivative dh. Notice that h iscompletely determined by dh and h((0,0)).

Unless mentioned otherwise, for the rest of the chapter, X will be a proper geodesic metric space.

The following proposition states that for any x ∈ X, if h(x) > λ for some λ ∈R, then there exists a point y in

the level set h−1(λ) realizing the distance between x and the level set h−1(λ). Therefore, one gets a geodesic

connecting x to h−1(λ) realizing the distance d(x,h−1(λ)). We will eventually iterate this process to obtain

a geodesic ray starting at x which is “perpendicular" to all of the h-level sets it crosses:

Proposition 3.1.1. If h is a distance-like function and x ∈ X, λ ∈R with h(x) > λ, then ∃y ∈ X with d(x,y) =

h(x)−h(y), and h(y) = λ.

Proof. Notice that since h is continuous, Y := h−1(λ) must be a closed set and hence, d(x,Y) = d(x,y)

for some y ∈ Y . Since h is a distance-like function, we get that h(x) = λ+d(x,h−1(λ)) = h(y)+d(x,Y) =

h(y)+ d(x,y) which implies the conclusion of the proposition. �

Definition 3.1.2 (gradient arc). Let h be a distance-like function, an h-gradient arc is a path g : I → X

parameterized by the arc’s length such that h(g(t))− h(g(s)) = s− t for all s, t ∈ I. An h-gradient ray is a

geodesic ray whose restriction to any interval I ⊂ [0,∞) is an h-gradient arc.

The following three lemmas are stated and proved in [10] for the case where X is δ-hyperbolic. However,

the authors only use that X is a proper geodesic metric space which is the case here:

Lemma 3.1.3 (concatenation of gradient arcs). Let I1 and I2 be closed intervals of the real line such that I2

begins where I1 ends. If I = I1∪ I2 and g : I→ X is a path whose restrictions to I1 and I2 are h-gradient arcs,

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3.1. Constructing Gradient Rays 29

p

p1

p2

p3

g(t)

Figure 3.3: Constructing a gradient ray.

then g itself is an h-gradient arc.

Lemma 3.1.4. (characterization of h-gradient arcs) Let X be a proper geodesic metric space and let h be a

distance-like function:

• If g is an h-gradient ray then g is a geodesic;

• If x,y ∈ X are points so that h(x)−h(y) = d(x,y), and g is a geodesic connecting x to y then g is an

h-gradient arc.

Lemma 3.1.5 (gradient rays). For any distance-like function h and any p ∈ X, there exists an h-gradient ray

starting at p.

Proof. This follows from Lemmas 3.1.1, 3.1.3 and 3.1.4 of this section. The idea of the proof is to set

λ1 = h(p)−1, take a projection of p on h−1(λ1) and call this projection p1. And then we repeat (see Figure

3.3), in other words, we set λ2 = h(p1)− 1, take a projection of p1 on h−1(λ2) and call this projection p2.

This process, yields an h-gradient ray g. For more details see Proposition 2.13 in [10]. �

Definition 3.1.6. Let c : [0,∞)→ X be a geodesic ray. Define the Busemann function associated to c by:

bc(x) = limt→∞

[d(x,c(t))− t].

We remark that the above limit exists by the triangle inequality. Also, notice that a Busemann function is

1-Lipschitz.

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30 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

Proposition 3.1.7. If h is a Busemann function and h(x) ≥ λ for some x ∈ X and λ ∈ R, then there must

exist some p ∈ X with h(x) = λ+ d(x, p), furthermore, h(p) = λ.

Proof. Since h is a Busemann function, we have h = bc for some geodesic ray c. For each large enough

integer t, there exists some pt ∈ [x,c(t)] such that d(x,c(t)) = d(x, pt)+d(pt,c(t)), and d(x, pt) = h(x)−λ.

Properness of X implies the desired result. �

Proposition 3.1.8. Any Busemann function is distance-like.

Proof. Let h be a Busemann function and let x ∈ X, λ ∈ R with h(x) ≥ λ. By the previous Proposition,

there must exist some p ∈ X such that h(x) = λ+ d(x, p) with h(p) = λ. Therefore, we need only to

show that d(x, p) = d(x,h−1(λ)). First notice that using the exact same argument we used in Proposition

3.1.1, the distance d(x,h−1(λ)) must be realized by a point in h−1(λ). But since a Busemann function

is 1-Lipschitz, any p′ ∈ X with h(p′) = λ must satisfy h(x)− h(p′) = h(x)− h(p) = d(x, p) ≤ d(x, p′).

Therefore, d(x, p) = d(x,h−1(λ)). �

Lemma 3.1.9. Let X be a proper geodesic metric space, let p ∈ X and let c be a geodesic ray in X. If

h = bc is the Busemann function associated to c, then the sequence of geodesics [p,c(n)] has a subsequence

converging to some h-gradient ray c′ with c′(0) = p.

Proof.

The proof follows easily using Arzel’a-Ascoli and the definition of a Busemann function.

3.2 Continuous Surjection onto the Morse Boundary

In section, we prove the first main result of this chapter.

Definition 3.2.1. Let X be a geodesic metric space and let δ ≥ 0. Given x,y,z ∈ X. Let ∆ := [x,y] ∪

[y,z]∪ [x,z] be a geodesic triangle with vertices x,y and z. We say that ∆ is δ-thin-1 if the union of the δ-

neighborhoods of any two sides of the triangle contains the third. We refer to this condition as slim condition

1.

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3.2. Continuous Surjection onto the Morse Boundary 31

u

v

δ

iz

a a

b c

b c

iy

ix

x

y z

Figure 3.4: Geodesic triangle with internal points ix, iy and iz.

Definition 3.2.2. A geodesic α is said to be δ-slim if every triangle based on α is δ-thin-1.

Remark 3.2.3. For any proper geodesic metric space X and any x,y,z ∈ X, if ∆ := [x,y]∪ [y,z]∪ [x,z] is a

geodesic triangle with vertices x,y and z, there must exist three non-negative real numbers a,b,c and three

points ix, iy, iz such that a+ b = |[x,y]|, a+ c = |[x,z]| and b+ c = |[y,z]| where ix ∈ [y,z], iy ∈ [x,z] and

iz ∈ [x,y] satisfying that a = d(x, iz) = d(x, iy), b = d(y, iz) = d(y, ix) and c = d(z, ix) = d(z, iy). The points

ix, iy and iz are called the internal points of ∆. See Figure 3.4.

Definition 3.2.4. Let X be a geodesic metric space and let δ ≥ 0. Given x,y,z ∈ X, define ∆ := [x,y]∪ [y,z]∪

[x,z] to be a geodesic triangle with vertices x,y and z. We say that ∆ is δ-thin-2 provided that for any point

w ∈ {x,y,z}, if we consider the two subgeodesics α1 and α2 connecting w to the internal points on those two

geodesics, then whenever u ∈ Im(α1), there must exist some v ∈ Im(α2) with d(u,v) ≤ δ. See Figure 3.4

with w= z, α1 = [z, iy], and α2 = [z, ix].

We refer to this condition as slim condition 2.

It is clear that if a triangle is thin with slim condition 2, then it is also thin with slim condition 1. But we

also have:

Lemma 3.2.5. Slim condition 1 and slim condition 2 are equivalent.

Proof. The argument in Proposition 1.17 in the Reformulations of the Hyperbolicity section of [5] shows

the above Lemma provided that X is a δ-hyperboic metric space, however, since their argument doesn’t use

thinness of any triangle in X aside from the x,y,z one, the argument is still valid here. �

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32 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

The following lemma states that for any δ-thin triangle ∆ with vertices x,y and z, the distance function

d(x,−) restricted to the opposite edge [y,z] is quasi-convex, where the quasi-convexity constant depends

only on δ.

Lemma 3.2.6. Let X be a geodesic metric space, and let x,y,z be the vertices of a δ-thin triangle in the sense

of slim condition 2. Then for any t ∈ [0,1] and any xt ∈ [y,z] with d(y, xt) = td(y,z), we must have:

d(x, xt) ≤ (1− t)d(x,y)+ td(x,z)+ 2δ.

Proof. Let ix, iy, iz,a,b and c be as in Remark 3.2.3. Let xt ∈ [y,z], with d(y, xt) = td(y,z). Then xt ∈

[y, ix]∪ [ix,z]. Therefore, we have two cases to consider. If xt ∈ [y, ix], then by δ-slimness, we must have

some w ∈ [y, iz] such that d(xt,w) ≤ δ. This and triangle inequality implies that:

d(y, xt)−δ ≤ d(y,w) ≤ d(y, xt)+ δ;

d(x,w)−δ ≤ d(x, xt) ≤ d(x,w)+ δ.

By the above two equations, we get that:

d(x, xt) ≤ d(x,w)+ δ

= (a+ b)−d(y,w)+ δ

≤ (a+ b)+ δ−d(y, xt)+ δ

= (a+ b)− t(b+ c)+ 2δ

= a+(1− t)b− tc+ 2δ

≤ a+(1− t)b+ tc+ 2δ

= (1− t)(a+ b)+ t(a+ c)+ 2δ.

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3.2. Continuous Surjection onto the Morse Boundary 33

Now for the second case, if xt ∈ [ix,z], by δ-slimness of the triangle, there must exist some u ∈ [z, iy] with

d(xt,u) ≤ δ. By the triangular inequality, we get that:

d(z, xt)−δ ≤ d(z,u) ≤ d(z, xt)+ δ;

d(x,u)−δ ≤ d(x, xt) ≤ d(x,u)+ δ.

Therefore, we get that:

d(x, xt) ≤ d(x,u)+ δ

= (a+ c)−d(z,u)+ δ

≤ (a+ c)+ δ−d(z, xt)+ δ

= (a+ c)− (1− t)(b+ c)+ 2δ

= a− (1− t)b+ tc+ 2δ

≤ a+(1− t)b+ tc+ 2δ

= (1− t)(a+ b)+ t(a+ c)+ 2δ.

Let Lip(X,R) be defined to be the set of all 1-Lipschitz maps from X to R. Let Lip(X,R) denote the

quotient of Lip(X,R) which identifies maps that differ by a constant. Notice that for a fixed p ∈ X, the space

Lip(X,R) is homeomorphic to the space Lipp(X,R) consisting of all f ∈ Lip(X,R) such that f (p) = 0. It

is an easy exercise to see that the space Lipp(X,R) is compact (alternatively, this is Proposition 3.1 in [18]).

Notice that if h1, h2 are distance-like function that differ by a constant, then g is a gradient ray for h1 if and

only if it is a gradient ray for h2. Therefore, it is more natural to think of gradient rays as being associated

to h rather than being associated to a specific h.

Intuitively, a distance-like function can be thought of as a function measuring distance from points in the

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34 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

(n,n) (0,n)

(n,0) (0,0)

c

Figure 3.5: Standing at the hyperbolic-direction c, but looking at some non-Morse directions.

space to a fixed point at infinity. In the case where X is δ−hyperbolic, such a distance function can be shown

to be 4δ-quasi-convex [10]. For a general metric space however, we don’t expect such a distance function to

be quasi-convex, even when measuring distance from a Morse point at infinity. To make this more precise,

consider Figure 3.5. This space is the Cayley graph of Z⊕Z with a geodesic ray c attached at the origin.

The geodesic ray c is Morse, however, for a fixed large enough t ∈ [0,∞), the map d(c(t)),−) is not quasi-

convex: think of how the values of d(c(t),−) change while travelling along the highlighted black geodesics;

in the beginning and end of each such black geodesic d(c(t),−) takes the same value, but it takes arbitrarily

large values in between, and hence, d(c(t),−) can’t be K-quasi-convex for any K. This implies that bc, the

Busemann function of c, is not quasi-convex. Hence, even though c is a Morse geodesic in this example, bc

was shown to not be quasi-convex.

The following definition is a weaker form of a quasi-convex function (Definition 2.3.5) and it is meant to

capture distance-like functions that are quasi-convex once restricted to travelling between two hyperbolic-

like directions. The main example we will give for such functions are Busemann functions {bc} whose

defining rays {c} are Morse. The intuition is this: Let c be a Morse ray in a proper geodesic metric space X,

this space will have hyperbolic and non-hyperbolic directions, whereas the distance function to the Morse

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3.2. Continuous Surjection onto the Morse Boundary 35

(n,n) (0,n)

(n,0)(0,0)

c

Figure 3.6: Black geodesics are travelling between the two non-Morse blue geodesics, this is prohibited inthe definition of a Morse-horounction above.

ray c might not be quasi-convex (as shown in Figure 3.5), its restriction to geodesics in the hyperbolic-like

directions will be (this will be proven in Lemma 3.2.15).

Definition 3.2.7 (Generalized horofunctions). A distance-like function h is said to be a generalized horo-

function if it satisfies the following quasi-convexity property: For any Morse rays α, β, there exists some

K > 0 which depends on the Morse gauges of α and β, such that given x0 ∈ Im(α) and x1 ∈ Im(β), if xt

satisfies d(x0, xt) = td(x0, x1), we must have: h(xt) ≤ (1− t)h(x0)+ th(x1)+K, for all t ∈ [0,1].

The above definition is meant to capture functions that will be quasi-convex once prohibited from travelling

between non-hyperbolic directions. In fact, the reason to failure in quasi-convexity in Fig 3.5, is precisely

that the black geodesics were travelling between the two (blue) non-hyperbolic directions in Figure 3.6.

Remark 3.2.8. Since in a δ-hyperbolic space every geodesic is N-Morse for a uniform N, our definition

of a generalized horofunction generalizes the quasi-horofunction’s Definition 2.3.6 for the case where X is

hyperbolic.

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36 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

Remark 3.2.9. We remark that in Definition 3.2.7, the constant K does not depend on x0 or x1.

The aim of the next few lemmas is to prove that each generalized horofunction h defines a unique point

at the Morse Boundary, in other words, we will show that if g1 and g2 are two Morse gradient rays, then

there must exist some C ≥ 0 such that d(g1(t),g2(t)) < C for all t ∈ [0,∞). We will also give an example

demonstrating that the above statement is not true if we don’t restrict our attention to Morse geodesics.

Definition 3.2.10 (thin polygons). An n-polygon is said to be δ-thin if the union of the δ-neighborhoods of

any n−1 sides of it contains the n-th side.

Lemma 3.2.11 (thin Morse-quadrilaterals). Let X be a proper geodesic metric space and let [x,z1], and [y,z2]

be N-Morse geodeiscs, then for any geodesics [x,y] and [z1,z2], the polygon [x,y]∪ [y,z2]∪ [z2,z1]∪ [z1, x]

is δ-thin where δ depends only on N and on d(x,y).

Proof. The proof follows easily from Lemma 2.8 in [11]. �

The following key lemma states that for a generalized horofunction function h, if α and β are two h-gradient

rays that are Morse, then d(α(t),β(t)) ≤C for all t ∈ [0,∞). It makes it possible to give a well-defined map

from the space of Generalized horofunctions to the Morse boundary:

Lemma 3.2.12 (Morse gradient rays define the same point). Let l be the equivalence class of a Generalized

horofunction l. If α,β are two Morse gradient rays associated to l, then ∃C > 0 such that d(α(t),β(t)) ≤ C

for all t > 0.

Before we give the proof, we give a non-example for the case where α and β are not Morse:

Example 3.2.13. Let X be the Cayley graph of Z⊕Z with the standard generators. Consider the two

geodesic rays α and β in X given in Figure 3.7. If we take h = bα, the Busemann function of α, then h is a

Generalized horofunction (since Z⊕Z has no Morse geodesic rays) and both α and β are h-gradient rays.

However, d(α(t),β(t)) is unbounded.

Proof. Let N := max{M, M′} where M and M′ are the Morse gauges for α and β respectively. First we

treat the case where α and β start at the same point. Let x := α(0) = β(0) and let γ be a geodesic connecting

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3.2. Continuous Surjection onto the Morse Boundary 37

α

β

Figure 3.7: The Busemann function of α given by h = bα is a Generalized horofunction and both α and βare h-gradient rays but d(α(t),β(t)) is unbounded.

α(t) to β(t) for some t ≥ 0. Let h ∈ l be normalized so that h(x) = 0. Notice that by assumption, since h is a

Generalized horofunction, there must exist some K such that h|γ is K-convex. Let m be the middle point of

γ, then we have h(m) ≤ 12 h(α(t))+ 1

2 h(β(t))+K = −t2 −

t2 +K = −t+K. This implies that d(x,α(t)) = t ≤

−h(m)+K = h(x)−h(m) ≤ d(x,m)+K. Since α and β are both N-Morse, then by Lemma 2.2 in [11], the

triangle [x,α(t)]∪γ∪ [x,β(t)] is 4N(3,0) thin. Therefore, there must exist some point w ∈ [x,α(t)]∪ [x,β(t)]

such that d(m,w) ≤ 4N(3,0)), without loss of generality, assume that w ∈ [x,α(t)]. Notice that since m is

the midpoint of γ, we must have d(α(t),β(t)) = 2d(α(t),m) ≤ 2(4N(3,0)+ d(α(t),w)). Thus, in order to

put a bound on d(α(t),β(t)), it is enough to put a bound on d(α(t),w). Notice that we have the following:

d(x,w)+ d(w,α(t)) = d(x,α(t)) ≤ d(x,m)+K ≤ d(x,w)+ d(w,m)+K ≤ d(x,w)+ 4N(3,0)+K. There-

fore, d(w,α(t)) ≤ 4N(3,0)+K. This gives that d(α(t),β(t)) = 2d(α(t),m) ≤ 2(4N(3,0)+ d(α(t),w)) ≤

2(4N(3,0)+ 4N(3,0)+K).

And thus, letting C = 2(4N(3,0)+ 4N(3,0)+K) yields that d(α(t),β(t)) ≤C for all t ∈ [0,∞).

Now we show it more generally, in other words, now we don’t assume that α and β start at the same point

(see Figure 3.8). Up to replacing α by some α′ defined by α′(t) = α(a+ t) for some a ≥ 0, we may choose

a representative h so that h(α(0)) = h(β(0)) = 0. Let t > 0 and let γ be a geodesic connecting α(t) to β(t).

Notice that by Lemma 3.2.11, the polygon [α(0),β(0)]∪ [β(0),β(t)]∪ [β(t),α(t)]∪ [α(0),α(t)] must be

δ-thin where δ depends only on d(α(0),β(0)) and on N. This implies that the middle point of γ, denoted

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38 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

γ(t)m

α(0)

β(0)

α(t)

β(t)

Figure 3.8: Morse gradient rays are convergent.

by m, is δ-close to [α(0),β(0)]∪ [β(0),β(t)]∪ [α(0),α(t)]. We claim that if t is large enough, then m can’t

be δ-close to [α(0),β(0)]. First notice that since h is a Generalized horofunction, we must have h(m) ≤

−t2 + −t

2 +K = −t+K. This gives d(α(0),α(t)) = t ≤ −h(m)+K = h(α(0))−h(m)+K ≤ d(α(0),m)+K.

Suppose for the sake of contradiction that there exist w ∈ [α(0),β(0)] with d(w,m) ≤ δ. Let m′ be the mid-

dle point of [α(0),β(0)], without loss of generality, we may assume that w lies between α(0) and m′. But

this yields: d(α(0),β(0)) = 2d(α(0),m′) ≥ 2d(α(0),w) ≥ 2(d(α(0),m) − d(m,w)) ≥ 2(d(α(0),α(t)) −

K − δ) ≥ 2t − 2K − 2δ which is a contradiction since t is arbitrary. This implies the existence of w ∈

[α(0),α(t)]∪ [β(0),β(t)], with d(w,m) ≤ δ, without loss of generality, assume that w ∈ [α(0),α(t)]. Again,

notice that d(α(t),β(t)) = 2d(α(t),m) ≤ d(α(t),w) + d(w,m) ≤ d(α(t),w) + δ. Therefore, in order to

bound d(α(t),β(t), we only need to bound d(α(t),w). But since d(α(t),w)+d(w,α(0)) = d(α(0),α(t)) ≤

d(m,α(0)+K ≤ d(m,w)+d(w,α(0))+K ≤ δ+d(w,α(0))+K. This implies that d(α(t),w)≤ K+δwhich

in turns gives us that d(α(t),β(t)) = 2d(α(t),m) ≤ d(α(t),w)+d(w,m) ≤ d(α(t),w)+ δ ≤ K +2δ for all t

large enough. �

We remark that the previous lemma does not imply that if h is a Generalized horofunction then all of its

gradient rays converge to the same point. The lemma only states that the Morse gradient rays of h must

converge. A natural question to ask at this point is:

Question 1. If h is a generalized horofunction and h has some Morse gradient ray. Are all other h-gradient

rays Morse?

Definition 3.2.14. Let p ∈ X. Define HNp to be the space of all generalized horofunctions h such that h has

an N-Morse gradient ray starting at p. We equipe HNp with the topology of uniform convergence on compact

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3.2. Continuous Surjection onto the Morse Boundary 39

sets.

Lemma 3.2.15. Let c be an N-Morse ray starting at p. Then the Busemann function h = bc is a generalized

horofunction and c is a Morse gradient ray for h. In other words, we have h ∈ HNp .

Before we start the proof, we give an easy corollary of Lemma 2.8 in [11].

Corollary 3.2.16. Let α be some N-Morse geodesic starting at p and let p′ be any other point in the space

X. Then, there exists some N′ ≥ N such that any geodesic starting at p′ and ending at some point on α is

N′-Morse, where N′ depends only on N and on d(p, p′).

Now we prove Lemma 3.2.15:

Proof. Let c be an N-Morse ray starting at some point p and set h = bc. The first thing to show is that

h has an N-Morse ray starting at p, but c clearly does the job. Now, given any two N-Morse rays α, β,

starting at p1 and p2 respectively, we want to show that there exists some K > 0 such that if [x1, x2] is a

geodesic connecting x1 ∈ Im(α) to x2 ∈ Im(β), then for any s ∈ [0,1], if xs satisfies d(x1, xs) = sd(x1, x2),

then h(xs) ≤ (1− s)h(x1) + sh(x2) + K. Let t ∈ [0,∞), we claim that any geodesics [x1,c(t)], [x2,c(t)]

must be M-Morse where M depends only on N and on the distances between the base points {p, p1, p2}.

Consider some geodesic triangle with vertices c(t), p, p1. See Figure 3.9. By the previous corollary, any

geodesic [p1,c(t)] must be N′-Morse where N′ depends only on N and on d(p, p1). Now, applying Lemma

2.3 in [11], to a geodesic triangle with vertices p1, x1 and c(t) yields that [x1,c(t)] must be N′′-Morse where

N′′ depends only on N and on d(p, p1). Similarly, we get that [x2,c(t)] is M′-Morse, where M′ depends

only on N and d(p, p2). Therefore, if M := max{N′′, M′}, then the geodesics [x1,c(t)], [x2,c(t)] are both M-

Morse. By Lemma 2.2 in [11], we get that the triangle ∆ given by [x1,c(t)]∪ [x2,c(t)]∪ [x1, x2] is 4M(3,0)

thin, but then by Lemma 3.2.5, and Lemma 3.2.6, we get that:

d(c(t), xs) ≤ (1− s)d(c(t), x1)+ sd(c(t), x2)+ 32M(3,0).

But then, if we substract t from both sides we get that:

d(c(t), xs)− t ≤ (1− s)d(c(t), x1)+ sd(c(t), x2)+ 32M(3,0)− t,

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40 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

p

x2

x1

p2

p1

β(t)

c(t)

α(t)

Figure 3.9: A Busmeann function is a Generalized horofunction.

and hence:

d(c(t), xs)− t ≤ (1− s)(d(c(t), x1)− t)+ s(d(c(t), x2)− t)+ 32M(3,0).

Now, if we let t→∞, we get that bc(xs) ≤ (1− s)bc(x1)+ sbc(x2)+ 32M(3,0). �

We remind the reader that the topology on the Morse boundary is described in Section 2.5.

Let HNp be the quotient of HN

p identifying functions that differ by a constant given with the quotient topology

of the compact convergence topology on HNp . Observe that HN

p is homeomorphic to the subspace {h ∈

HNp |h(p) = 0} (with the subspace topology).

Notation 3.2.17. Denote the space {h ∈ HNp |h(p) = 0} by HN

p,p.

Theorem 3.2.18. The natural map ϕN : HNp → ∂N Xp, is well-defined, continuous, and surjective.

Proof. We showed in Lemma 3.2.12 that if g1 and g2 are two Morse gradient rays associated to the same

Generalized horofunction h, then g1 and g2 must define the same point at infinity, this shows that ϕN is well

defined.

Since HNp is homeomorphic to HN

p,p, we may work with the later space instead. To show that ϕN is con-

tinuous, let hn,h ∈ HNp,p with hn → h. We need to show that ϕN(hn)→ ϕN(h). Let gn ∈ ϕN(hn) be a se-

quence of N-Morse rays starting at p. By definition of convergence in ∂N Xp, we need to show that any

subsequence gnk of gn has some subsequence gnkithat converges uniformly on compact sets to some ray

β, where β ∈ ϕN(h). Notice that in order to show that β ∈ ϕN(h) we need to show that β is an h-gradient

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3.2. Continuous Surjection onto the Morse Boundary 41

ray and that β is N-Morse. By Arzela’-Ascoli, any subsequence gnk contains a subsequence gnkiconverg-

ing to some β uniformly on compact sets. Now we are left to show two things, first, we need to show

that β is a gradient ray for h and then we should show that β is in fact N-Morse. To show that β is an

h-gradient ray, we need to show that for a given t1, t2 ∈ [0,∞), we must have h(β(t1))− h(β(t2)) = t2 − t1.

Let ε > 0, we will show that |h(β(t1)) − h(β(t2)) − (t2 − t1)| < ε giving the desired conclusion. Notice

that the two sets K1 = {gnki(t1)} and K2 = {gnki

(t2)} live on circles of radius t1, t2 respectively. Since

X is proper, then there exist a compact set K with K1, K2 ⊆ K. Since hn → h uniformly on compact

sets, then hn → h uniformly on K. Therefore, there must exist a positive integer s1 such that for each

i ≥ s1, we have |hnki(gnki

(t1))− h(gnki(t1))| < ε

4 and |hnki(gnki

(t2))− h(gnki(t2))| < ε

4 . Also, since h is uni-

formly continuous and gnki(t1)→ β(t1) and gnki

(t2)→ β(t2), there must exist s2 such that if i ≥ s2, then

both of the quantities |h(gnki(t1))− h(β(t1))| and |h(gnki

(t2))− h(β(t2))| are less than ε4 . Now notice that

|[t2− t1]− [h(β(t1))−h(β(t2))]|= |[hnki(gnki

(t1))−hnki(gnki

(t2))]− [h(β(t1))−h(β(t2))]| ≤ |[hnki(gnki

(t1))−

h(gnki(t1))] + [h(gnki

(t2)) − hnki(gnki

(t2))] + [h(gnki(t1)) − h(β(t1))] + [h(β(t2)) − h(gnki

(t2))]| < 4 ε4 = ε.

This shows that β is a gradient ray for h. Now we need to show that β is N-Morse, but this is Lemma

2.10 in [11]. �

If we define Hp :=⋃

N∈MHN

p taken with the direct limit topology, where M is the collection of all Morse

gauges, we get the following corollary:

Corollary 3.2.19. There exists a continuous surjection ϕ : Hp� ∂?Xp.

Remark 3.2.20. We remark that one can use a very similar argument to the continuity argument given

in the previous theorem to show that the space HNp = HN

p,p is a closed subspace of Lipp(X,R) = { f ∈

Lip(X,R)| f (p) = 0}.

For the case where X is δ-hyperbolic, the analogus space of distance-like functions that surjects on the

Gromov boundary is compact, so it is natural to ask whether that is still the case. We in fact prove the

following:

Corollary 3.2.21. The space Hp is compact if and only if X is hyperbolic.

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42 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

Before we give the proof, to be as self-contained as possible, we state the following Lemma from [12]:

Lemma 3.2.22. Let X be a proper geodesic metric space and fix a base point p. If the Morse boundary ∂?X

is compact, then there exists a Morse gauge N such that every geodesic in X is N-Morse. In particular, using

Lemma 2.5.2, the space X must be hyperbolic.

Now we give the proof:

Proof. If the space X is δ-hyperbolic, then by the stability Lemma in [5], there exists an N such that

every geodesic ray is N-Morse, and our definition of Hp coincides with the definition of horofunctions given

in [10] where they prove such a space is compact (Proposition 3.9 of [10]). For the converse, if Hp is

compact, then by continuity of ϕ, its image ∂?Xp must also be compact, but then Lemma 3.2.22 implies that

the space X must be hyperbolic. �

3.3 Symbolic Coding of the Morse Boundary

In his famous paper [16], Gromov gives a construction describing the boundary of a hyperbolic group G in

terms of “derivatives" on its Cayley graph. In this section, we develop a similar construction for the Morse

boundary. More precisely, Gromov shows that there exists a finite set of symbols, denoted by A, some

G-invariant subset Y ⊆AG and a continuous G-equivariant surjection ϕ : Y � ∂G. This is called a symbolic

coding or symbolic presentation of ∂G. In other words, if a group G acts on a topological space X, we say

that this action admits a symbolic coding if there exists a finite set A, some G-invariant subset Y of AG

along with a continuous G-equivariant surjection of Y onto X.

The purpose of this section is to use the map developed in the previous section to show that the action of a

finitely generated group G on its Morse boundary admits a symbolic coding.

For the rest of the section, we fix G to be a finitely generated group with a generating set S , and we let

X = Cay(G,S ).

Definition 3.3.1. Let A be a finite set and let G be a finitely generated group. The shift space, denoted by

AG, is defined to be the collection of all maps from G to A. The space AG is equipped with the product

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3.3. Symbolic Coding of the Morse Boundary 43

topology. Notice that this topology is the same as the topology of pointwise convergence: a sequence

σn→ σ if and only if for any g ∈G, there exists n0 ∈N such that σn(g) = σ(g) for all n ≥ n0.

Remark 3.3.2. If we fix a finite generating set S for G and we equip G with the word metric distance,

then, since G is locally finite, the point-set convergence in AG described above agrees with the uniform

convergence on compact sets.

Notice that the above shift spaceAG admits a natural action of G, called the Bernoulli shift action given by

the following: If g ∈G and σ : G→A, we have (gσ)(h) = σ(g−1h).

Given a finitely generated group G and a finite set of symbolsA, the shift spaceAG defined above is simply

the collection of all decorations of the Cayley graph’s vertices by symbols fromA. For example, if G = Z2

andA= {X,O}, then the setAG is simply the set of all tilings of the grid Z2 with X and O.

Definition 3.3.3. Let A be a finite set and let F be a finite subset of G. A pattern is a map σ : F →A.

Let F := {σi : F → A} be a finite set of patterns all defined on the same finite set F, we define the set

X|F := {σ ∈ AG | (gσ)|F < F for allg ∈G}. In this case we say that F is a finite set of forbidden patterns.

Definition 3.3.4. A subset Y of AG is said to be a subshift if it is closed and G-invariant. A subshift Y of

AG is said to be a subshift of finite type if Y = Clo(X|F ) for some F as above, where Clo(X|F ) denotes the

closure of X|F inAG.

The following is an example of a subshift of finite type. Let G = Z2 and takeA= {X,O}. Define Y ⊂AG to

be the subset of all σ : G→ {X,O} such that neither symbol ever shows up three times in a row, horizantally,

vertically or diagonally. This subshift corresponds to the set of all possible Z2 tic-tac-toe games with no

winner.

Definition 3.3.5. Let BNp be the space of all distance-like functions h : X → R satisfying the following

conditions:

1. The distance-like function h is a Busemann function of some Morse geodesic ray α;

2. There exists an h-gradient ray β ∈ [α] such that β is N-Morse with β(0) = p.

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44 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

Notice that G admits a natural action on the space of all distance-like functions on X by gh(x) = h(g−1x).

Also, every finitely generated group G admits a natural action on its Morse boundary ∂?Xp given as follows:

If g ∈ G and [β] ∈ ∂?Xp with β(0) = p, we define (gβ)(t) = g.β(t). Let M denote the set of all Morse

gauges, we have the following:

Lemma 3.3.6. The set Bp :=⋃

N∈MBN

p is G-invariant. Furthermore, if p is a vertex in X, then h(X0) ⊆Z for

all h ∈ Bp, where X0 denotes the set of all vertices in the Cayley graph X.

Proof. For any h ∈ Bp, we have h = bα for some Morse ray α. Now, if g ∈ G, then gbα = bgα and gα

is Morse. Now, using Lemma 3.1.9, the sequence of geodesics [p,α(n)] has a subsequence converging to

some h-gradient ray β with β(0) = p. By Lemma 2.5.6, the geodesic ray β must be M-Morse for some

Morse gauge M. This shows that Bp is G-invariant. The second assertion is clear. �

We remark that since Bp is a G-invariant subset of distance-like functions that take integer values on the

vertices of the Cayley graph X, then Bp ⊆ Lip(G,Z) = {The set of all 1-Lipschitz maps from G toZ}.

Lemma 3.3.7. Fix a vertex p ∈ X, the natural map ϕN : BNp → ∂N Xp is well-defined, continuous and surjec-

tive.

Proof. The proof is the exact same as the one given in Theorem 3.2.18. �

Corollary 3.3.8. If we give Bp the direct limit topology, then we get the following: There exists a continuous

G-equivariant surjection ϕ : Bp→ ∂?Xp.

Proof. This follows from Lemma 3.3.7, Lemma 3.3.6. �

Remark 3.3.9. Now, let G and H be finitely generated groups and let S be a generating set for G. By

Lipn(G, H) we denote the space of all n-Lipschitz maps from G to H. For any f ∈ Lipn(G, H), we define

d f : G→ B(1H ,n)S by g 7→ (s 7→ f (g)−1 f (gs)). See Figure 3.2. It is an easy exercise to show that the map

( f 7→ d f ) is G-equivariant. Let A = B(1H ,n)S . Notice that d f1 = d f2 if and only if f1 = h f2 or f1 = f2h

for some h ∈ H. Define Lipn(G, H) to be the set of all equivalence classes [ f ] were [ f1] = [ f2] if and only

if d f1 = d f2. Notice that using the above we get a map: φ : Lipn(G, H)→AG which is a G-equivariant

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3.4. The Behaviour of Horofunctions in CAT(0) Spaces 45

homeomorphism onto its image. This map was introduced (and discussed in greater details) by Cohen [8]

where he also shows that Lipn(G, H) is a subshift of finite type.

Now we specialize to the case where H = Z and n = 1. By Lemma 3.3.6, the space Bp is a G-invariant

subspace of Lip1(G,Z) or simply Lip(G,Z). Applying the previous remark to this special case, we get

a map φ : Lip(G,Z)→AG. Using Corollary 3.3.8 and letting Y? = φ(Bp) ⊆ AG, we get a G-equivariant

continuous surjection Φ = ϕ◦φ−1 : Y?→ ∂?Xp. In other words, we have the following:

Corollary 3.3.10. The action of a finitely generated group on its Morse boundary admits a symbolic coding.

Question 2. In the case where G is a hyperbolic group, the analogous map Φ : Y?→ ∂?G is finite-to-one.

Is that still the case here?

Question 3. By the previous corollary, the Morse boundary of a finitely generated group G can be thought

of as a subspace of maps from G to a finite setA. Can one give a characterization of Morse rays in terms of

those maps inAG?

3.4 The Behaviour of Horofunctions in CAT(0) Spaces

Throughout this section, We assume that X is a proper complete CAT(0) space. The goal is to study the be-

haviour of Busemann functions whose defining rays are Morse. We remark that Section 2.4 is a prerequisite

to full understanding of this section.

Lemma 3.4.1. Let h be a horofunction for X. Then h must satisfy the following:

For any x0 ∈ X and any r > 0, the function h attains its minimum on the sphere S r(x0) at a unique point y

with h(y) = h(x0)− r.

Proof. Let x0 ∈ X and let r > 0. First we show that there exists y on the sphere S r(x0) with h(y) = h(x0)−r,

but this is Proposition 3.1.1 with λ= h(x0)−r. Now we show that h(y) is in fact a minimum on the sphere. If

there exists some w ∈ S r(x0) with h(w)< h(y) = h(x0)−r, then one would have d(x0,w) = r < h(x0)−h(w),

but that is not possible as h is 1-Lipschitz. Now we are left to show uniqueness. Suppose for the sake of

contradiction that ∃y′ ∈ S r(x0) such that h(y′) = h(y) = h(x0)− r with y′ , y. Let [y,y′] be the unique

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46 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

geodesic segment connecting y and y′. Notice that by convexity of the CAT(0) metric, we have d(x0, x) ≤ r

for all x ∈ [y,y′]. But since projections to geodesics in a CAT(0) are unique, there must exist some point

x ∈ [y,y′] such that d(x, x0) < r. Since h is convex and since y and y′ satisfy h(y) = h(y′), we must have

that h(x) ≤ h(y). Let g be an h-gradient ray with an initial subsegment [x0,y]. Notice that h◦g is a strictly

decreasing function of t. Let t be so that g(0) = x0 and g(t) = y. Since h(x) ≤ h(y), there must exist some

s ≥ t with h(g(s)) = h(x). Now notice that r = d(x0,y) = d(g(0),g(t)) = t ≤ s = d(g(s),g(0)) = s−0 =

h(g(0))−h(g(s)) = h(x0)−h(x) ≤ d(x0, x) < r which is a contradiction. �

Let X be any metric space, and let C(X) be the collection of all continuous maps f : X→R. Let C?(X) be

the quotient of C(X) by all constant maps. There is a natural embedding i : X→ C?(X) by i(x) = d(x,−),

denote X the closure of i(X) in C?(X). Now let B be the collection of all continuous maps h : X → R

satisfying the following three conditions:

• h is convex;

• h is 1-Lipschitz;

• For any y0 ∈ X and any r > 0, the function h attains its minimum on the sphere S r(y0) at a unique

point y with h(y) = h(y0)− r.

Proposition 3.4.2. Let X be a proper complete CAT(0) space and fix x0 ∈ X. If h is a distance-like function

with h(x0) = 0 then the following are equivalent:

1. h is a horofunction;

2. h ∈ B;

3. h is a Busemann function for some geodesic ray c;

4. h ∈ X− i(X).

Proof. (1) =⇒ (2): Notice that since h is distance-like, then by Proposition 2.2.8 it is 1-Lipschitz. Now

Lemma 3.4.1 implies that h ∈ B.

(2) =⇒ (1): By definition of B.

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3.4. The Behaviour of Horofunctions in CAT(0) Spaces 47

The equivalence of (2), (3) and (4) is Corollary 8.20 and Proposition 8.22 in the Horofunctions and Buse-

mann functions section of [5]. �

Remark 3.4.3. One consequence of the previous proposition is that any horofunction h, is of the form h= bc

where c is a geodesic ray. By Corollary 8.20 in the Horofunctions and Busemann functions section of [5],

up to changing the base point, such a c is unique. Therefore, any horofunction h defines a unique point ζ at

the CAT(0) boundary ∂X and vice versa. This gives that, up to translation by constants, horofunctions and

points in the CAT(0) boundary ∂X are in a 1-1 correspondence. Hence, for any point ζ ∈ ∂X, we will denote

its corresponding horofunction by hζ +C or simply by hζ if h is normalized to vanish at a fixed base point

x0.

Definition 3.4.4. Let ζ be a point in the CAT(0) boundary ∂X and let h be one of its horofunctions. For each

r ∈R, we define the r-horosphere of h by Hr := h−1(−r). A horosphere of h is just an r-horosphere of h for

some r. A horosphere of ζ is a horosphere of some ζ-horofunction h.

Notice that for any C ∈R both h and h+C have the same set of horospheres.

Definition 3.4.5. Let ζ be a point in the CAT(0) boundary ∂X and let h be one of its horofunction. Let

{Hn}n∈N be a sequence of horospheres for h. We say that the sequence {Hn}n∈N converges to ζ if for any

sequence {xn}n∈N with xn ∈ Hn, such that xn→ η ∈ ∂X, we must have η= ζ.

Remark 3.4.6. Notice that convergence of horospheres is a hyperbolicity phenomena. For example, in

R2 if one takes the geodesic ray c to be the positive y-axis, then the associated horospheres, which are all

hyperplanes perpendicular to c, do not converge since we have two different sequences (the one in black

and the one in green in Figure 3.10) living on the same set of horospheres but yet defining different points

in ∂R2. However, as shown in Figure 3.11, if we consider horoshperes centered at ζ in the Poincaré disk

model, we can see that any convergent sequence living on those horospheres must converge to ζ.

Definition 3.4.7. Let ζ be a point in the CAT(0) boundary ∂X and let h be one of its horofunction. For each

r ∈R, we define the r-horoball of h by Br := h−1((−∞,−r]). A horoball of h is an r-horoball for some r. A

horoball of ζ is a horoball of some ζ-horofunction h.

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48 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

Figure 3.10: Horospheres of R2 do not converge.

ζ

Figure 3.11: Horospheres in H2 are convergent.

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3.4. The Behaviour of Horofunctions in CAT(0) Spaces 49

Theorem 3.4.8 (Morse horosphers are convergent). Let X be a CAT(0) space and let ζ be a point in the

Morse boundary of X. If hζ is a corresponding horofuntion, then any sequence of hζ-horospheres must

converge to ζ.

Proof. The idea of the proof is the following. First, we will show that if hζ is a corresponding horofunction

for a Morse point, then, for any horofunction h corresponding to a different point η (not necessarily a Morse

point) the sum hζ+hηmust be bounded below. Once that has been established, the result follows easily, as if

a sequence {xn}n∈N converges to η , ζ, we would have hη(xn)→−∞ and since we also have hζ(xn)→−∞,

the sum hζ+ hη can not bounded below. Now we give the proof:

Let ζ be a point in the Morse boundary and let hζ be one of its horofunctions. We need to show that any

convergent sequence xn ∈ {Hn}n∈N, where {Hn}n∈N are hζ horospheres, must converge to ζ. Suppose for the

sake of contradiction that xn→ ηwith η, ζ. By [7], the Morse boundary of a CAT(0) space is totally visible,

hence, there must exist a geodesic line c : R→ X with c(∞) = ζ and c(−∞) = η. Define c1(t) = c|[0,∞)(t)

and c2(t) = c|(−∞,0](−t). Fix p = c(0). By Remark 3.4.3, we have hζ = bc1 +C1 and hη = bc2 +C2. For

now, lets assume that C1 = C2 = 0. The assumption that xn→ η implies that there exists a K > 0 such that

xn ∈ NK(Im(c2)) for all n ∈N. Since hη is distance-like, and up to possibly passing to a subsequence, we get

that hη(xn)→−∞. But by assumption, xn ∈Hn where Hn are horospheres of hζ ; this gives that hζ(xn)→−∞.

Therefore, hζ + hη is not bounded below. Now we argue that this can’t be the case. It is an easy exercise

to show that for any geodesic line c : R→ X and any x ∈ X, we have bc(p(x)) ≤ bc(x) where p(x) is the

projection of x on the closed convex subspace c(R). Now, notice that for all t ∈R, we have hζ(c(t)) = −t

and hη(c(t)) = t. Thus, hζ(c(t)) + hη(c(t)) = 0 for all t ∈ R. As noted above, for all x ∈ X, we have

hζ(p(x)) ≤ hζ(x) and hη(p(x)) ≤ hη(x). Therefore, we get that hζ(x)+ hη(x) ≥ hζ(p(x))+ hη(p(x)) = 0

for all x ∈ X. This show that if ζ is a Morse point and η is any other point in the CAT(0) boundary, then the

sum of their corresponding horofunctions, hζ+hη must be bounded below by zero which is a contradiction.

Now if C1 and C2 are not zero, we still get that hζ + hη is bounded below but now by −C1 −C2, which is

also a contradiction. �

False-Converse: We remark that convergence of horospheres does not characterize horofunctions whose

gradient rays are Morse due to the following example.

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50 Chapter 3. The Morse Boundary as a Quotient of Generalized horofunctions

T 2 ∨ S 1

Figure 3.12: Torus wedge a circle.

Example 3.4.9. Consider the topological space X given by a Torus wedge a circle, X = T 2 ∨ S 1. The

fundemental group of X is given by G = 〈a, b, c| [a,b]〉, where a and b correspond to the torus’s meridian

and longitude where c corresponds to the circle. Each edge of the 1-skeleton of the universal cover inherits

a labelling by a, b or c. Consider the geodeisc ray α given by α = aca2ca3ca4c... and let ζ be the point in

the CAT(0) boundary defined by α. Now, taking h = bα, the Busemann function of α, we can see that any

sequence of h-horospheres {Hn}n∈N converges to ζ. However, the point ζ is not Morse.

Conjecture 3.4.10. The CAT(0) assumption in the previous theorem can be dropped.

Lemma 3.4.11. Let ζ be a point in the Morse boundary of a CAT(0) space X and let hζ be one of its

horofunctions. If c is a geodesic ray with c(∞) , ζ then hζ(c(t))→∞ as t→∞.

Proof. The proof is easy and left as an exercise for the reader. �

Lemma 3.4.12. Any horoball is convex.

Proof. This follows easily by convexity of the corresponding horofunction. �

The following corollary states that a horoball corresponding to a point in the Morse boundary looks similar

to horoballs in Hn. More precisely, we show that a Morse horoball can’t intersect any other horoball in an

unbounded set.

Corollary 3.4.13. Let ζ be a point in the Morse boundary of a proper complete CAT(0) space X, and let

B1 be a horoball of ζ. If η is any other point in the CAT(0) boundary of X, and B2 is a horoball of η, then

B1∩B2 must be bounded.

Proof. We remark that this proof is similar to the one in Proposition 9.35 of [5], the key is that the Morse

boundary of a CAT(0) space is totally visible. Notice that since X is proper X ∪ ∂X must be compact [5].

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3.4. The Behaviour of Horofunctions in CAT(0) Spaces 51

Since B1 and B2 are horoballs for ζ and η respectively, then by definition, we get two horofunctions h1 and

h2 such that B1 is a horoball of h1 and B2 is a horoball of h2. If B1∩B2 is unbounded we get an unbounded

sequence xn ∈ B1 ∩ B2. But as X ∪ ∂X is compact, up to passing to a subsequence, we may assume that

xn → γ ∈ ∂X. Notice that by the previous Lemma, B1 and B2 are both convex and therefore B1 ∩ B2 is

convex. Hence, if p ∈ B1 ∩ B2, then [p, xn] ⊆ B1 ∩ B2 for all n. Thus [p, xn] converges to a geodesic ray

c ∈ B1∩B2. Since c ∈ B1, h1(c(t)) must be bounded above. Similarly, h2(c(t)) must also be bounded above.

But Lemma 3.4.11 implies that c(∞) = ζ = η which is a contradiction. �

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Chapter 4

Regular Languages for Contracting

geodesics

We first remark that Section 2.6 is a prerequisite to this chapter. The main theorem we prove in this chapter

is the following:

Theorem 4.0.1. Let G be a finitely generated group, and let A be any finite generating set. Then the language

LD consisting of all D-contracting geodesics in Cay(G, A) is a regular language for any D.

Since a finitely generated group G is hyperbolic if and only if every geodesic in G is D-contracting for

a uniform D, the above theorem recovers a classic result by James Cannon where he shows that for a

hyperbolic group G, and for a finitely generated set A, the language consisting of all geodesics in Cay(G, A)

is a regular language.

Corollary 4.0.2. If G is a hyperbolic group with a finite generating set A, then the language L consisting of

all geodesics in Cay(G, A) is a regular language.

Since the generating function counting the number of words of length n in a regular language is always

rational, this opens a host of combinatorial questions. For any finitely generated group, choice of generating

set, and parameter D, we can ask how many geodesics of length n are D-prefix-contracting. We can also ask

how many group elements in a ball of radius n can be reached by a D-prefix-contracting geodesic. All of

these questions can be answered with a rational generating function.

As an interesting application to the above theorem, we answer the following question posed by Denis Osin

52

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4.1. Prefix-Contracting Geodesics 53

positively: Does the existence of an infinite contracting geodesic in a finitely generated group implies that

the group is acylindrically hyperbolic?

The previous theorem does indeed yield a positive answer to this question:

Theorem 4.0.3 (Eike-Zalloum). Let G be a finitely generated group with a generating set A such that

Cay(G, A) contains an infinite contracting geodesic, then G must be acylindrically hyperbolic.

4.1 Prefix-Contracting Geodesics

Definition 4.1.1 (projection). Let C be a closed subset of a metric space X. We define the projection of a

point x onto C to be

πC(x) = {p ∈C | d(x, p) = infc∈C

d(x,c)}.

In general πC(x) may contain more than one point. Also, if B is a subset of X, we define πC(B) :=⋃x∈B

πC(x)

Definition 4.1.2 (contracting). Let c be a continuous quasi-geodesic (possibly infinite). We say that c is

D-contracting if for any closed metric ball B disjoint from c, diam(πc(B)) ≤ D. We say it is contracting if it

is D-contracting for some D.

Definition 4.1.3 (prefix-contracting). Let c be a continuous quasi-geodesic in a metric space X. We say

that c is D-prefix-contracting if every sub-segment of c is D-contracting in the above sense. That is, if

for any subsegment γ ⊆ c and any closed metric ball B disjoint from γ, diam(πγ(B)) ≤ D. We say c is

prefix-contracting if it is D-prefix-contracting for some D.

Notice that projections on contracting/prefix-contracting quasi-geodesics are coarsely well defined. More

precisely, for each x ∈ X we must have diam(π(x)) < D.

It is clear from the above definitions that if a geodesic is D-prefix-contracting, then it is D-contracting.

Section 4.3 is devoted to proving the converse of the above statement. In other words, we prove that the

notions of contracting and prefix-contracting are indeed the same for any proper geodesic metric space.

In Hn all infinite geodesics are contracting and Morse whereas in Rn, no infinite geodesic is contracting or

Morse. Therefore, both the contracting and the Morse property can be thought of as hyperbolicity properties,

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54 Chapter 4. Regular Languages for Contracting geodesics

Definition 4.1.4. Let u ∈ F(A) be a geodesic word in the Cayley graph of G. We define the D-prefix-

contracting cone of u, denoted by PconeD(u), to be all w ∈ F(A) so that the concatenation uw is a D-prefix-

contracting geodesic in the Cayley graph. If u and v have the same D-prefix-contracting cone, we will say

that u and v have the same D-prefix cone type.

Example 4.1.5. In the free group on two letters F2 = 〈a,b〉, if we take D = 1, the D-prefix-contracting cone

of a consists of all geodesic words in the group that don’t start with a−1 whereas in Z⊕Z = 〈a,b|[a,b]〉, if

D = 1, the D-prefix-contracting cone of a is empty. If we take D = 2, then in the free group example, the

D-prefix-contracting cone of a will still be the set of all geodesic words in the group that don’t start with a−1

whereas in the Z⊕Z example the D-prefix-contracting cone of a is now the set {a,b,b−1}.

Definition 4.1.6. Given an element u in F(A) representing a geodesic in the Cayley graph, let u be the

unique group element that u maps to. Given k ∈N, we define the k-tail of u to be the set of all elements

h ∈G with |h| ≤ k such that |uh| < |u|. We denote the k-tail of u by Tk(u).

The k-tail of a geodesic word u is all group elements in a ball of radius k of the identity that move u closer

to the identity in the Cayley graph.

Definition 4.1.7. Given u ∈ F(A) representing a geodesic in the Cayley graph and given t ∈N, we define

the t-local contracting type of u to be all words w ∈ F(A) with |w| ≤ t such that the edge path uw is a

D-prefix-contracting geodesic in the Cayley graph of G.

Lemma 4.1.8. Let α be an M-Morse geodesic in the Cayley graph of G starting at the identity. If γ is any

geodesic in the Cayley graph starting at the identity and ending 1 apart from α then γ is N-Morse where

N = 2M+ 1.

Proof. The proof of this lemma follows easily from Lemma 2.1 in [11]. �

The following is Lemma 2.7 in [11]. It basically states that if you have two N-Morse geodesics with the

same origin that end close to each others then they have to be roughly uniformly close.

Lemma 4.1.9. If X is any geodesic metric space and α1,α2 : [0, A]→ X are N-Morse geodesics with α1(0) =

α2(0) and d(α1(s),α2) < K for some s ∈ [0, A] and some K > 0, then d(α1(t),α2(t)) ≤ 8N(3,0) for all

t < s−K −4N(3,0).

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4.1. Prefix-Contracting Geodesics 55

Remark 4.1.10. A D-prefix-contracting geodesic is D-contracting and therefore M-Morse (Lemma 3.3

in [22]). Combining this with Lemmas 4.1.8 and 4.1.9 we get the following. If α is any D-prefix-contracting

geodesic in the Cayley graph of G starting at e , then α has to be M-Morse where M depends only on D. By

Lemma 4.1.8, if β is any other geodesic in the Cayley graph, starting at e and ending 1 apart from where α

ends, then β has to be N-Morse where N depends only on M which depends only on D. Now Lemma 4.1.9

gives us that d(α(t),β(t)) < 8N(3,0) for all t < |α| −1−4N(3,0).

Lemma 4.1.11 (bounded jumps). Let X be a geodesic metric space and let γ : [a,b]→ X be a continuous,

D-prefix-contracting quasi-geodesic. Let c ∈ [a,b]. Denote by α and β the subsegments of γ from γ(a) to

γ(c) and from γ(c) to γ(b) respectively. For any x ∈ X, if πγ(x)∩α , ∅, then every y ∈ πβ(x) satisfies

d(y,z) ≤ D, where z ∈ πγ(x)∩α.

Proof. Let x ∈ X be such that there exists a point γ(p) ∈ πγ(x)∩α , ∅. Since γ is continuous, the map

f (t) = d(x,γ(t)) is a continuous real-valued function. Let γ(q) ∈ πβ(x) and let γ(q′) be the point in α

closest to γ(c) satisfying d(x,γ(q′)) = d(x,γ(q)). That is, let q′ = sup{s ≤ c | f (s) = f (q)}. Such a point

exists by the intermediate value theorem applied to the inequality f (p) ≤ f (q) ≤ f (c). The intermediate

value theorem also guarantees that no point between γ(q′) and γ(c) is closer to x than γ(q′) is. Thus, γ(q)

and γ(q′) are both in the projection of x onto γ|[q′,q]. Since γ is D-prefix-contracting, d(γ(q′),γ(q)) ≤ D.

Therefore d(γ(p),γ(q)) ≤ D. �

Definition 4.1.12 (acylindrical action). The action of a group G on a metric space X is called acylindrical

if for every ε > 0, there exist R, N > 0 such that for every two points x,y ∈ X with d(x,y) > R, there are at

most N elements g ∈G satisfying:

d(x,gx) < ε and d(y,gy) < ε.

Definition 4.1.13. A group G is said to be acylindrically hyperbolic if it admits an acylindrical action on a

hyperbolic space.

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56 Chapter 4. Regular Languages for Contracting geodesics

4.2 The Prefix-Contracting Language is Regular

Let G be a group with a finite generating set A. The goal of this section is to show that the languages LD

consisting of all D-prefix-contracting geodesics are all regular languages regardless of the chosen gener-

ating set A. Before doing so, we will state an important key theorem that will provide us with the states

needed for our FSA, we will refer to this theorem by “the cone types theorem”. The theorem states that

in order to determine D-prefix-contracting cone type of a geodesic word u, you need only to understand

the local geometry around the vertex u. To be more precise, it says that there exist a uniform m such that

the m-neighborhood around a vertex u encodes the information needed to determine what elements are in

PconeD(u). This will imply that we have only finitely many cone types because there are only finitely many

types of m-neighborhoods in Cay(G, A).

Theorem 4.2.1. There exists a uniform m such that if u,v are two geodesic words in Cay(G, A) with the

same m-tail and the same m-local contracting type, then PconeD(u) = PconeD(v). In particular, there are

only finitely many such cones.

We will prove this theorem, but for now, let us show how it implies our first main theorem about the existence

of a regular language for all D-prefix-contracting geodesics:

Corollary 4.2.2. Let G be a group and A any finite generating set. Then for any fixed D, the language LD

consisting of all D-prefix-contracting geodesics in Cay(G, A) is a regular language.

Proof. Consider the finite graph whose vertices are the D-contracting cone types of G and which has

a directed edge labeled a ∈ A∪ A−1 connecting the D-prefix-contracting cone type of a geodesic word u

to the D-prefix-contracting cone type of ua if and only if a belongs to the D-prefix-contracting cone of u.

Otherwise the edge labeled a goes to the unique fail state, which can be thought of as the empty cone type.

All non-empty D-prefix-contracting cone types are accept states. The previous theorem shows that there

are finitely many vertices, and the D-prefix-contracting cone type was defined precisely to pick out those

continuations that are both geodesic and D-prefix-contracting. �

Now we prove Theorem 4.2.1:

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4.2. The Prefix-Contracting Language is Regular 57

Proof. We want to show that there exists some m large enough so that for any two geodesic words in

the Cayley graph u and v with the same m-tail and the same m-local contracting type then PconeD(u) =

PconeD(v). Let u, v be the unique group elements represented by u and v respectively. Note that since u,v

are assumed to be geodesic words, we must have |u|= |u| and |v|= |v|. Let N be as in Remark 4.1.10. Choose

m > max{8N(3,0)+ 1,2D}.

We proceed by induction on the length of the words in the cone. Since u and v have the same m-tail and

m-local contracting type, if w ∈ F(A) with |w| ≤ m, then w ∈ PconeD(u) if and only if w ∈ PconeD(v). This

covers the base cases. For the induction step, let w ∈ PconeD(u)∩PconeD(v) and consider wa for a ∈ A. We

want to show that wa ∈ PconeD(u) if and only if wa ∈ PconeD(v).

Suppose for contradiction that wa ∈ PconeD(v) but wa < PconeD(u). By the previous observation, we must

have |wa| ≥ m+ 1. Since wa < PconeD(u), then by definition, either the word uwa is not a geodesic word or

uwa is a geodesic word that is not D-prefix-contracting.

First we show that uwa must be geodesic. We remark that this part of the proof closely follows a proof in the

cone types section of [5]. If uwa is not a geodesic word, then there must exist some geodesic word ` of length

strictly less than |u|+ |w|+1 such that ¯ = uwa. Write ` as a product `1`2 such that |`1|= |u|−1 = |u|−1 and

|`2| ≤ |w|+ 1. Note that since w ∈ PconeD(u) then by Lemma 3.3 in [22], uw has to be M-Morse where M

depends only on D. Also since ` ends 1 apart from where w ends, Lemma 4.1.8 gives us that ` is N-Morse

(where N = 2M+ 1). In particular, both of the geodesic words uw and ` are N-Morse.

Since uw and ` end 1 apart from each other, Lemma 4.1.9 gives us that d( ¯1, u) < 8N(3,0) + 1. Define

z := u−1`1, hence, the group element z = u−1`1 satisfies |z| ≤ 8N(3,0)+ 1 ≤ m and |uz| < |u| which implies

that z ∈ Tm(u). Recall that Tm(u) = Tm(v) by assumption, so z ∈ Tm(v) and |vz| < |v|. Let α be any geodesic

word connecting e to the group element vz, so |α| < |v|. Now consider the concatenation of the geodesic α

with the edge path labeled `2. On one hand, you get α`2 = vz`2 = vu−1`1`2 = vu−1uwa = vwa. Therefore,

the edge path α`2 ends at the same vertex as the geodesic word vwa. Consequently, since vwa is a geodesic,

we have |α`2| ≥ |v|+ |w|+1. But on the other hand, if we concatenate the geodesic word α with the edge path

labeled `2 we get |α`2| ≤ |α|+ |`2| < |v|+ |w|+1 which is a contradiction. Therefore, uwa must be a geodesic

word.

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58 Chapter 4. Regular Languages for Contracting geodesics

id

u

v

uwuwa

vw

vwa

uz

vz

`2 `2

α`1

Figure 4.1: Geodesic cone is determined locally.

Now we consider the other possibility, i.e. if uwa is a geodesic word that is not D-prefix-contracting. Denote

this geodesic by q1 and its geodesic subsegment labeled wa by σ1. Similarly, denote the geodesic vwa by q2

and its geodesic subsegment labeled wa by σ2. Notice that if we let g := vu−1, then we have σ2 = gσ1. So

the assumption is that q1 is not a D-prefix-contracting geodesic but q2 is. This implies the existence of x,y

in Cay(G, A) whose projections on a subsegment of q1, say γ1, are more than D apart. In other words, we

get px ∈ πγ(x) and py ∈ πγ(y) with d(px, py) > D.

Note that since the geodesic uw is assumed to be D-prefix-contracting, then at least one of the points px and

py, say px, is on the edge labeled a at the end of the geodesic q1.

The first thing to observe is that, since g is an isometry taking σ1 to σ2, gpx ∈ πσ2(gx) which is at the edge

labeled a at the end of σ2.

Now we consider the two different possibilities for the projection of y on the subsegment γ of q1. If

all projections of y on the γ-subsegment of q1 are on σ1, then by our assumption, ∃py ∈ πσ1∩γ(y) with

d(px, py) > D. Since g is an isometry, gpx ∈ πσ2(gx), gpy ∈ πσ2∩gγ(gy) and d(gpx,gpy) = d(px, py) > D

which contradicts the assumption that q2 is D-prefix-contracting. Now the other possibility is for y to have

a projection point on the subsegment of q1 given by u. If that is the case, then by removing this subsegment

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4.2. The Prefix-Contracting Language is Regular 59

id

u

v

uw

uwa

vw

vwax

y

px

py

gx

gy

gpx

gpy

σ1

σ2

Figure 4.2: Contracting cone is determined locally.

from q1, using Lemma 4.1.11, there must exist some point py ∈ πσ1(y) which is at most D away from u. This

implies that d(px, py) ≥ (|σ1| −1)−D = |w| −D ≥ m−D > D. Again, since g is an isometry, gpx ∈ πσ2(gx),

gpy ∈ πσ2(gy) and d(gpx,gpy) = d(px, py) > D which contradicts the fact that q2 is D-prefix-contracting.

As a consequence of the above theorem, we will show that any finitely generated group which contains an

infinite prefix-contracting geodesic must be acylindrically hyperbolic answering a question posed by Denis

Osin, but first, we state an easy corollary of Theorems H and I in [2]:

Corollary 4.2.3. Let G be a finitely generated group which is not virtually cyclic, and let A be a gener-

ating set for G. If G contains an element g with a prefix-contracting axis in Cay(G, A), then G must be

acylindrically hyperbolic.

Now we show how Corollary 4.2.2 along with the above corollary imply the following:

Corollary 4.2.4. Let G be a finitely generated group which is not virtually cyclic, and let A be a generating

set for G. If Cay(G, A) contains an infinite prefix-contracting geodesic, then G must be acylindrically

hyperbolic.

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60 Chapter 4. Regular Languages for Contracting geodesics

Proof. Let γ be an infinite D-prefix-contracting geodesic. Using Corollary 4.2.2, there exists a finite state

automaton M that accepts every initial subsegment of γ. Denote its accepted language by LD. Choose a

large enough initial subsegment w of γ whose length is larger than the number of states in M. This implies

the existence of a state that w passes twice. In other words, w has an initial subsegment of the form uv such

that uvn ∈ LD for all n ∈N. Hence, v is an element whose axis is prefix-contracting, and therefore, using the

previous corollary, the group G must be acylindrically hyperbolic. �

4.3 Contracting is Equivalent to Prefix-Contracting

The goal of this section is to prove the following.

Theorem 4.3.1. Let α be a D-contracting geodesic in a proper geodesic metric space X. There exists a D′,

depending only on D so that α is D′-prefix-contracting.

First we state and reprove Lemma 3.1 of [23] in a more general setting. The proof is the same.

Lemma 4.3.2. Let a,b,c be any three points in a geodesic metric space X. Let [b,c] be any geodesic between

b and c and let p ∈ π[b,c](a). Let γ be a concatenation of a geodesic from a to p with a geodesic from p to c,

then γ is a (3,0)-quasigeodesic.

Proof. Let m be such that γ(m) = p. Let x and y be points on γ. If they are on the same geodesic segment,

then the quasi-geodesic inequalities hold trivially. So without loss of generality, suppose x = γ(s) for s ≤m

and y= γ(t) for t > m. Immediately from the definitions d(x,y) ≤ 3|s− t|. For the other direction, note that

p ∈ π[b,c](x), so d(x, p) ≤ d(x,y). Thus

|s− t|= d(x, p)+ d(p,y)

≤ d(x, p)+ d(p, x)+ d(x,y)

≤ 3d(x,y)

Therefore 13 |s− t| ≤ d(x,y) ≤ 3|s− t| as desired. �

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4.3. Contracting is Equivalent to Prefix-Contracting 61

We remind the reader of the definition of a slim geodesic 3.2.2. Using the above, we can see that contracting

geodesics are slim.

Lemma 4.3.3. Let α be a D-contracting geodesic in a proper geodesic metric space. Then α is also δ-slim

where δ depends only on D.

Proof. The proof is the same as in Theorem 3.5 of [23]. �

Remark 4.3.4. Let α be a D-contracting (and thus δ-slim by lemma 4.3.3) geodesic. Let x be any point in

X and p = πα(x). If pt is a point on α at distance t away from p, then

d(x, pt) ≥ d(x, p)+ t−2δ.

Lemma 4.3.5. Let γ : [a,b]→ X be a D-contracting geodesic and σ any subsegment of γ. Set δ to be the

constant from Lemma 4.3.3 so that γ is δ-slim. Suppose we have a point x and a projection point q ∈ πγ(x)

so that q < σ. If u is the endpoint of σ closest to q, then for any p ∈ πσ(x), d(p,u) ≤ 2δ.

Proof. If the length of σ is ≤ D, the result follows trivially, because D ≤ δ. So suppose the length is more

than D. Since πγ(x) has diameter at most D, it must be entirely on one side of σ.

Note that d(x, p) ≤ d(x,u). The idea is that a geodesic from x to p is similar in length to the path x→ q→

u→ p, so for p to be in the projection, it can’t be too far from u. More precisely, starting from the inequality

in Remark 4.3.4, we have

d(p,q)+ d(q, x) ≤ d(p, x)+ 2δ

d(p,u)+ d(u,q)+ d(q, x) ≤ d(u, x)+ 2δ

d(p,u)+ d(u,q)+ d(q, x) ≤ d(u,q)+ d(q, x)+ 2δ

d(p,u) ≤ 2δ.

Lemma 4.3.6. Let γ be a δ-slim geodesic in X. Then projection on γ is coarsely distance decreasing. That

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62 Chapter 4. Regular Languages for Contracting geodesics

is, for all x,y ∈ X and any points p1 ∈ πγ(x) and p2 ∈ πγ(y) :

d(p1, p2) ≤ d(x,y)+ 2δ.

Proof. Without loss of generality, suppose d(p1, x) ≤ d(p2,y). By Remark 4.3.4, d(p1, p2)+ d(p2,y) ≤

d(p1,y)+ 2δ. So we have

d(p1, p2) ≤ d(p1, p2)+ d(p2,y)−d(p1, x)

d(p1, p2) ≤ d(p1,y)+ 2δ−d(p1, x)

d(p1, p2) ≤ d(p1, x)+ d(x,y)+ 2δ−d(p1, x)

d(p1, p2) ≤ d(x,y)+ 2δ.

Lemma 4.3.7 (distance decreasing). Let γ be a D-contracting geodesic and σ any subsegment of γ. Let δ be

the slimness constant for γ, which depends only on D. Projections onto σ are coarsely distance decreasing

in the sense that for any p1 ∈ πσ(x) and p2 ∈ πσ(y):

d(p1, p2) ≤ d(x,y)+ 6δ.

Proof. Since γ is D-contracting, it is also δ-slim where δ comes from Lemma 4.3.3 and depends only on

D. Immediately from Lemma 4.3.6 we have d(q1,q2) ≤ d(x,y)+ 2δ for any q1 ∈ πγ(x) and q2 ∈ πγ(y). We

want to show that projections onto the subsegment σ are also coarsely distance decreasing, so let p1 ∈ πσ(x)

and p2 ∈ πσ(y) be projection points for an arbitrary pair of points x,y ∈ X. There are several cases, but none

of them are difficult.

If p1 ∈ πγ(x) and p2 ∈ πγ(y), then we immediately have d(p1, p2) ≤ d(x,y)+ 2δ. Otherwise, at least one

of the points is not already in a projection on to γ. Without loss of generality, suppose it is p1. Then there

is a projection point q1 ∈ πγ(x) that is not in σ. Denote by u the endpoint of σ nearest to q1. Note that by

Lemma 4.3.5, d(p1,u) ≤ 2δ.

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4.3. Contracting is Equivalent to Prefix-Contracting 63

If p2 ∈ πγ(y), then

d(p1, p2) ≤ d(p1,u)+ d(u, p2)

≤ 2δ+ d(q1, p2)

≤ d(x,y)+ 4δ.

Otherwise p2 is not in πγ(y) and there exists a q2 ∈ πγ(y) and not in σ. Let v denote the endpoint of σ

opposite u. Either u or v is closer to q2. If v is closer to q2, then

d(p1, p2) ≤ d(p1,u)+ d(u,v)+ d(v, p2)

≤ 2δ+ d(q1,q2)+ 2δ

≤ d(x,y)+ 6δ.

If u is closer to q2 we have instead

d(p1, p2) ≤ d(p1,u)+ d(u, p2)

≤ 4δ

≤ d(x,y)+ 4δ.

We are now ready to prove that subsegments of a contracting geodesic are also contracting. Our proof is

essentially the same as that of Lemma 3.2 in [3]. We include it only to verify that is still valid assuming only

that X is a proper geodesic metric space, using Lemma 4.3.7 in place of their distance decreasing axiom.

Lemma 4.3.8. Let γ be a D-contracting geodesic. There is a constant D′, depending only on D, so that any

subsegment σ of γ is D′-contracting. Consequently, γ is D′-prefix-contracting.

Proof. Let δ= δ(D) be the slimness constant from Lemma 4.3.3. Let u and v be the endpoints of σ.

For especially short subsegments we can simply use a D′ large relative to the length of σ. So we will assume

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64 Chapter 4. Regular Languages for Contracting geodesics

that length(σ) > D. This ensures that no point’s projections onto γ can fall on both sides of σ, because the

diameter of a point’s projection is at most D. Let B be a ball centered at a point z and disjoint from our

subsegment σ. From here we divide the proof into two cases. The constant D′ = 2D+ 12δ+ 3 is sufficient

for both.

Case 1. πγ(z)∩σ , ∅. In this case, the ball B is also disjoint from γ, so we know that the diameter of πγ(B)

is at most D. Let I = [s1, s2] denote the smallest interval containing all of the points in πγ(B). The length

of I is bounded above by D. We will argue that every point in πσ(B) is within a distance 2δ of I∩σ.

If x ∈ B has projection πγ(x)∩σ , ∅, then πσ(x) = πγ(x)∩σ. In particular, πσ(x) ⊂ I. This is true for z, so

we know this holds for at least one point. Suppose instead πγ(x) contains a point q < σ. For concreteness,

suppose u is the nearest endpoint to q (otherwise apply the same argument with v instead of u). Then u ∈ I

and d(u, p) ≤ 2δ for any p ∈ πσ(x) by Lemma 4.3.5. So everything in πσ(B) is within 2δ of I, an interval of

length at most D. Therefore the diameter of πσ(B) is at most D+ 4δ for Case 1.

Case 2. πγ(z)∩ [u,v] = ∅. In this case it is possible that B intersects γ. The projection of z, the center, onto

γ must be entirely on one side of σ, because its diameter is no more than D. In fact, nothing in πγ(B) can

be on the opposite side of σ for the same reason. Again for concreteness assume u is the endpoint closest

to πγ(z). In this case, it will turn out that everything in B projects close to u. Let px be in πγ(x) for some

x ∈ B. If px is not in σ, then d(πγ(x),u) ≤ 2δ by Lemma 4.3.5. When px ∈ σ, we will show by contradiction

that px still can’t be too far from u.

Suppose for contradiction that d(px,u) > D + 6δ+ 3. Pick a geodesic [x,z] and let y be the first point

travelling from x to z so that there is a py ∈ πγ(y)∩σ. Take a point y′ less than distance 1 from y along [x,y]

with py′ ∈ πγ(y′). By applying Lemmas 4.3.6 and 4.3.5, we get d(py,u) ≤ d(py, py′) ≤ 1+ 2δ. Let B′ be

the closed ball of radius r′ = d(y,γ)− 1. This is the ball we will use to draw a contradiction with γ being

D-contracting. The point x is not necessarily in B′, however, we can show that it is not too far from B′.

Recall that both x and y are in a ball around z that is disjoint from [u,v] by assumption. From this we get

that

d(y,u) ≥ d(z,u)−d(z,y) = d(z,u)−d(z, x)+ d(x,y) ≥ d(x,y).

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4.4. Generating Functions for Regular Languages are Rational 65

On the other hand,

d(y,u) ≤ d(y, py)+ d(py,u) ≤ (r′+ 1)+ (1+ 2δ) ≤ r′+ 2δ+ 2.

Combining the two we see that d(x,y) ≤ r′+2δ+2. This means we can find x′ ∈ B′ with d(x′, x) ≤ 2δ+2.

Now we project. Let px′ be in πγ(x′). By the distance decreasing Lemma 4.3.6, d(px, px′) ≤ 4δ+ 2. To

show a contradiction, we now have

d(px′ , py) ≥ d(px′ ,u)−1−2δ

≥ d(px,u)−6δ−3

> D.

Now we’ve established that every point p ∈ πσ(B) is within a distance of D+ 6δ+ 3 of u, so the diameter

of the projection is bounded above by twice that. �

Using the previous lemma and Corollary 4.2.2 and Corollary 4.2.4 we get the following two corollaries:

Corollary 4.3.9. Let G be a finitely generated group and A any finite generating set. Then for any fixed D,

the language LD consisting of all D-contracting geodesics in Cay(G, A) is a regular language.

Corollary 4.3.10. Let G be a finitely generated group which is not virtually cyclic, and let A be a generating

set for G. If Cay(G, A) contains an infinite contracting geodesic, then G must be acylindrically hyperbolic.

4.4 Generating Functions for Regular Languages are Rational

In this section, we give some background on the generating functions corresponding to regular languages,

and as a consequence of the first main theorem of this chapter, we show that the generating function corre-

sponding to the growth of all contracting geodesics is a rational function.

Lemma 4.4.1. Let L be a regular language, and let ai be the number of words of length i in L, then the

sequence {ai} satisfies a linear recursion.

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66 Chapter 4. Regular Languages for Contracting geodesics

Proof. The proof follows easily using the adjacency matrix of a finite state automaton describing L and

Cayley-Hamilton’s Theorem. �

Definition 4.4.2. Given a sequence of real numbers {an}with n ∈N, the generating function of the sequence

ai is defined to be the formal sum:

g(x) :=∞∑

i=0anxn.

If in addition g(x) = P(x)Q(x) where P(x) and Q(x) are real polynomials in x, then we say that the generating

function g(x) is rational.

The following is a well known fact stating that the generating function for a sequence is rational if and only

if the sequence satisfies a linear recurrence.

Lemma 4.4.3. Given a sequence of real numbers {ai}with i ∈N, then the corresponding generating function

g(x) is rational if and only if there exist c1, . . . ,ck such that an = c1an−1 + c2an−2 + ...+ ckan−k ∀n ∈N.

The above two lemmas show that if L is a regular language then the generating function corresponding to

the sequence {an} whose nth-term is the number of elements of L of length n must be rational. Sometimes,

it is useful to consider the sequence {bn} where bn is the number of words in L of length at most n. Since

an = bn−bn−1 it follows that {an} satisfies a linear recursion if and only if {bn} does. Therefore, the generating

function ga corresponding to {an} is rational if and only if the generating function gb corresponding to {bn}

is rational.

Definition 4.4.4. Let L be a regular language and let {bn} be the sequence defined as above. The growth

function for L, p : N→N is defined to be:

p(n) := bn.

Corollary 4.4.5. Fix D ≥ 0. If bn is the number of D-contracting geodesics of length at most n, then bn

satisfies a linear recursion. Furthermore, the generating function corresponding to bn must be rational.

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