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c 2009 Sylvia E. B. Carlisle

c 2009 Sylvia E. B. Carlislehenson/cfo/carlisle... · 2009. 8. 13. · SYLVIA E. B. CARLISLE B.A., Carleton College, 2002 DISSERTATION Submitted in partial ful llment of the requirements

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Page 1: c 2009 Sylvia E. B. Carlislehenson/cfo/carlisle... · 2009. 8. 13. · SYLVIA E. B. CARLISLE B.A., Carleton College, 2002 DISSERTATION Submitted in partial ful llment of the requirements

c© 2009 Sylvia E. B. Carlisle

Page 2: c 2009 Sylvia E. B. Carlislehenson/cfo/carlisle... · 2009. 8. 13. · SYLVIA E. B. CARLISLE B.A., Carleton College, 2002 DISSERTATION Submitted in partial ful llment of the requirements

MODEL THEORY OF REAL-TREES AND THEIR ISOMETRIES

BY

SYLVIA E. B. CARLISLE

B.A., Carleton College, 2002

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2009

Urbana, Illinois

Doctoral Committee:

Professor S lawomir Solecki, ChairProfessor C. Ward Henson, Director of ResearchAssociate Professor Ilya KapovichProfessor Emeritus Carl G. Jockusch

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To my family.

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ACKNOWLEDGMENTS

Most importantly, many thanks to my adviser Ward Henson for all his guidance and support.

This thesis certainly would not have happened without his help. I would also like to acknowledge

the great mathematics teachers I was lucky enough to learn from. There are too many to list

here. Finally, thanks to my family and friends for their encouragement. This includes especially

my mom Barbara, my dad Tom, my brother Ben and my partner Will.

iii

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PREFACE

Continuous logic as developed by Ben Yaacov, Berenstein, Henson and Usvyatsov in [1] is an

extension of first-order logic that expands the scope of model-theoretic tools to include structures

from analysis and geometry in a natural way. It grew out of several earlier approaches to this

challenge, including Henson’s positive bounded logic for Banach spaces, Ben Yaacov’s metric

compact abstract theories and [0, 1]-valued logics studied by Chang and Keisler.

An R-tree is a uniquely geodesic metric space such that between any two points there is

exactly one arc. In this thesis, we begin by exploring the continuous theory of R-trees, denoted

RT. We show this theory has a model companion, denoted rbRT, which has nice model theoretic

properties. Since R-trees may be unbounded, we use a many-sorted approach. The idea is to take

a pointed metric space (M,d, p) and have signature with a sort for each closed ball of diameter

n centered at p. We must also have functions between the sorts that will be interpreted as

inclusion maps, preserving the relationships between the balls.

We also study theories of R-trees equipped with isometries, and find model companions

to those theories. By an isometry of an R-tree M , we mean a surjective distance preserving

function from M to M . Isometries of R-trees fall into two categories. If an isometry f of

an R-tree M has a fixed point it is called elliptic, otherwise it is hyperbolic. The quantity

||f || := infx∈M d(x, f(x)) is called the translation distance of f . If ||f || = 0, then f is elliptic.

If ||f || > 0, then f is hyperbolic and acts as a translation along an axis, which is a copy of R in

M . The points on this axis are moved by exactly distance ||f ||. (See [7, 1.3])

In Chapter 1 we outline the basics of continuous logic and model theory of metric structures

necessary for this thesis. Much of this material is adapted from [1]. In Section 1.2 we give a

many-sorted, bounded continuous signature Lp which we will use to study R-trees. In Section 1.8

we explain more about this signature and its structures, and set up our many-sorted approach

to studying unbounded metric spaces. We also address some issues with definability inherent

to this many-sorted approach.

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In Chapter 2 we define an Lp-theory RT which axiomatizes the class of R-trees (Theorem

2.2.3) and discuss some useful properties of this theory. We then define the class of richly

branching R-trees (Definition 2.3.1). We axiomatize this class and denote its theory by rbRT

(Theorem 2.3.6). Next, we prove that rbRT is the model companion of RT (Theorem 2.5.4). In

Section 2.6, we prove some model theoretic properties of rbRT, including quantifier elimination,

completeness and stability.

In Chapter 3 we expand the signature Lp to a signature Ls which is suitable for studying

R-trees equipped with isometries. Then, for each r ∈ R>0 we consider a specific class of R-trees

equipped with a hyperbolic isometry f , such that in every member of the class, the translation

distance satisfies ||f || ≥ r (Definition 3.3.1). We give an Ls-theory HRTr,s which axiomatizes

this class (Lemma 3.3.3). Then, we show that a model of HRTr,s is existentially closed if and

only if its underlying R-tree is a model of rbRT (Theorem 3.4.5). This lets us find the model

companion theory rbHRTr,s (Theorem 3.4.7). In Section 3.5 we prove some model theoretic

properties of rbHRTr,s and its completions, including quantifier elimination and stability.

In Chapter 4, we study the class of R-trees equipped with an elliptic isometry. We give an

Ls-theory ERTs which axiomatizes this class (Lemma 4.1.3). Then, we give axioms for a theory

rbERTs (Definition 4.3.3,) and show it is the model companion of ERTs (Theorem 4.3.13). In

Section 4.4, we prove some model theoretic properties of rbERTs and its completions, including

quantifier elimination and stability.

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TABLE OF CONTENTS

CHAPTER 1 CONTINUOUS LOGIC BACKGROUND . . . . . . . . . . . . . . . . . . 11.1 Metric structures and signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Main example: pointed metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Continuous logic: syntax and semantics . . . . . . . . . . . . . . . . . . . . . . . 31.4 Model theory basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Types and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 More model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.8 Lp-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

CHAPTER 2 MODEL THEORY OF R-TREES . . . . . . . . . . . . . . . . . . . . . . 272.1 Introduction to R-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 The theory of R-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Richly branching R-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Example: Universal R-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5 The model companion of RT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.6 Properties of the model companion . . . . . . . . . . . . . . . . . . . . . . . . . . 45

CHAPTER 3 HYPERBOLIC ISOMETRIES OF R-TREES . . . . . . . . . . . . . . . . 503.1 Introduction to isometries of R-trees . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 The signatures Ls and Ls-structures . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Theories of hyperbolic isometries of R-trees . . . . . . . . . . . . . . . . . . . . . 523.4 Model companions: hyperbolic case . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5 Properties of the model companions . . . . . . . . . . . . . . . . . . . . . . . . . 63

CHAPTER 4 ELLIPTIC ISOMETRIES OF R-TREES . . . . . . . . . . . . . . . . . . . 684.1 Theories of elliptic isometries on R-trees . . . . . . . . . . . . . . . . . . . . . . . 684.2 Orbits under an elliptic isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Model companions: elliptic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4 Properties of the model companions . . . . . . . . . . . . . . . . . . . . . . . . . 89

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

AUTHOR’S BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

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CHAPTER 1

CONTINUOUS LOGIC BACKGROUND

1.1 Metric structures and signatures

This section sets out the definitions of continuous structures and signatures used in this thesis.

The material here is adapted and summarized from [1]. An alternative presentation is given in

[2].

First, some preliminary definitions. Let (M,dM ) and (N, dN ) be metric spaces. Given a

function f : M → N , a function ∆f : (0, 1]→ (0, 1] is called a modulus of uniform continuity for

f if, for any ε ∈ (0, 1], whenever dM (x, y) < ∆(ε) then dN (f(x), f(y)) ≤ ε. Also, if p ∈ M we

will denote the closed ball in (M,dM ) of radius n centered at p by BMn (p). If the metric space

in which this ball resides is clear, we will omit the superscript M .

Note: In this thesis, if (Mi, di) are metric spaces for i = 1, ..., n, then unless otherwise specified

the metric on M1 × ...×Mn is taken to be the maximum metric

d((x1, ..., xn), (y1, ..., yn)) := max{d(xi, yi) | i = 1, ..., n}.

A many-sorted, bounded metric structure

M =(

((M (s), d(s)) | s ∈ S), (Fi | i ∈ I), (cj | j ∈ J), (Rk | k ∈ K))

consists of:

• a family ((M (s), d(s)) | s ∈ S) of complete bounded metric spaces, with S 6= ∅, called the

sorts of the structure;

• a family of functions (Fi | i ∈ I), where for each i ∈ I, there are s0, ..., sn ∈ S such that

Fi : M (s1) × ...×M (sn) →M (s0) is uniformly continuous;

• a family of constants (cj | j ∈ J), each from a specific sort;

1

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• a family of predicates (Rk | k ∈ K), where for each k ∈ K, there are s1, ..., sn ∈ S such

that Rk : M (s1) × ...×M (sn) → R≥0 is a bounded uniformly continuous function.

In order to work with these metric structures, we need appropriate continuous signatures.

A many-sorted, bounded continuous signature L consists of:

• a non-empty sort index set S and a family of non-negative real numbers (Cs | s ∈ S);

• a family of function symbols (Fi | i ∈ I), each with an arity (s1, ..., sn; s0) where s0, ...sn ∈

S, and a modulus of uniform continuity ∆Fi: (0, 1]→ (0, 1];

• a family of constant symbols (cj | j ∈ J), each with a given arity s ∈ S;

• a family of predicate symbols (Rk | k ∈ K), each with an arity (s1, ..., sn) where s1, ..., sn ∈

S, a closed bounded interval IRkin R≥0, and a modulus of uniform continuity ∆Rk

;

• an infinite number of variables of each sort;

• a metric symbol d(s) for each s ∈ S;

where the sets I, J and K are possibly empty.

Given a continuous signature L. An L-structure

M =(

((M (s), d(s)) | s ∈ S), (FMi | i ∈ I), (cMj | j ∈ J), (RM

k | k ∈ K))

consists of:

• for each s ∈ S, a metric space (M (s), d(s)) with diameter less than or equal to Cs;

• for each i ∈ I, a function FMi : M (s1)×M (s2)× ...×M (sn) →M (s0) that satisfies the given

modulus of uniform continuity ∆Fi, where (s1, ..., sn; s0) is the arity of Fi;

• for each j ∈ J , an element cMj of M (s), where s is the arity of cj ;

• for each k ∈ K, a predicate RMk : M (s1)×...×M (sn) → IRk

that satisfies the given modulus

of uniform continuity ∆Rk, where (s1, ..., sn) is the arity of Rk.

1.2 Main example: pointed metric spaces

A pointed metric space is a metric space with a specified basepoint. As an example of a many-

sorted continuous signature and its structures, in this section we define a signature Lp suitable

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for studying pointed metric spaces. It is this signature that we use in this thesis to study R-

trees. The pointed metric spaces are allowed to be unbounded, and the signature Lp provides a

framework for studying them using many-sorted bounded continuous logic. Section 1.8 contains

more about this signature and a discussion of its use.

The continuous signature Lp consists of:

• the sort index set S = N and a family of positive real numbers (Cn = 2n | n ∈ N);

• for each m,n ∈ N with m ≤ n, a function symbol Im,n with arity (m;n) and modulus of

uniform continuity ∆m,n(ε) = ε;

• constant symbols (pn | n ∈ N), where each pn has arity n;

• metric symbols d(n) for n ∈ N.

Given a pointed metric space (M,d, p), the corresponding Lp-structure is

M =(

(M (n), d(n))|n ∈ N), pMn , I

Mm,n

)

where:

• the nth sort (M (n), d(n)) is the closed ball BMn (p) of radius n centered at p in (M,d) with

d(n) equal to d restricted to BMn (p);

• for each m,n ∈ N with m ≤ n, the function Im,n : M (m) →M (n) is the inclusion map;

• the constants pn are all interpreted as the point p.

1.3 Continuous logic: syntax and semantics

This section presents the syntax and semantics of continuous logic. The material here is adapted

from [1]. For this section, let L be a continuous signature and let

M =(

((M (s), d(s)) | s ∈ S), (Fi | i ∈ I), (cj | j ∈ J), (Rk | k ∈ K))

be an L-structure. Let M stand for the underlying collection of sorts ((M (s), d(s)) | s ∈ S).

Syntax

In continuous logic, L-terms are built up inductively from variables and constants using function

3

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symbols in the same way as in first order logic. Each term has an associated sort s ∈ S. The

definition is as follows:

• any variable or constant of a given sort s is a term of sort s;

• if F is a function symbol of arity (s1, ..., sn; s0), and t1, ..., tn are terms of sort s1, ..., sn

respectively, then F (t1, ..., tn) is a term of sort s0.

Formulas are built inductively as in first order logic, starting with atomic formulas and applying

connectives and quantifiers to obtain new formulas. Note that every formula ϕ will have range

contained in a closed bounded interval Iϕ contained in R≥0, which we must specify. For a

formula ϕ, call Iϕ the range interval of ϕ. Atomic L-formulas are defined as follows:

• d(s)(t1, t2) is an atomic L-formula with range interval [0, Cs], if t1 and t2 are terms of sort

s;

• R(t1, ..., tn) is an atomic L-formula with range interval IR if R is a predicate symbol with

arity (s1, ..., sn) and ti is a term of sort si for all i ∈ {1, 2, ...n}.

Any continuous function u : [0,∞)n → [0,∞) is a connective, and for each sort s ∈ S and each

variable x of sort s we have quantifiers sups,x

and infs,y

. The formal definition of L-formula is as

follows:

• any atomic L-formula is an L-formula, with range interval as specified above;

• given L-formulas ϕ1, ..., ϕn with range intervals Iϕ1 , ..., Iϕn respectively, if u : [0,∞)n →

[0,∞) is a connective, then u(ϕ1, ..., ϕn) is an L-formula with range interval u(Iϕ1 , ..., Iϕn);

• given an L-formula ϕ(x) with free variable x of sort s, infs,x

ϕ(x) and sups,x

ϕ(x) are both

L-formulas with range interval Iϕ.

If the free variables are among x1, ..., xn, we often denote a term by t(x1, ..., xn) to emphasize

that fact. Likewise, we may write formulas ϕ(x1, ..., xn) if the free variables in a term are among

x1, ..., xn. An L-sentence is an L-formula with no free variables. A formula is called quantifier

free if it is built via the definition of L-formula without using quantifiers.

Semantics

The family A = (A(s) | s ∈ S) is called a subset of ((Ms, d(s)) | s ∈ S) if A(s) ⊆ M (s) for

all s ∈ S. We will often denote this by A ⊆ M . Let A be a subset of ((M (s), d(s)) | s ∈ S).

We extend the signature L to a signature L(A) by adding a new constant symbol c(a) of the

4

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appropriate sort for each a ∈ A. Every L-structure extends canonically to an L(A) structure by

taking the interpretation of c(a) to be a for each a ∈ A. We denote this extension by (M, a)a∈A.

For the sake of readability, we will write a instead of c(a) when it does not cause confusion.

Let t(x1, ..., xn) be an L(M)-term of sort s0, where xi is of sort si for i = 1, ..., n. Then tM

denotes the interpretation of t in M. This interpretation tM is defined as in first order logic (by

induction on the definition of L-term) and is a function

tM : M (s1) × ...×M (sn) →M (s0).

Next, for each L(M)-sentence σ, we define the value of σ in M by induction. This value is a

real number in Iσ and is denoted σM. Note, the terms in the definition below must have no

variables. (So they are terms built up from constants only.)

1.3.1 Definition. 1. (d(s)(t1, t2))M = d(s)(tM1 , tM2 ) for any terms t1, t2 of sort s;

2. (P (t1, ..., tn))M = PM(tM1 , ..., tMn ) for terms of appropriate sorts;

3. for any L(M)-sentences σ1, ..., σn and any connective u : [0,∞)n → [0,∞),

(u(σ1, ..., σn))M = u(σM1 , ..., σM

n );

4. for any L(M) formula ϕ(x), where x is of sort s,

(sups,x

ϕ(x))M

is the supremum in Iϕ of the set {ϕ(a)M | a ∈M (s)};

5. for any L(M) formula ϕ(x), where x is of sort s,

(infs,x

ϕ(x))M

is the infimum in Iϕ of the set {ϕ(a)M | a ∈M (s)}.

Given an L(M)-formula ϕ(x1, ..., xn), we use the notation ϕM to denote the function from

M (s1) × ...×M (sn) to Iϕ defined by

ϕM(a1, ...., an) = (ϕ(a1, ..., an))M.

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Two L-formulas ϕ(x1, ..., xn) and ψ(x1, ..., xn) are called logically equivalent if

ϕN(x1, ..., xn) = ψN(x1, ..., xn)

for any L-structure N. It is a fact that the interpretation tM(x1, ..., xn) of a term t(x1, ..., xn), and

the interpretation ϕM(x1, ..., xn) of a formula ϕ(x1, ..., xn) are uniformly continuous functions.

Moreover (see [1, Theorem 3.5]), for each L-term t(x1, ..., xn) and each L-formula ϕ(x1, ..., xn)

there exist functions ∆t and ∆ϕ from (0, 1] to (0, 1] such that in any L-structure N, the func-

tion ∆t is a modulus of uniform continuity for tN(x1, ..., xn) and ∆ϕ is a modulus of uniform

continuity for ϕN(x1, ..., xn).

As we saw above, in continuous logic, the interpretation ϕM of a formula ϕ(x1, ..., xn) in

a structure M will be a function mapping n-tuples from the underlying metric spaces into a

closed bounded interval Iϕ. This is in contrast to the situation of first order logic, where the

interpretation of a formula is a function that maps an n-tuple from the underlying set of a

structure N into the discrete set {true, false}.

We make the following syntactic definition. An L-condition is a statement of the form ϕ = r

where ϕ is a formula and r ∈ R≥0. A closed L-condition is an L-condition ϕ = r such that ϕ is an

L-sentence. We often denote an L-condition ϕ(x1, ..., xn) = r by E(x1, ..., xn) if the free variables

of the formula are among x1, ..., xn. For an L-condition E(x1, ..., xn) =(ϕ(x1, ..., xn) = r

)and a1, ..., an ∈ M (s1) × ... × M (sn), we say that E(x1, ..., xn) is true of a1, ..., an in M if

ϕM(a1, ..., an) = r. This is denoted M |= E[a1, ..., an].

1.4 Model theory basics

In this section, we present some basic model theoretic concepts, adapted from [1]. For this

section, let L be a continuous signature and let M and N be L-structures with sorts (M (s) | s ∈ S)

and (N (s) | s ∈ S) respectively. We say M is a substructure of N, denoted M ⊆ N, if:

• M (s) ⊆ N (s), and d(s) on M (s) is the restriction of d(s) on N (s) for all s ∈ S;

• FN extends FM for every function symbol F in L;

• RN extends RM for every predicate symbol R in L;

• cN = cM for every constant symbol c in L.

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The structures M and N are isomorphic if there is a family of surjective isometries

(g(s) : M (s) → N (s) | s ∈ S) with the following properties:

• for each function symbol F of arity (s1, ..., sn; s0) in L, and any a1 ∈M (s1), ..., an ∈M (sn)

g(s0)(FM(a1, ..., an)) = FN(g(s1)(a1), ..., g(sn)(an));

• for each constant symbol c with arity s in L,

g(s)(cM) = cN;

• for each predicate symbol R in L of arity (s1, ..., sn) in L and any a1 ∈ M (s1), ..., an ∈

M (sn),

RN(g(s1)(a1), ..., g(sn)(an)) = RM(a1, ..., an).

Such a family of maps is called an isomorphism from M onto N.

An L-theory is a set of closed L-conditions. If T is an L-theory, then an L-structure M is

called a model of T , denoted M |= T , if M |= E for every E ∈ T . If M is an L-structure, we

define the L-theory of M to be the set of closed L-conditions Th(M) := {E | M |= E}. An

L-theory T that equals Th(M) for some L-structure M is called a complete theory. (Note: this

is not to be confused with the fact that the underlying metric spaces for our structures are all

metrically complete.) Given a class K of L-structures, we often study the theory of that class,

Th(K) := {E |M |= E for all M ∈ K}. A class of L-structures K is axiomatizable in L if there

exists a set Σ of closed L-conditions such that K is exactly the class of all models of Σ.

The L-structures M and N are called elementarily equivalent, denoted M ≡ N, if for every

closed L-condition E, M |= E if and only if N |= E. In other words, σM = σN for every

L-sentence σ. A family of maps (g(s) : M (s) → N (s) | s ∈ S) is called an elementary embedding

if

• each g(s) is an isometric embedding;

• for all formulas ϕ(x1, ..., xn) where x1, ..., xn are suitable variables, and for any a1, ..., an

from M (s1), ...,M (sn) respectively, we have

ϕM(a1, ..., an) = ϕN(g(s1)(a1), ..., g(sn)(an)).

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Note that an isomorphism is always an elementary embedding. If M ⊆ N and the inclusion maps

on the sorts constitute an elementary embedding, then M is called an elementary substructure

of N, and N is an elementary extension of M, denoted M � N.

Instead of using cardinality to measure the size of structures, in continuous logic the density

character of the underlying metric space is used. In our many-sorted setting, we will define the

density character of an L-structure to be the list κ := (κs | s ∈ S) of the density characters of

its sorts. An L-theory T is κ-categorical if any two L-structures with the same density character

κ are isomorphic.

An L-theory T has quantifier elimination if, for any formula ϕ(x1, ..., xm), there exist quan-

tifier free formulas {ψn(x1, ..., xm) | n ∈ N} such that for any ε > 0, there exists N ∈ N such

that for all n ≥ N and any M |= T

|ϕM(x1, ..., xm)− ψMn (x1, ..., xn)| ≤ ε.

1.5 Ultraproducts

In this section we define the ultraproduct of a family of L-structures, and give some results

concerning of ultraproducts. For this section, let L be a continuous signature.

Let X be a topological space. Let xi ∈ X for all i ∈ I and x ∈ X. Let U be an ultrafilter

on I. Then, the U-ultralimit of (xi)i∈I is defined to be

limU,i

xi = x

if for any open set O containing x, the set {i ∈ I | xi ∈ O} is in U .

Next, the ultraproduct for bounded metric spaces is defined. (See [1, Chapter 5].) Say

((Mi, di) | i ∈ I) is a family of uniformly bounded metric spaces indexed by the infinite set

I, and let U be an ultrafilter on I. We define∏U Mi = (N, d), the ultraproduct over U of

((Mi, di) | i ∈ I) as follows.

Let N be the cartesian product∏i∈I

Mi. Define a pseudometric d on N by d((ai), (bi)) =

limU,i

di(ai, bi). Let (N, d) be the result of taking the quotient of (N , d) by the equivalence rela-

tion defined by x ∼U y if and only if d(x, y) = 0. Let [ai]U denote the equivalence class of the

sequence (ai)i∈I .

Now that we have a notion of ultraproduct for metric spaces, we define how to take ultra-

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products of metric structures. Say (Mi | i ∈ I) is a family of L-structures indexed by an infinite

set I. Let U be a non-principal ultrafilter on I. We define N =∏U Mi, the ultraproduct over

U of metric structures (Mi | i ∈ I) as follows.

• For each sort s ∈ S, let (N (s), d(s)) be the ultraproduct∏U M

(s)i . (Note that this is an

ultraproduct of bounded metric spaces.)

• For each function symbol F ∈ L and (aji )i∈I for j = 1, ..., n (where the aji are from

appropriate sorts) let

FN([a1i ]U , ..., [a

ni ]U ) = [FMi(a1

i , ..., ani )]U .

Note that this is a function from∏U M

(s1)i × ... ×

∏U M

(sn)i = N (s1) × ... × N (sn) to∏

U M(s0)i = N (s0), and it satisfies modulus of uniform continuity ∆F , because each of the

factors FMi satisfies that modulus (see [1, Chapter 5]).

• For each constant symbol c ∈ L, let cN = [cMi ]U .

• For each predicate symbol R ∈ L and (aji )i∈I for j = 1, ..., n (where the aji are from

appropriate sorts) let

RN(([a1i ]U , ..., [a

ni ]U ) = lim

U,iRMi(a1

i , ..., ani ).

This predicate is a uniformly continuous function from∏U M

(s1)i × ... ×

∏U M

(sn)i =

N (s1) × ... × N (sn) to IR which satisfies the modulus of uniform continuity ∆R (see [1,

Chapter 5]).

As demonstrated by the following theorem and its corollary, ultraproducts of metric struc-

tures are useful in continuous logic for building elementary extensions. An ultrapower of a

structure M is an ultraproduct of the family (Mi | i ∈ I) where Mi = M for all i ∈ I.

1.5.1 Theorem (Fundamental theorem of ultraproducts). ([1, Theorem 5.4]) Let (Mi | i ∈ I) be

a family of L-structures. Let U be an ultrafilter on I and let M =∏U Mi be the U -ultraproduct of

(Mi | i ∈ I). Let ϕ(x1, ..., xn) be an L-formula. If aj = [aji ]U are elements of M for j = 1, ..., n,

then

ϕM(a1, ..., an) = limi,U

ϕMi(a1i , ..., a

ni ).

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1.5.2 Corollary. ([1, Corollary 5.5]) Let N be an L-structure and∏U N be the ultrapower of

N. Let j be the embedding of N into∏U N which sends x to [x]U . Then j is an elementary

embedding of N into∏U N.

1.6 Types and stability

This section contains definitions and results adapted from Chapters 8 and 14 of [1]. Let L be

a continuous signature and M an L-structure. Let A ⊆ M . Let b1 ∈ M (s1), ..., bn ∈ M (sn).

The type of b1, ..., bn over A in M, denoted tpM(b1, ..., bn/A), is the set of L(A)-conditions

E(x1, ..., xn) such that (M, a)a∈A |= E[b1, ..., bn]. We say this type has arity (s1, ..., sn). Let TA

be a complete L(A)-theory. Denote by Ss1,...,sn(TA) the collection of types

{tpM(b1, ..., bn/A) | (M, a)a∈A |= TA, b1 ∈M (s1), ..., bn ∈M (sn)}.

If A = ∅ we omit it from the notation, and where T is clear we may write Ss1,...,sn(A) for

Ss1,...,sn(TA). Given a tuple of sorts (s1, ..., sn), we define a metric on Ss1,...,sn(A) as follows:

d(p, q) = inf{maxjdMA(bj , cj) |MA |= p[b1, ..., bn], MA |= p[c1, ..., cn]}

where MA varies over all models of TA. It is straightforward to see that this is a metric on

Ss1,...,sn(A). We call it the d-metric on the type space Ss1,...,sn

(A). The set Ss1,...,sn(A) with

the corresponding d-metric is a complete metric space. (See [1, Proposition 8.8].)

Let T be a complete L-theory and let λ be an infinite cardinal. The theory T is λ-stable if

for any M |= T , for any A ⊆ M with |A| ≤ λ, for every sort s the space Ss(TA) has density

character ≤ λ with respect to the d-metric. We say T is stable if it is λ-stable for some λ. For

more about types and stability in continuous logic see [1] and [2].

1.7 More model theory

This section presents more definitions and results of model theory for metric structures, adapted

from [1]. For this section let L be a continuous signature, T an L-theory and M an L-structure.

A set Σ(x1, ..., xn) of L-conditions (with free variables among x1, ..., xn)is called satisfiable in

M if there exist a1, ..., an in M such that M |= E[a1, ..., an] for every E(x1, ..., xn) ∈ Σ. Let κ

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be a cardinal. A model M of T is called κ-saturated if for any set of parameters A ⊆ M with

cardinality < κ and any set Σ(x1, ..., xn) of L(A)-conditions, if every finite subset of Σ(x1, ..., xn)

is satisfiable in (M, a)a∈A, then the entire set Σ(x1, ..., xn) is satisfiable in (M, a)a∈A.

1.7.1 Proposition. ([1, Proposition 7.6]) For any countably incomplete ultrafilter U on I, the

U -ultraproduct of a family of L-structures (Mi | i ∈ I) is ω1-saturated.

Note that any non-principal ultrafilter on N is countably incomplete.

1.7.2 Proposition. ([1, Proposition 7.10]) For any cardinal κ, any L-structure M has a κ-

saturated elementary extension.

Saturated structures have many useful properties. For example, in an ω-saturated structure

all quantifiers are realized exactly. The proposition below captures this idea. It is stated for

structures with just one sort, but an analogous statement holds for many-sorted structures.

1.7.3 Proposition. ([1, Proposition 7.7]) Let M be an L-structure and suppose E(x1, ..., xm)

is the L-condition

(Q1y1 ...Q

nynϕ(x1, ..., xm, y1, ..., yn)) = 0

where each Qi is either inf or sup and ϕ is quantifier free. Let E(x1, ..., xm) be the mathematical

statement

Q1y1 ...Q

nyn

(ϕ(x1, ..., xm, y1, ..., yn) = 0)

where each Qi is ∃yi if Qiyiis infyi

and is ∀yi if Qiyiis supyi

. If M is ω-saturated, then for any

elements a1, ..., am of M , we have M |= E[a1, ..., am] if and only if E(a1, ..., an) is true in M .

If Λ is a linearly ordered set, a Λ-chain of L-structures is a family of L-structures (Mλ | λ ∈

Λ) such that Mα ⊆Mβ for all α < β < λ. The L-structure M with each sort M (s) equal to the

completion of the union⋃λ∈ΛM

(s)λ is called the union of the chain and denoted

⋃λ∈Λ Mλ. A

chain of structures (Mλ | λ ∈ Λ) is an elementary chain if Mα � Mβ for all α < β. A class K

of L-structures is called inductive if it is closed under unions of chains. We say T is a ∀∃-theory

if T is an L-theory axiomatized by closed L-conditions of the form

sups1,x1

... supsk,xk

inft1,y1

... inftl,yl

ψ(x1, ..., xk, y1, ..., yl)

where ψ is quantifier free.

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1.7.4 Proposition. The class of models Mod(T ) of a ∀∃-theory T is an inductive class.

Proof. Let T be an ∀∃-theory. Let sups1,x1

... supsk,xk

inft1,y1

... inftl,yl

ψ(x1, ..., xk, y1, ..., yl) = 0 be an axiom

of T , where ψ is quantifier free. Let Λ be a linearly ordered set and let (Mλ | λ ∈ Λ) be a

chain of L-structures from Mod(T ). Let M =⋃λ∈Λ Mλ be the union of the chain. For any

a = a1, ..., ak ∈ M from suitable sorts, there exists λ ∈ Λ so that a1, ..., ak ∈ Mλ. The L(a)-

condition inft1,y1

... inftl,yl

ψ(a1, ..., ak, y1, ..., yl) = 0 is true in Mλ, since Mλ |= T . Then in Mλ there

exist witnesses to the fact that inft1,y1 ... inftl,ylψ(a1, ..., ak, y1, ..., yl) = 0 in Mλ. That is for all

ε > 0 there exist bε1, ..., bεl ∈ Mλ such that ψM(a1, ..., ak, b

ε1, ..., b

εl ) ≤ ε. We know Mλ ⊆ M, and

therefore these witnesses are also in M. Thus, inft1,y1 ... inftl,ylψ(a1, ..., ak, y1, ..., yl) = 0 is true

in M. The a1, ..., ak ∈ M were arbitrary, so sups1,x1

... supsk,xk

inft1,y1

... inftl,yl

ψ(x1, ..., xk, y1, ..., yl) = 0 is

true in M. Therefore, M |= T .

1.7.5 Proposition. ([1, Proposition 7.2]) If (Mλ | λ ∈ Λ) is an elementary chain and λ ∈ Λ,

then Mλ �⋃λ∈Λ Mλ.

Next, we discuss existential closure, model completeness, and the definition of a model

companion.

1.7.6 Definition. An inf-formula of L is a formula of the form

infs1,y1

... infsn,yn

ϕ(x1, ..., xk, y1, ..., yn)

where ϕ(x1, ..., xk, y1, ..., yn) is quantifier-free.

1.7.7 Definition. Let T be an L-theory for a given continuous signature L. Let M |= T . We

say M is an existentially closed (e.c.) model of T if, for any inf-formula

infs1,y1

... infsn,yn

ϕ(x1, ..., xk, y1, ..., yn)

any a1, ..., ak ∈M and any N |= T that is an extension of M we have:

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N = infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M.

An L-theory T is model complete if any embedding between models of T is an elementary

embedding.

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The following lemma and proposition are stated for a signature L with a single sort, but are

easily extended to the many-sorted case.

1.7.8 Lemma. If M ⊆ N are models of T so that for any inf-formula

infs1,y1

... infsn,yn

ϕ(x1, ..., xk, y1, ..., yn)

and any a1, ..., ak ∈M

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N = infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M

(that is, M is existentially closed in N), then there exists an elementary extension M′ of M and

an embedding f : N→M′ so that f(a) = a for all a ∈M.

Proof. Let M ⊆ N such that M and N are models of T . Assume, for any inf-formula

infs1,y1

... infsn,yn

ϕ(x1, ..., xk, y1, ..., yn)

and any a1, ..., ak ∈M

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N = infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M.

Let κ be a cardinal larger than the cardinality of the underlying set N of N. Let M′ be a

κ-saturated elementary extension of M. Then, (M′, a)a∈M ≡ (M, a)a∈M , and (M′, a)a∈M is

κ-saturated (since κ > card(N) ≥ card(M)). Moreover, any closed condition

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn) = 0

(where a1, ..., ak are in appropriate sorts of M and ϕ is quantifier-free) that is true in N must

also be true in M, and hence in M′. This allows us to build an embedding of N in M′ over

M.

1.7.9 Proposition. The L-theory T is model complete if and only if every model of T is an

existentially closed model of T .

Proof. The left to right direction is immediate from the definitions. Now, assume that every

model M of T is an existentially closed model of T . Let M1 ⊆ N1 be models of T . We will show

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that M1 is an elementary substructure of N1, from which it follows that T is model complete. By

Lemma 1.7.8, there exists an elementary extension M2 of M1 and an embedding f : N1 →M2 so

that f(a) = a for all a in the underlying metric space M1 of M1. By renaming the elements of

the image of N1, we may assume M1 ⊆ N1 ⊆M2. Since f is the identity on M1, this renaming

does not change the fact that M1 �M2.

Next, again by Lemma 1.7.8, there exists an elementary extension N2 of N1 and an embedding

f : M2 → N2 so that f(a) = a for all a in the underlying metric space N1 of N1. By renaming

as above we may assume N1 ⊆M2 ⊆ N2. Since f is the identity on N1, this renaming does not

change the fact that N1 � N2.

We proceed in this fashion to get an increasing chain of models M1 ⊆ N1 ⊆ M2 ⊆ N2 ⊆ ...

so that Mi � Mi+1 and Ni � Ni+1 for all i ∈ N. Let W be the union of the elementary chain

(Mn | n ≥ 1), which is also the union of the elementary chain (Nn | n ≥ 1). By Proposition

1.7.5 we have M1 �W and N1 �W. Since M1 ⊆ N1, this implies M1 � N1, as claimed.

1.7.10 Definition. A model companion of T is an L-theory S such that:

• every model of T extends to a model of S;

• every model of S extends to a model of T ;

• S is model complete.

1.7.11 Proposition. If the L-theories S and S′ are both model companions of T , then S is

equivalent to S′, that is, Mod(S) = Mod(S′).

Proof. Assume S and S′ are both model companions of T . It suffices to show Mod(S) ⊆

Mod(S′), since then we could switch S and S′ in that argument to get Mod(S′) ⊆ Mod(S). Let

M |= S. Then, there exists an extension W of M so that W |= T , and there exists an extension

N of W so that N |= S′. Therefore, there exists an extension N of M that is a model of S′.

Similarly, for any N |= S′ there exists an extension M of N so that M |= S.

We use these facts to build an increasing chain M1 ⊆ N1 ⊆ M2 ⊆ N2 ⊆ ... so that each Mi

is a model of S and each Ni is a model of S′. Since both S and S′ are model complete, we

know Mi � Mi+1 for all i ∈ N>0 and Ni � Ni+1 for all i ∈ N>0. Let W be the union of the

elementary chain (Mn | n ≥ 1), which is also the union of the elementary chain (Nn | n ≥ 1)..

By Proposition 1.7.5 we have M1 � W and N1 � W. Since M1 ⊆ N1, this implies M1 ≡ N1.

Therefore any model of S is elementarily equivalent to a model of S′, so Mod(S) ⊆ Mod(S′).

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1.7.12 Definition. We say the L-theory T has amalgamation over substructures if: for any

substructures M0, M1 and M2 of models of T and embeddings f1 : M0 → M1, f2 : M0 → M2,

there exists a model N of T and embeddings gi : Mi → N such that g1 ◦ f1 = g2 ◦ f2.

1.7.13 Proposition. Let S and T be L-theories such that S is the model companion of T .

Assume T has amalgamation over substructures. Then S has quantifier elimination.

Proof. Let L, S and T be as above. A theory S has quantifier elimination if and only if the

following holds: if M,N |= S, then every embedding of a substructure of M into N can be

extended to an embedding of M into an elementary extension of N. (See [1] Proposition 13.6.)

It is straightforward to show this property holds for S. Let M and N be models of S. Let A ⊆M

and let f : A→ N be an embedding. Let W be an amalgam of M and N over A as models of T .

Without loss of generality, we may assume N ⊆W. Let W′ |= S be an extension of W. Since S

is model complete we know W′ is an elementary extension of N. And, clearly M is embedded

in W′ by a map that extends f .

Definability is one of the central notions in model theory. The following definition is adapted

from [1, Chapter 9]. Let IP be a bounded interval in R≥0. A predicate P : M (s1)× ...×M (sn) →

IP is definable in M over A if an only if there is a sequence (ϕk | k ≥ 1) of L(A)-formulas

such that the predicates ϕMk converge to P uniformly on M (s1) × ... × M (sn). A closed set

D ⊆ M (s1) × ... × M (sn) is definable in M over A if and only if the predicate dist(x,D) is

definable in M over A. A function f : M (s1) × ... ×M (sn) → M (s0) is definable in M over A

if and only the predicate d(s0)(f(x1, ..., xn), y) is definable in M over A. We will use the term

0-definable when the set of parameters A is empty.

1.7.14 Definition. Let T be an L-theory and ϕ(x1, ..., xn) an L-formula. By the zerosets of

the formula ϕ(x1, ..., xn), we mean the sets

{(x1, ..., xn) ∈M (s1) × ...×M (sn) | ϕM(x1, ..., xn) = 0}

as M varies over the models of T .

The zerosets of ϕ(x1, ..., xn) are said to be uniformly definable in models of T if and only if

there exists a definable predicate ψ(x1, ..., xn) such that for all models M of T and all a ∈M of

appropriate sort

ψM(a1, ..., an) = dist((a1, ..., an), {(b1, ..., bn) ∈Mn | ϕM(b1, ..., bn) = 0}).

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That is, for each model M of T the distance to the zeroset of ϕ(x1, ..., xn)M is given by the

interpretation of the same definable predicate ψ. In the rest of this thesis, if we say a certain

family of sets is uniformly definable in all models of a theory T , we mean that there exists

an L-formula ϕ(x1, ..., xn) so that the given family of sets is exactly the family of zerosets of

ϕ(x1, ..., xn), and that they are uniformly definable as defined above.

The utility of definable sets is demonstrated by the following theorem.

1.7.15 Theorem. ([1, Theorem 9.17]) For a closed set D ⊆M (s), the following are equivalent:

1. D is definable in M over A.

2. For any predicate

P : (M (s1) × ...×M (sn))×M (s) → IP

that is definable in M over A, the predicate Q : M (s1) × ...×M (sn) → IQ defined by

Q(x) = inf{P (x1, ..., xn, y) | y ∈ D}

is definable in M over A.

It is straightforward to extend this theorem to closed sets D ⊆M (s1) × ...×M (sn).

As in first order logic, in continuous logic there is a notion of an extension by definition

whereby we extend a signature and its structures by adding a symbol for a definable predicate

or function without really changing the expressive power of that signature. See [1, Chapter 9]

for more about definability and extension by definition.

1.8 Lp-structures

This section presents the method used in this thesis for studying certain kinds of unbounded

metric spaces using many-sorted bounded continuous logic. There is an alternative approach in

[3], which is equivalent to what we do here for the structures considered in this thesis.

Recall from Section 1.2 that the continuous signature Lp consists of:

• a non-empty sort index set S = N and the family of positive real numbers (2n | n ∈ N);

• for each m,n ∈ N with m ≤ n, a function symbol Im,n with arity (m;n) and modulus of

uniform continuity ∆m,n(ε) = ε;

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• constant symbols (pn | n ∈ N), where pn has arity n;

• metric symbols d(n) for each n ∈ N.

In addition, recall that given a pointed metric space (M,d, p), we define the corresponding

Lp-structure

M =(

((M (n), d(n))|n ∈ N), pMn , I

Mm,n

)so that:

• the nth sort (M (n), d(n)) is the closed ball of radius n centered at p in (M,d) with the

metric restricted to that ball;

• for each m,n ∈ N with m ≤ n, the function Im,n : M (m) →M (n) is the inclusion map;

• the constants pn are all interpreted as the point p.

Thus, given a pointed metric space, we know how to construct the (unique) corresponding

Lp-structure. Next, we define an Lp-theory Tp so that for every M |= Tp, there is a unique

pointed metric space (M,d, p) such that M is isomorphic to the Lp structure corresponding to

(M,d, p). First, we need some more definitions and lemmas.

A metric space M is a length space if, for any points x, y ∈ M , the distance between x and

y is equal to the infimum of the lengths of the rectifiable paths between x and y.

1.8.1 Definition. ([5, Chapter 3, Section 1]) We say a metric space M has the approximate

midpoint property if: for any x, y ∈M , for any ε > 0, there exists z ∈M such that

|d(x, z)− d(x, y)2| ≤ ε and |d(y, z)− d(x, y)

2| ≤ ε.

1.8.2 Fact. ([5, Chapter 3, Section 1]) A complete metric space M is a length space if and only

if it has the approximate midpoint property.

1.8.3 Definition. ([5, Chapter I, 1.3]) A metric space (X, d) is geodesic if for any x, y ∈ X

there exists an isometric embedding γ : [0, d(x, y)] → X with γ(0) = x and γ(d(x, y)) = y. A

geodesic segment is the image of such a path.

1.8.4 Fact. ([5, Chapter I, 1.4]) A complete metric space M is a geodesic space if and only if

for any two points x, y ∈M there exists a midpoint z between x and y. That is, there exists z

such that:

d(x, z) =d(x, y)

2, and d(y, z) =

d(x, y)2

.

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1.8.5 Lemma. If a complete metric space M is a length space, then for any r ∈ R and any

a ∈M , the open ball Or(a) with center a and radius r is dense in the closed ball Br(a).

Proof. Let M be a complete length space and let r ∈ R and a ∈ M . Towards contradiction,

assume that Or(a) is not dense in Br(a). Then there exists y ∈ Br(a) \ Or(a) and ε > 0 such

that for all x ∈ Or(a), we know d(x, y) > ε. Because M is a length space and d(a, y) = r, we

may find a path γ between a and y with length l(γ) < r+ ε2 . (Let γ refer to both the path and

its image.) By the connectedness of γ, we may find x ∈ γ such that ε2 ≤ d(x, y) < ε. Then,

d(a, x) + d(x, y) ≤ l(γ) < r +ε

2

and because ε2 ≤ d(x, y), we conclude d(a, x) < r. So, x ∈ Or(a) and d(x, y) < ε. This is a

contradiction.

The theory Tp defined below gives axioms for a class of Lp-structures for which we may

identify a unique underlying pointed metric space.

1.8.6 Definition. Let Tp be the following Lp-theory:

1. for all m,n ∈ N with m ≤ n the axiom

d(n)(Im,n(pm), pn) = 0;

2. for all m,n ∈ N with m ≤ n the axiom

supm,x

supm,y|d(n)(Im,n(x), Im,n(x))− d(m)(x, y)| = 0;

3. for all n ∈ N the axiom

supn,x

d(n)(x, In,n(x)) = 0;

4. for all m,n ∈ N with m < n the axiom

supn,y

min{m−· d(n)(y, pn), infm,x

d(n)(Im,n(x), y)} = 0;

18

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5. for all j,m, n ∈ N with j ≤ m ≤ n, the axiom

supj,x

d(n)(Im,n ◦ Ij,m(x), Ij,n(x)) = 0;

6. for all n ∈ N the axiom

supn,x

supn,y

infn,z

max{|d(x, z)− d(x, y)2|, |d(y, z)− d(x, y)

2|} = 0.

1.8.7 Lemma. If (M,d, p) is a pointed metric space and M is the corresponding Lp-structure,

then M is a model of the axioms in (1)-(5) from the definition of Tp.

Proof. This is clear from the definition of the Lp-structure corresponding to (M,d, p).

1.8.8 Lemma. Let (M,d, p) be a pointed metric space such that each ball closed ball Bn(p) is

a length space. Let M be the corresponding Lp-structure. Then, M |= Tp.

Proof. This follows from Lemma 1.8.7 and the fact that if each Bn(p) = M (n) is a length space,

then the axioms in (6) from the definition of Tp are true in M.

1.8.9 Lemma. Let M =(

((M (n), d(n)) | n ∈ N), pMn , I

Mm,n

)be a model of Tp. Then,(

(M (n), d(n), pMn ), IM

m,n

)is a directed system of pointed metric spaces indexed by N. If (W,dW , q)

is the direct limit of this directed system, then the corresponding Lp-structure

W =(

((W (n), d(n)W ) | n ∈ N), pW

n , IWm,n

)

is isomorphic to M.

Proof. Assume M is as above. By the axioms in (1) and (2) in Definition 1.8.6, we know that

each function IMm,n : M (m) → M (n) is an isometric embedding which takes pM

m to pMn . The

axioms in (3) imply that In,n is the identity on (M (n), d(n)). The axioms in (4) imply that for

each m ≤ n, for every y ∈ M (n) if d(y, pMn ) < m, then infm,x d(n)(IM

m,n(x), y) = 0. Therefore,

the closure of the image of Im,n contains the open ball centered at pMn of radius m in M (n). By

the axioms in (6), each sort (M (n), d(n)) is a length space, and therefore the open ball centered

at pMn of radius m in M (n) is dense in the closed ball centered at pM

n of radius m in M (n). By

uniform continuity of IMm,n, and the completeness of each sort we conclude that IM

m,n maps M (m)

isometrically onto the closed ball of radius m and center p contained in M (n). The axioms in

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(5) imply that IMj,m ◦ IM

m,n = IMj,n for j ≤ m ≤ n. Therefore,

((M (n), d(n), pM

n ), IMm,n

)is a directed

system.

Suppose W is as in the statement of the lemma. Then, (W,dW , q) is the direct limit of((M (n), d(n), pM

n ), IMm,n

), and we have isometric embeddings gn : M (n) →W for n ∈ N such that

gn(pMn ) = q, and for all m ≤ n we know gn = gm ◦ IM

m,n. Note that gn maps onto W (n). The

family of functions (gn | n ∈ N) is an isomorphism between the Lp-structures M and W.

1.8.10 Corollary. Assume (M,d, p) is a pointed metric space such that the corresponding Lp-

structure M is a model of Tp. Then((M (n), d(n), pM

n ), IMm,n

)is a directed set of pointed metric

spaces indexed by N. If (W,dW , q) is the direct limit of this directed set, then (M,d, p) and

(W,dW , q) are isomorphic as pointed metric spaces. That is, there exists an isometry from

(M,d) onto (W,dW ) which sends p to q.

Proof. Take the union of the family of functions (gn | n ∈ N) from the proof above.

So, for a model M of Tp, there is a unique underlying pointed metric space (up to isomor-

phism), equal to the direct limit of the sorts of M. For any M |= Tp we will call this space the

underlying metric space of M. From now on, all of the pointed metric spaces considered in this

thesis will be such that their corresponding Lp-structures are models of Tp. In this many-sorted

setting, the proliferation of sorts and the sort by sort way we defined notions such as embedding,

subset and isomorphism often gets in the way of the clarity of statements and arguments. Here

are some conventions intended to help avoid some of the notational and expositional complica-

tions of the many-sorted setting.

Let (M,d, p) be a pointed metric space and M |= Tp its corresponding Lp-structure.

• For legibility, the inclusion map symbols Im,n will be left out of formulas.

• Because the metrics d(n) on the sorts are all just restrictions of the metric on M , we will

leave off the superscripts.

• We will often use p instead of pn, and often use p instead of pM.

• We will often refer to the underlying metric space of M by M instead of referring to the

sorts.

• Let (A(n) | n ∈ N) ⊆ (M (n) | n ∈ N) be such that A(m) = A(n) ∩M (m) for all m < n.

Then, there exists a unique subset A ⊆ M such that A(n) = A ∩M (n) for each n ∈ N.

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In the other direction, clearly any subset A ⊆ M gives a unique family (A ∩M (n) | n ∈

N) ⊆ (M (n) | n ∈ N). This implies that the underlying metric space of any substructure

of M |= Tp is uniquely determined up to isomorphism. For models of Tp, we will often

simply refer to a subset A of the underlying space instead of referring to the images of A

in each sort.

• Say M |= Tp and N |= Tp are two Lp-structures with underlying metric spaces (M,dM , pM )

and (N, dN , pN ) respectively. Let f : M → N be an isometric embedding such that

f(pM ) = pN . Then the collection of maps

(f (n) : M (n) → N (n) | n ∈ N)

where f (n) is the restriction of f to M (n), is an embedding of Lp-structures. If f : M → N

is onto, then (f (n) : M (n) → N (n) | n ∈ N) is an isomorphism of Lp-structures. Moreover,

every embedding or isomorphism between Lp structures is given by an embedding on the

underlying metric spaces in this manner. When it is not confusing to do so, we will use f

as an abbreviation for (f (n) : M (n) → N (n) | n ∈ N).

• If the underlying metric space M of M has density character κ, then we say M has density

character κ. The inclusion maps between the sorts guarantee that the density characters of

the sorts are increasing. The density of the underlying space M is equal to the supremum

of the densities of the sorts.

• The following convention will be useful in Chapters 3 and 4 when we want to add functions

to our Lp-structures. Let k, s ∈ N. Let Ls be an extension of Lp by a family of function

symbols {fn | n ∈ N} where fn has arity (n;n+s). Let T ′ be an extension of the Lp-theory

Tp by a family of axioms

{supm,x

d(fn(Im,n(x)), Im+s,n+s(fm(x))) = 0 | m,n ∈ N; m ≤ n}.

Let M |= T ′ be an Ls-structure. Then, when we take the direct limit of the sorts to find

the underlying metric space for M, we may also take the direct limit (union) of each of

the functions {fMn | n ∈ N} and get a function fM on the underlying metric space such

that fMn is the restriction of f to Bn(p) = M (n). We will call this the underlying function

of M. As with the metric, in Ls-formulas we may use the symbol f instead of fn.

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Next, the ultraproduct for pointed (possibly unbounded) metric spaces is defined. Say

((Mi, di, pi) | i ∈ I) is a family of pointed metric spaces, and let U be an ultrafilter on I.

We define∏U (Mi, di, pi) = (N, d, p), the U -ultraproduct of ((Mi, di, pi) | i ∈ I) as follows.

Let

N = {(xi)i∈I | xi ∈Mi, and there exists c ∈ R, ∀i ∈ I di(xi, pi) ≤ c}.

Define a pseudometric d on N by d((ai), (bi)) = limU,i

di(ai, bi). Let (N, d, p) be the result of

taking the quotient of (N , d) where p is the image of (pi) in this quotient. In the case that

the pointed metric spaces in question are uniformly bounded, this ultraproduct is isometric to

the ultraproduct of bounded metric spaces as defined in Section 1.5, and the basepoint and the

condition that sequences in N be bounded are superfluous.

1.8.11 Lemma. Say ((Mi, di, pi) | i ∈ I) is a family of pointed metric spaces such that the

corresponding Lp-structures Mi are all models of Tp. Let U be an ultrafilter. Then the Lp-

structure N corresponding to the ultraproduct∏U (Mi, di, pi) is isomorphic to the Lp-structure∏

U Mi.

Proof. Let ((Mi, di, pi) | i ∈ I) and Mi be as above. By the Fundamental Theorem of Ultra-

products (Theorem 1.5.1) we know that∏U Mi is a model of Tp. To make the notation nicer,

let∏U Mi = M. By the definition of the ultraproduct of many-sorted metric structures (see

Section 1.5),

M =∏U

Mi =((

(∏U

M(n)i , δ(n)) | n ∈ N

), pMn , I

Mm,n

)where the metric δ(n) = lim

i,Ud

(n)i , the basepoint pM = [pMi ]U and IM

m,n([ai]U ) = [IMim,n(ai)]U . Let

(N, d, p) be the ultraproduct∏U (Mi, di, pi) of pointed metric spaces, and denote elements of

N by [ai]NU . It suffices to show that for each n ∈ N, there is an isometry from∏U M

(n)i onto

BNn (p) which sends pMn to p. If such isometries exist, then they form an isomorphism between

the Lp-structure∏U Mi = M and the Lp-structure N corresponding to the pointed metric space∏

U (Mi, di, pi) = (N, d, p). Note that for all [ai]U ∈∏U M

(n)i , the representative (ai)i∈I must

be such that ai ∈Mi, and di(ai, pi) ≤ n for all i ∈ I. Thus,

(ai)i∈I ∈ N = {(xi)i∈I | xi ∈Mi, and there exists c ∈ R, di(xi, pi) ≤ c ∀i ∈ I}.

which means that [ai]NU is an element of the pointed ultraproduct (N, d, p) =∏U (Mi, di, pi).

Also, since di(ai, pi) ≤ n for all i ∈ I, clearly d([ai]NU , [bi]NU ) = lim

U,idi(ai, bi) ≤ n. Thus,

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[ai]NU ∈ BNn (p). Therefore, we may define f :∏U M

(n)i → BNn (p) by f([ai]U ) = [ai]NU .

For all [ai]U , [bi]U ∈∏U M

(n)i ,

δ(n)([ai]U , [bi]U ) = limU,i

di(ai, bi)

by the definition of the ultraproduct of bounded metric structures. We also know

limU,i

di(ai, bi) = d([ai]NU , [bi]NU )

by the definition of the ultraproduct for pointed metric spaces. Therefore

δ(n)([ai]U , [bi]U ) = d([ai]NU , [bi]NU ) = d(f([ai]U ), f([bi]U ))

which proves f is well defined, since if δ(n)(f([ai]NU ), f([bi]NU )) = 0, then d([ai]NU , [bi]NU ) = 0, and

also shows f is an isometric embedding.

It remains to show that f is onto. It suffices to show that for any [bi]NU ∈ BNn (p), there is

a representative (ai)i∈I of [bi]NU so that for every i ∈ I, we know ai ∈ M (n)i . In that case, we

know [ai]U ∈∏U M

(n)i and f([ai]U ) = [ai]NU = [bi]NU .

Let [bi]NU ∈ BNn (p). If {i | di(bi, pi) ≤ n} ∈ U , then we are done. So, assume {i | di(bi, pi) >

n} ∈ U . It follows from this assumption that d(p, [bi]NU ) = n, and we may assume without loss

of generality that 2n > di(bi, pi) > n for all i ∈ I.

For each i ∈ I, let γi be a rectifiable path from pi to bi contained in M(2n)i such that

d(2n)i (pi, bi) +

12i≥ length(γi) ≥ d(2n)

i (pi, bi).

These paths exist because for each i ∈ I, the Lp-structure Mi is a model of Tp, and therefore

each M (2n)i is a length space. (Note that Mi |= Tp also implies d(2n)

i is a restriction of the metric

di on Mi to the closed 2n-ball M (2n)i ⊆ Mi.) The image of each γi is connected, so for each

i ∈ I we may find a point ai on γi such that d(2n)i (ai, pi) = n. Therefore ai ∈M (n)

i .

Let γ1i denote γi restricted to [0, γ−1

i (ai)], so γ1i is a rectifiable path from pi to ai. Let γ2

i

denote γi restricted to [γ−1i (ai), γ−1

i (bi)], so γ2i is a rectifiable path from ai to bi. Then,

d(2n)i (ai, bi) ≤ length(γ2

i ) = length(γi)− length(γ1i ) ≤ d(2n)

i (pi, bi) +12i− d(2n)

i (pi, ai).

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Therefore,

limU,i

d(2n)i (ai, bi) ≤ lim

U,id

(2n)i (pi, bi) + lim

U,i

12i− lim

U,id

(2n)i (pi, ai) = n+ 0− n = 0.

So,

d([ai]NU , [bi]NU ) = lim

U,idi(ai, bi) = lim

U,id

(2n)i (ai, bi) = 0

and for each i ∈ I the distance di(ai, pi) = d(2n)i (ai, pi) ≤ n, which implies that ai ∈M (n)

i .

We now discuss a notion of definability for subsets of and functions on the underlying metric

space M of a model M of Tp. A closed set D that is a subset of M (n1) × ... ×M (nk) can be

viewed as a subset of (M (m))k for any m ≥ max{n1, ..., nk}, and D also corresponds to a unique

closed subset of the underlying metric space Mk.

1.8.12 Definition. Let M |= Tp with underlying metric space (M,d, p). Let A ⊆ M be a

subset and let D ⊆Mk be a closed subset. We say D is definable in M over A if the intersection

D ∩ (M (n))k is definable in M over A for each n ∈ N.

If for each n ∈ N the sets D ∩ (M (n))k are uniformly definable in models M of Tp, then we

say D is uniformly definable in models of Tp.

1.8.13 Definition. Let M |= Tp with underlying metric space (M,d, p). Let A ⊆ M be a

subset and f : Mk →M a uniformly continuous function. We say f is definable in M over A if

the restriction f |(M (n))k is definable in M over A for each n ∈ N.

If the functions f |(M (n))k are uniformly definable in models of Tp, then we say f is uniformly

definable in models of Tp.

In formulating the signature Lp we made an arbitrary choice to use closed balls of integer

radius as the sorts of our many-sorted structures. It is natural to ask if this choice has any

effect on the model theoretic properties of the structures we study (especially, of R-trees). In

fact it has no such effect, as the following discussion (in conjunction with Theorem 1.7.15 above)

indicates.

We show in what follows that closed balls centered at the basepoint are uniformly definable

in models of Tp via quantifier free formulas. We will need the following useful connective.

1.8.14 Definition. ([1, Definition 6.1]) Define the binary function −· : [0,∞)× [0,∞)→ [0,∞)

by x−· y = max{x− y, 0}.

24

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1.8.15 Lemma. Let r ∈ R>0 and n ∈ N with r ≤ n. Let ϕn(x) be the quantifier-free formula

d(x, p) −· r where x has sort n. Suppose M |= Tp and (M,d, p) is the underlying metric space

of M. If M is a geodesic space, then the closed ball Br(p) ⊆ M (n) is 0-definable. Indeed, for

x ∈M (n) we have dist(x,Br(p)) = ϕn(x)M.

Proof. Assume the situation in the hypotheses of the lemma. We show that ϕMn (x) is equal to

the distance function dist(x,Br(p) ∩M (n)). For x ∈ M (n) we know ϕMn (x) = 0 if and only if

d(x, p) ≤ r, i.e. if and only if x ∈ Br(p). Now, let x /∈ Br(p). Then, ϕMn (x) = d(x, p)− r. Let γ

be a geodesic segment from p to x with γ(0) = p. Then,

dist(x,Br(p) ∩M (n)) ≤ d(x, γ(r)) = d(x, p)− d(p, γ(r)) = d(x, p)− r = ϕMn (x).

Now towards contradiction, assume d(x, p)− r > dist(x,Br(p)∩M (n)). Then, there exists a

point in c ∈ Br(p) with d(c, x) < d(x, p)− r. So,

d(x, p) ≤ d(p, c) + d(c, x) ≤ r + d(c, x) < r + d(x, p)− r = d(x, p)

and thus d(x, p) < d(x, p), which is a contradiction. Therefore, dist(x,Br(p)∩M (n)) = ϕn(x)M.

1.8.16 Lemma. Assume M |= Tp. Let N be an ω-saturated elementary extension of M. Then

the underlying metric space N of N is a geodesic space.

Proof. Assume M and N are as specified above. For each n ∈ N let θn be the Lp-sentence

supn,x

supn,y

infn,z

max{|d(x, z)− d(x, y)2|, |d(y, z)− d(x, y)

2|}.

For each n we know that θMn = 0 because M |= Tp. Because N is an elementary extension of

M, we know θNn = 0 is true. Since N is ω-saturated, Proposition 1.7.3 yields that in each of the

θn the infn,z quantifier is realized exactly in N (n). Thus, between any x and y in N (n) there

is a midpoint. Therefore between any x and y in N there is a midpoint. By Fact 1.8.4, N is

geodesic.

25

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1.8.17 Theorem. For any r ∈ R≥0 and n ∈ N the family of sets M (n) ∩ BMr (p) is uniformly

definable in models of Tp.

Proof. Let M |= Tp and take an ω-saturated elementary extension N. Then the underlying

metric space N of N is a geodesic space by Lemma 1.8.16. By Lemma 1.8.15, the intersection

N (n) ∩BNr (p) is a definable subset in N over {p}. Moreover, the distance to N (n) ∩BNr (p) in N

is given by interpreting the quantifier-free formula ϕn(x) = d(x, p)−· r in N, where the variable

x is from the sort N (n). Then, since ϕn(x) was quantifier free, the distance in M to BMr (p)

is given by the interpretation of the formula ϕn(x) = d(x, p) −· r in M (where the variable x

is from the sort M (n) ⊆ N (n).) Since the distance to the ball is given by the same formula in

every model of Tp, the balls are uniformly definable in models of Tp.

26

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CHAPTER 2

MODEL THEORY OF R-TREES

2.1 Introduction to R-trees

This section contains a number of basic definitions and facts about R-trees. Much of this

material can be found in [5], [6] and [13].

First, a standard bit of notation. Given a metric space X, a subset Y of X and δ > 0, we

denote the set {x ∈ X | dist(x, Y ) ≤ δ} by Y δ.

An R-tree is a metric space X such that between any two points in X there is a unique

arc, and that arc is a geodesic segment. (An arc is the image of a topological embedding

f : [a, b]→ X of [a, b] ⊂ R where f(a) = x and f(b) = y.) In an R-tree, [a, b] denotes the unique

geodesic segment between a and b. Setwise, [a, b] = [b, a], but when we write the segment as

[a, b], we mean that the corresponding isometric embedding γ : [0, d(a, b)]→ X is such that the

initial point γ(0) = a and the terminal point γ(d(a, b)) = b. When it does not cause confusion,

we may use [a, b] to refer to the isometric embedding as well as its image. It is a fact (see [6,

Lemma 2.4.14]) that the completion of an R-tree is an R-tree. In this thesis most of the R-trees

we consider are complete, since metric structures are required to be based on complete metric

spaces.

Let M be an R-tree and a ∈ M . Call the connected components of M \ {a} branches at a.

Let the degree of a point a ∈ M be the cardinal number of branches at a. If there are three

or more branches at a ∈ M , then we call a a branch point. The height of a branch β at a is

sup{d(a, x)|x ∈ β} if that supremum exists, and is∞ otherwise. A subtree of M is any subspace

of M that is itself an R-tree. Any connected subspace of M is a subtree. Moreover, subtrees

are convex. That is, given any two points a, b in a subtree N , the geodesic segment [a, b] ⊆ M

is contained in N . Thus, the intersection of two subtrees is a subtree. A ray in an R-tree is an

isometric copy of R≥0. If a ∈ M a ray at a is a ray so that the image of 0 under the isometric

embedding of R≥0 into M is a.

27

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2.1.1 Lemma. (See [6, Lemma 1.9]) If M is an R-tree and T1, T2 are disjoint, closed, non-

empty subtrees of M , then there exists a unique shortest geodesic segment with its initial point

in T1 and its terminal point in T2. Moreover, for all b ∈ T1 and c ∈ T2, the geodesic segment

from b to c must contain this segment.

2.1.2 Lemma. (See [6, Chapter 2]) If M is an R-tree and a, b, c ∈M , then

1. d(a, b) + d(b, c) = d(a, c) + 2 dist(b, [a, c]).

2. b ∈ [a, c] if and only if d(a, c) = d(a, b) + d(b, c).

3. b ∈ [a, c] if and only if a and c are in different branches at b.

2.1.3 Lemma. Let T be a closed subtree of M and let a, b ∈M . Let ea ∈ T be the closest point

to a, and let eb ∈ T be the closest point to b. If ea 6= eb, then

d(a, b) = d(a, ea) + d(ea, eb) + d(b, eb).

Proof. Assume T is a closed subtree of M and a, b ∈ M . Let ea ∈ T be the closest point to a,

and let eb ∈ T be the closest point to b. Assume ea 6= eb. First we show ea ∈ [a, b]. Otherwise, a

and b are on the same branch at ea, which implies ea must be the closest point to both a and b

in T, a contradiction. So, ea ∈ [a, b], which implies d(a, b) = d(a, ea) + d(b, ea) by Lemma 2.1.2.

Because eb is the closest point to b in T and ea ∈ T, by Lemma 2.1.1 we know that eb ∈ [b, ea].

Therefore, d(a, b) = d(a, ea) + d(b, ea) = d(a, ea) + d(ea, eb) + d(b, eb).

2.1.4 Definition (Gromov product). For a metric space M and x, y, w ∈M , define

(x · y)w =12

[d(x,w) + d(y, w)− d(x, y)].

2.1.5 Definition. Let δ > 0. A metric space M is δ-hyperbolic if, for all x, y, z, w ∈M

min{(x · z)w, (y · z)w} − δ ≤ (x · y)w.

If M is geodesic, then M is δ-hyperbolic if and only if, given a, b, c ∈ M and any geodesic

segments [a, b], [b, c], and [c, a], the segment [a, b] is contained in ([b, c] ∪ [c, a])δ. A metric space

is 0-hyperbolic if it is δ-hyperbolic for all δ > 0.

28

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2.1.6 Lemma. ( [13, Proposition 6.13]) Any R-tree is 0-hyperbolic. Moreover, any 0-hyperbolic

metric space embeds isometrically in an R-tree.

Next, we define what it means for an R-tree to be finitely generated, and prove some prop-

erties of finitely generated R-trees.

2.1.7 Definition. If A ⊆ M is a subset of the R-tree M , let EA denote the smallest subtree

containing A. We call this the R-tree generated by A. Note that

EA =⋃{[a1, a2] | a1, a2 ∈ A}.

The closure EA of EA is the smallest closed subtree containing A.

2.1.8 Definition. An R-tree M is finitely generated if there exists a finite subset A ⊆M such

that M = EA.

Note that if A is finite, EA = EA. Therefore, finitely generated R-trees are complete.

2.1.9 Definition. Let M be an R-tree. If c ∈M is such that there do not exist a, b ∈M \ {c}

with c ∈ [a, b], then c is called an endpoint of M . Equivalently, an endpoint is a point with

degree one.

2.1.10 Lemma. If an R-tree M is finitely generated and C is the set of endpoints of M , then

1. if B is a generating set, then C ⊆ B;

2. the set C generates M .

Thus, C is the unique least set of generators for M .

Proof. Let M be a finitely generated R-tree. Then M is complete and the diameter D of M is

finite. Let B be a generating set for M .

Proof of (1): Assume there is an endpoint c ∈ M not contained in B. Then, there must exist

a, b ∈ B such that c ∈ [a, b]. But, this is a contradiction because c is an endpoint.

Proof of (2): Let a ∈ M . Let Sa be the set of all segments [b, c] ⊆ M such that a ∈ [b, c] and

order Sa by inclusion. This is a partial ordering on Sa. Let {[bi, ci] | i ∈ α} be a chain in this

partial ordering of segments containing a, where α is some cardinal. Let I be the closure of⋃i∈α[bi, ci]. Then, I is a geodesic segment in M . Clearly a ∈ I, and the length of I is at most

D. Therefore I ∈ Sa, and I is an upper bound for the chain, since [bi, ci] ⊆ I for all i ∈ N.

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The chain was arbitrary, so any chain has an upper bound. Therefore, by Zorn’s Lemma there

exists a maximal element of Sa. Let [ba, ca] denote such a maximal element. The elements ba

and ca must be endpoints of M . (Say, for instance, that ba is not an endpoint. Then there exist

e, f ∈M such that ba ∈ [e, f ], and either [e, ca] or [f, ca] will contain [ba, ca]. This would mean

[ba, ca] was not maximal in Sa.) Therefore, for each a ∈ M , there exist endpoints ba and ca so

that a ∈ [ba, ca]. So, M is generated by the set of its endpoints, and this generating set is as

small as possible by (1).

2.2 The theory of R-trees

In this section we give axioms in the signature Lp for the theory of pointed R-trees and investigate

some properties of this theory.

2.2.1 Definition. Let K be the class of models M of Tp whose underlying metric space (M,d, p)

is a pointed R-tree.

2.2.2 Definition. Let RT be the following collection of axioms:

1. the axioms of Tp from Definition 1.8.6; recall specifically the axioms from item (6) of

Definition 1.8.6: for each n ∈ N, the axiom

supn,x

supn,y

infn,z

max{|d(x, z)− d(x, y)2|, |d(y, z)− d(x, y)

2|} = 0;

2. for each n ∈ N the axiom

supn,x

supn,y

supn,z

supn,w

(min{(x · z)w, (y · z)w} −· (x · y)w

)= 0.

2.2.3 Theorem. The class K is exactly the class of models of the Lp-theory RT. That is, RT

axiomatizes K.

Proof. Let M |= RT. Then M |= Tp and therefore has a unique underlying metric space (M,d, p)

by Lemma 1.8.9. If we let x, y, z, w ∈M and let m = max{d(x, p), d(y, p), d(z, p), d(w, p)}, then

the axiom

supm,x

supm,y

supm,z

supm,w

(min{(x · z)w, (y · z)w} −· (x · y)w

)= 0

from (2) implies that the metric condition for 0-hyperbolicity given in 2.1.5 holds for x, y, z, w.

The points x, y, z, w were arbitrary, so we may conclude that (M,d, p) is 0-hyperbolic. Then by

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Lemma 2.1.6 we know that (M,d, p) embeds isometrically in an R-tree. The axioms in item (6)

of Definition 1.8.6 guarantee that for any x, y ∈ M , for any ε > 0, there exists a point z ∈ M

such that ∣∣∣∣d(x, z)− d(x, y)2

∣∣∣∣ ≤ ε, and∣∣∣∣d(y, z)− d(x, y)

2

∣∣∣∣ ≤ ε.Given a pair of points x, y in M , the existence of these approximate midpoints and the com-

pleteness of M may be used to construct a path between x and y. So M is path-connected.

A path-connected subspace of an R-tree is an R-tree. Therefore (M,d, p) is a pointed R-tree.

Conversely, let (M,d, p) be a pointed R-tree and let M be the corresponding Lp-structure. Then

M clearly satisfies the axioms in (1)-(5) of the definition of Tp. The fact that M is a uniquely

geodesic space with convex balls implies that M satisfies the axioms in (6). So, M |= Tp, and

the fact that M is 0-hyperbolic implies that M satisfies axioms in (2).

For a model M of RT, we will refer to the underlying metric space (M,d, p) of M as the

underlying R-tree of M.

Now, we present a short discussion of some definability issues in models of RT. Let M be

an R-tree and for r ∈ [0, 1] define the function νr : M ×M → M by: νr(x1, x2) = the point in

[x1, x2] with distance rd(x1, x2) from x1 and distance (1 − r)d(x1, x2) from x2. Define ν(n)r to

be the restriction of νr to Bn(p).

2.2.4 Lemma. Let r ∈ [0, 1]. The function νr : M×M →M is a uniformly 0-definable function

in models of RT.

Proof. Let M |= RT, and let (M,d, p) be the underlying R-tree of M. Let r ∈ [0, 1]. For n ∈ N

let ψn be the formula

max{d(x1, y)−· rd(x1, x2), d(x2, y)−· (1− r)d(x1, x2)}

where the variables are from the sort M (n). In M, the distance d(ν(n)r (x1, x2), y) is equal to

ψMn (x1, x2, y). So for any n ∈ N the function ν

(n)r is 0-definable via this quantifier-free formula

in any model of RT.

Define µ = ν1/2. We will use the midpoint function µ extensively in Chapter 3.

Next, we discuss of amalgamation over substructures for the Lp-theory RT. We begin by

proving any substructure of a model of RT extends to a unique model of RT.

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2.2.5 Lemma. A substructure A of N |= RT extends to a model M |= RT such that any

embedding of A into W |= RT extends to an embedding of M into W.

Proof. Let (N, d, p) be the underlying R-tree of N. Let A be the underlying metric space of

the substructure A. Note that pM = pA ∈ A. Then, EA is a closed subtree of N , and since N

is complete, EA is complete. Let M be the Lp-structure corresponding to EA with basepoint

pA. Then M |= RT. If W |= RT, it is straightforward to extend an embedding A → W to an

embedding of M. First, extend it isometrically to each [a, b] for a, b ∈ A, then, extend to the

closure.

2.2.6 Theorem. The Lp-theory RT has amalgamation over substructures. That is, if M0, M1

and M2 are substructures of models of RT and f1 : M0 → M1, f2 : M0 → M2 are embeddings,

then there exists a model N of RT and embeddings gi : Mi → N such that g1 ◦ f1 = g2 ◦ f2.

Proof. Let M0,M1,M2 and the fi for i = 1, 2 be as above. Let (Mi, di, pi) be the underlying

metric space of Mi for i = 0, 1, 2. By Lemma 2.2.5 we may assume that each of the Mi is

actually a model of RT, and therefore Mi is path connected for i = 0, 1, 2. In addition, to

simplify notation we may assume M0 ⊆ M1, M0 ⊆ M2, and M1 ∩M2 = M0, that the fi are

inclusion maps and that p0 = p1 = p2. Let N = M1 ∪M2 and define d : N ×N → R by:

• if x, y ∈Mi for i = 1 or i = 2, then define d(x, y) = di(x, y);

• if x ∈M1 and y ∈M2, define

d(y, x) = d(x, y) = inf{d1(x, z) + d2(z, y) | z ∈M0}.

This is a standard construction for the amalgamation of metric spaces, and it is straightforward

to show the function d is a metric on N . Note that in the second case of the definition, if one

of x or y is in M0, then the infimum will be realized by setting z = x or z = y. Thus the two

cases of the definition agree. Also, for i = 0, 1, 2, the metric di is equal to the restriction of d to

Mi ⊆ N .

Let N be the Lp-structure corresponding to (N, d, p) for the (N, d) defined above, where

p = p0. Define gi : Mi → N by gi(x) = x for i = 1, 2. Then, gi(p0) = pi for i = 1, 2. So,

gi are embeddings of Mi into N for i = 1, 2. It is clear that g1 ◦ f1 = g2 ◦ f2, since on the

level of underlying spaces, both maps are just inclusion maps. Because it is the Lp-structure

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corresponding to a metric space, N satisfies the axioms from (1)-(5) of Definition 1.8.6 by Lemma

1.8.7. Thus, to show N |= RT, it suffices to show that N is an R-tree by Theorem 2.2.3.

Claim: For x ∈M1 and y ∈M2

d(x, y) = d(x, ey) + d(ey, y)

where ey is the closest point to y in M0.

Proof of claim: Because ey ∈M0, by the definition of d we know,

d(x, y) ≤ d1(x, ey) + d2(ey, y) = d(x, ey) + d(ey, y).

Towards contradiction, assume d(x, y) < d(x, ey) + d(ey, y). Then there exists z ∈M0 such that

d(x, z) + d(z, y) = d1(x, z) + d2(z, y) < d1(x, ey) + d2(ey, y) = d(x, ey) + d(ey, y).

Because z ∈M0 and therefore ey ∈ [z, y] ⊆M2 we conclude

d(z, y) = d2(ey, z) + d2(y, ey) = d(ey, z) + d(y, ey).

Therefore

d(x, z) + d(ey, z) + d(y, ey) = d(x, z) + d(z, y) < d(x, ey) + d(ey, y)

which implies

d(x, z) + d(ey, z) < d(x, ey).

This contradicts the triangle inequality for d. Therefore, it must be true that

d(x, y) = d(x, ey) + d(ey, y).

So, the claim is proved.

By our construction, N = M1 ∪ M2. Let x, y ∈ N . If x ∈ M1 and y ∈ M2, then since

d(x, y) = d(x, ey) + d(ey, y), we may concatenate [x, ey] and [ey, y] to get a geodesic segment

between x and y in N . Denote this geodesic segment by [x, y]. We must show [x, y] is the unique

arc connecting x and y in N . Assume α : [0, t] → N is a continuous map with α(0) = x and

α(t) = y. For convenience let α also refer to the image of α. Towards contradiction, assume that

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setwise, α 6= [x, y]. We conclude that either α∩M1 6= [x, y]∩M1 or α∩M2 6= [x, y]∩M2 because

if both of these were equal it would contradict our assumption. Without loss of generality, say

α∩M1 6= [x, y]∩M1. Let a, b ∈M1 such that a ∈ [x, y]\α and b ∈ α\[x, y]. Now, γ = [x, a]∪[a, b]

is an arc in M1 from x to b. If we take α restricted to the interval [0, α−1(b)] we get another arc

α′ in M1 from x to b. Since a ∈ γ but is not on α′ we know these two arcs are distinct. Then,

g−11 (γ) and g−1

1 (α′) are distinct arcs in M1, which can’t happen because M1 is an R-tree. This

is our contradiction. So, we can’t have more than one arc from x to y in N .

If x, y ∈ M1, then since M1 is an R-tree there is a geodesic segment [x, y] in Mi ⊆ N and

moreover this is the only path between x and y contained in M1. Towards contradiction, suppose

there is an arc α between x and y in N not equal to [x, y]. As before let α denote both the

map and its image in N . Since α is a different arc, α ∩M1 6= [x, y] ∩M1, and we may proceed

as above. If x, y ∈ M2 the same argument works, just switch the subscripts 1 and 2. Thus,

between any x, y ∈ N there is a unique arc, and that arc is a geodesic segment. Therefore N is

an R-tree.

Intuitively, it is clear that one may build an R-tree by “gluing” copies of intervals in R and

other R-trees together. The following lemmas capture these ideas.

2.2.7 Lemma. Let Λ be a linearly ordered set. If (Mλ | λ ∈ Λ) is a chain of models of RT,

then the union of this chain is a model of RT.

Proof. Note that RT is an ∀∃-theory. Therefore, the conclusion is immediate from Proposition

1.7.4.

This lemma implies that the union of any increasing chain of R-trees is an R-tree. In [6,

Lemma 2.1.14], this was proved for countable increasing chains of R-trees, and the proof there

actually also works for chains of arbitrary length.

2.2.8 Lemma. Let M be a model of RT with underlying R-tree (M,d, p). Let (ai | i ∈ I) for

an ordinal I be a list of elements of M , which are not necessarily distinct. For each i ∈ I, let

Ti = (Ti, di, pi) |= RT. Then we may construct an extension N |= RT of M, such that there is

a copy of Ti isometrically embedded in (N \M) ∪ {ai} via an isometry taking pi to ai for all

i ∈ I. Moreover, if i 6= j and ai 6= aj, then the images of Ti and Tj in N are disjoint, and if

i 6= j but ai = aj, then the images of Ti and Tj intersect only at {ai}.

Proof. Assume the situation described in the hypotheses. Define M0 to be the result of amalga-

mating M with T0 over a substructure consisting of a point. The point in M is a0, and the point

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in T0 is p0. Then, by the proof of Lemma 2.2.6, there is a copy of T0 isometrically embedded in

(M0 \M) ∪ {a0} via an isometry taking p0 to a0. Note that this copy of T0 does not intersect

M except at a0.

For i ∈ I, define Mi+1 to be the result of amalgamating Mi with Ti+1 over the point

ai+1 ∈ M ⊂ Mi, and the point pi+1 in Ti+1. Then, by the proof of Lemma 2.2.6 there is a

copy of Ti+1 isometrically embedded in (Mi+1 \Mi) ∪ {ai+1} ⊆ (Mi+1 \M) ∪ {ai+1}. Note

that if ai+1 is equal to aj for some j < i + 1, then the images of Ti+1 and Tj overlap only at

ai+1 = aj . If ai+1 6= aj for all j < i + 1, then the image of Ti+1 is disjoint from the images of

the Tj for j < i + 1, because all those images are contained in M \ {ai+1}. If i ∈ I is a limit

ordinal, first let Mi be the union of the chain of models (Mj | j < i). Then, define Mi to be

the result of amalgamating Mi with Ti over the point pi ∈ Ti and the point ai ∈ Mi. Each Mj

is an extension of M, and an extension of Mi for all i < j. Thus, we get an increasing chain

of models extending M indexed by I. Let N be the completion of⋃i∈I Mi. Then N |= RT by

Lemma 2.2.7 and clearly N extends M.

2.2.9 Lemma. There exists an R-tree N such that every point in N has degree three.

Proof. Let N0 be the R-tree R with basepoint 0. Let N0 be the corresponding Lp-structure,

which is a model of RT. Let M = N0 and let (ai | i ∈ I) be an enumeration of N0. For each i,

let the R-tree Ti be R≥0 with basepoint 0, and let Ti be the corresponding model of RT. Define

N1 to be the extension of N0 that results from the application of Lemma 2.2.8 and let (N1, d, p)

be its underlying R-tree. There are now three branches in N1 at each point of N0. However,

the points on the rays we added (with the exception of the basepoints) all still have only two

branches. Let {a′i | i ∈ I ′} be an enumeration of N1 \N0 and apply Lemma 2.2.8, now with and

M = N1, and the same Ti, but now defined for all i ∈ I ′. Call the resulting extension N2. In

the underlying R-tree N2 of N2 there are exactly three branches in N2 out of each point of N1.

Generally, for n ≥ 1, enumerate Nn \ Nn−1 and apply Lemma 2.2.8 to get an extension Nn+1

of Nn such that in the R-tree Nn+1, all the points in Nn have exactly three branches and all

the points in Nn+1 \Nn have exactly two branches. Define N =⋃n∈ω Nn. This will be a model

of RT such that there are exactly three branches at every point in its underlying metric space

N .

2.2.10 Remark. The lemma above just is one example of how amalgamation may be used to

construct an example of an R-tree with a specific branching pattern. Obviously, the construction

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could be modified to produce R-trees with quite complicated patterns of branch points of varying

degrees. Another construction of an R-tree with three branches at every point is mentioned in

Example 2.4.2.

In the rest of this thesis, we will not always explicitly refer to the amalgamation construction

whenever we want to construct R-trees. We may use the more intuitive language which describes

“adding” rays or intervals, or “gluing” things together at a point.

2.3 Richly branching R-trees

In this section we find the model companion of RT.

2.3.1 Definition. Let h > 0. An R-tree (M,d) is h-richly branching if the set

Bh := {b ∈M | at b there are at least three branches of height ≥ h}

is dense in (M,d). If (M,d) is h-richly branching for some h, we say it is richly branching.

2.3.2 Lemma. Suppose (M,d) is an h-richly branching R-tree for some h > 0. Let a ∈M and

let β be a branch at a. Then β has infinite height.

Proof. Assume (M,d) is an h-richly branching R-tree, let a ∈M and β a branch at a. It suffices

to show that for any r ∈ R>0, there exists b ∈ β such that d(a, b) ≥ r. Let r ∈ R>0. Let k be

the smallest integer larger than rh . Let ε = h

2k . Let a1 ∈ Bh ∩ β be such that d(a, a1) ≥ h − ε.

Such an a1 exists by the density of Bh in M , the path-connectedness of β and the fact that the

height of β is at least h. There are at least three branches at a1 of length at least h. Let β1

be a branch at a1 that does not contain a. So setwise, β1 is contained in β. Let a2 ∈ Bh be a

point in β1 with d(a1, a2) ≥ h− ε. Note that a1 ∈ [a, a2] by how we chose a2. We may find a2

for the same reasons we may find a1. Continue like this to get a = a0, a1, a2, ..., ak+1 ∈ β such

that d(ai, ai+1) ≥ h − ε for all 0 ≤ i ≤ k and ai ∈ [ai−1, ai+1] for all 1 ≤ i ≤ k. Using Lemma

2.1.2 we know that

d(a0, ak+1) = d(a0, a1) + d(a1, a2) + ...+ d(ak−1, ak) + d(ak, ak+1).

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Then

d(a, ak+1) =( k−1∑i=0

d(ai, ai+1))

+ d(ak, ak+1) ≥ k(h− ε) + (h− ε)

= kh− kε+ (h− ε) ≥ r − h

2+ (h− h

2k) ≥ r.

Let b = ak+1. Then, d(a, b) ≥ r.

This lemma has the following immediate corollary.

2.3.3 Corollary. An R-tree (M,d) is h-richly branching for some h > 0 if and only if it is

h-richly branching for every h > 0.

Next we give axioms for the class of richly branching R-trees.

2.3.4 Definition. Given n ∈ N let ϕn(x) be the Lp-formula

inf2n,y1

inf2n,y2

inf2n,y3

max1≤i<j≤3

{|(d(x, yi) + d(x, yj))− d(yi, yj)|, |d(x, yi)−12|}

where the variable x is of sort M (n). Let ϕn be the closed formula supn,x

ϕn.

2.3.5 Definition. Let rbRT = RT ∪ {ϕn = 0 | n ∈ N}.

2.3.6 Theorem. The Lp-theory rbRT axiomatizes the class of models of Tp whose underlying

metric spaces are richly branching R-trees.

Proof. Let M |= Tp be such that its underlying metric space (M,d, p) is a richly branching

R-tree. Then M |= RT by Theorem 2.2.3. Also, for every h > 0 the set Bh is dense in M . Let

n ∈ N and let a ∈ M (n). Since B1/2 is dense in M , the set B1/2 ∩M (n) is dense in M (n). For

any b ∈ B1/2 ∩M (n), there exist c1, c2, c3 ∈ M (2n) that are each distance 12 from b and each

on a different branch out of b. These exist by the definition of B1/2, and by Lemma 2.1.2 they

witness the fact that ϕn(b)M = 0 for b ∈ B1/2 ∩M (n). Now, the uniform continuity of ϕn and

the density of B1/2 ∩M (n) in M (n) imply that ϕn(a)M = 0 for any a ∈ M (n). Therefore the

condition ϕn = 0 is true in M. Since n was arbitrary, we conclude that M |= rbRT.

Now, assume the Lp-structure M |= rbRT. Assume 116 > ε > 0 and a ∈M . Let Oε(a) denote

the open ball of radius ε in M centered at a. Let n be large enough so that Oε(a) ⊆M (n). By

our assumption, ϕn(a)M = 0. We will show that B1/4 ∩ Oε(a) 6= ∅.

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Because ϕn(a)M = 0, we know there exist distinct b1, b2, b3 ∈ M (2n) ⊆ M such that for

i 6= j ∈ {1, 2, 3}

|d(a, bi)−12| ≤ ε and 2 dist(a, [bi, bj ]) = |d(a, bi) + d(a, bj)− d(bi, bj)| ≤ ε.

Let z ∈ [b1, b2] be the closest point to a on [b1, b2]. Then, there are two cases.

Case I: The point z is also the closest point to b3 on [b1, b2]. Then we know that

d(z, a) = dist(a, [b1, b2]) ≤ ε

2.

We also know that

d(z, bi) ≥ d(a, bi)− d(z, a) ≥ (12− ε)− d(z, a) ≥ (

12− ε)− ε

2=

12− 3ε

2>

14

for i = 1, 2, 3, since we assumed ε < 116 . It is also clear that z ∈ [b1, b2] (since it is the closest

point to a on that segment), and z ∈ [b2, b3] and z ∈ [b1, b3] (since z is the closest point to b3 on

[b1, b2], which means the paths from b3 to b1 and from b3 to b2 must go through z.) So, there

are at least three distinct branches at z. Therefore in this case, the point z is in B1/4 ∩ Oε(a).

Case II: The point z is not the closest point to b3 on [b1, b2]. Then let y be the closest point

to b3 on [b1, b2]. Note that y ∈ [b1, b2], and y ∈ [b1, b3] and y ∈ [b2, b3]. Also, z ∈ [a, y].

Claim: The point y is either the closest point to a on [b1, b3] or the closest point to a on

[b2, b3].

Proof of claim: We know either y ∈ [z, b1] or y ∈ [z, b2]. Assume y ∈ [z, b1]. Then y ∈ [a, b1],

since [a, b1] must include the segment [z, b1]. Because z ∈ [a, y] and y ∈ [z, b3] we know [a, b3] ⊆

[a, y] ∪ [z, b3], and [a, y] ∩ [z, b3] = [y, z]. Thus [a, b3] = [a, z] ∪ [z, y] ∪ [y, b3], and y is on [a, b3].

So y ∈ [a, b1] ∩ [a, b3] ∩ [b1, b3], making it the point on [b1, b3] closest to a. The proof in the

case where y ∈ [z, b2] is exactly analogous to the above, with the conclusion that y is the closest

point to a on [b2, b3]. So the claim is proved.

Now, assume for convenience that y is the closest point to a on [b1, b3]. So, we know that

d(a, y) = dist(a, [b1, b3]) ≤ ε

2.

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We also know that

d(y, bi) ≥ d(a, bi)− d(y, a) ≥ (12− ε)− d(y, a) ≥ (

12− ε)− ε

2=

12− 3ε

2>

14.

Therefore, y ∈ B1/4 ∩Oε(a). (The proof that y ∈ B1/4 ∩Oε(a) in the case where y is the closest

point to a on [b2, b3] goes exactly the same way.) Thus, B1/4 ∩ Oε(a) 6= ∅. Our 116 > ε > 0 was

arbitrary, and therefore, B1/4 is dense in M . Therefore the underlying metric space of M is a

richly branching R-tree.

2.3.7 Lemma. Assume M |= rbRT.

1. Let M be ω-saturated. Then for any m ∈ N, there are at least m branches at any point.

2. Let κ be an infinite cardinal. If M is κ-saturated, then there are at least κ branches at

every point.

Proof. Let M |= rbRT.

Proof of (1): Assume M is ω-saturated. For m,n ∈ N, let σn,m(x) be the following Lp-formula,

where x is of sort n:

inf2n,y1

..., inf2n,ym

max1≤i<j≤m

{|(d(x, yi) + d(x, yj))− d(yi, yj)|, |d(x, yi)−12|}.

If the condition supn,x

σn,m(x) = 0 is true in M, then, by 1.7.3, for any a ∈ M there are points

b1, ..., bm ∈M (2n) so that

|(d(a, bi) + d(a, bj))− d(bi, bj)| = 0, and |d(a, bi)−12| = 0.

By Lemma 2.1.2, the points b1, ..., bm are distinct and in separate branches at a. Then there

must be at least m branches at a.

Thus it suffices to show each supn,x

σn,m(x) = 0 is true in M. Let a ∈M (n) and let 12 > ε > 0.

Since M |= rbRT we know the axiom ϕn = 0 is true in M, and therefore ϕMn (a) = 0. By ω-

saturation, we know that the quantifiers in this formula are realized exactly. Thus, by Lemma

2.1.2 there are at least 3 branches of height at least 12 at every point. Let c1, ..., cm be such that

ci ∈ [ci−1, ci+1] and d(ci, a) = ε2i . For each i = 1, ...,m let bi be a point on a branch at ci that

does not contain any of the cj for i 6= j. Such a branch exists at every ci because each of them

has at least 3 branches, and by how we chose them, only two of those branches will contain any

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cj for j 6= i. Note that this makes ci the closest point to a on [bi, bj ]. Let d(bi, ci) = 12 . The

distance

d(a, bi) = d(a, ci) + d(ci, bi) <12

2i

and clearly d(a, bi) > d(ci, bi) = 12 , so |d(a, bi)− 1

2 | < ε. Also,

|(d(a, bi) + d(a, bj))− d(bi, bj)| = 2 dist(a, [bi, bj ]) = 2d(a, ci) =ε

2i< ε.

Since 12 > ε > 0 was arbitrary, we conclude sup

n,xσMn,m(x) = 0

Proof of (2): Assume M is κ-saturated and towards contradiction assume there is a point

b ∈ M such that there are < κ branches at b. Let n be large enough so that b ∈ M (n).

Enumerate all the branches at b by {βi | i < α} for some α < κ. Let ai ∈ βi be the point on βi

with d(ai, b) = 12 , and let B be the set of parameters {b} ∪ {ai | i < α}. Note that |B| < κ. Let

Σ be the following set of Lp(B) conditions: for each i < α and ε > 0

Ei(x) := max{|(d(b, x) + d(b, ai))− d(ai, x)|, |d(b, x)− 12|} = 0

where x is a variable from sort M (2n). Take a finite subset Γ of Σ and let G = {i | Ei(x) ∈ Γ}. So,

G is finite. By the first part of this lemma there are at least |G|+1 branches at b. Therefore, Γ is

satisfiable. Simply take x on a branch β such that β 6= ai for any i ∈ G. Now, by the saturation

assumption we know there is an x that satisfies Σ. This x is on a branch not mentioned in our

list of branches. This is a contradiction. Therefore there must be at least κ branches at every

point in M .

2.4 Example: Universal R-trees

This section presents some examples of richly branching R-trees.

2.4.1 Definition. Let µ be a cardinal. An R-tree M is called µ-universal if, for any R-tree N

with ≤ µ branches at every point, there is an isometric embedding of N into M .

2.4.2 Example. ([10, Lemma 2.1.1]) In [10], the authors construct examples of complete,

homogeneous µ-universal R-trees Aµ for µ ≥ 2. These spaces are unique up to isometry in that

any complete homogeneous R-tree with µ branches at each point is isometric to Aµ. Clearly,

for µ ≥ 3, these spaces Aµ are examples of richly branching R-trees.

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A group is hyperbolic if its Cayley graph is a δ-hyperbolic metric space for some δ > 0 (see

[5, Chapter III.Γ Definition 2.2.1]). A non-elementary hyperbolic group is one that does not

have a cyclic subgroup of finite index.

2.4.3 Definition. Let (M,d, p) be a family of pointed metric spaces. Let U be a non-principal

ultrafilter on N and let (νm)m∈N be a sequence of positive integers such that limn→∞ νn = ∞.

The asymptotic cone of (M,d, p) with respect to (νm)m∈N and U is the ultraproduct of pointed

metric spaces∏U (M, d

νm, pm). Denote the asymptotic cone of (M,d, p) with respect to (νm)m∈N

and U by AU,(νm)(M,d, p).

The asymptotic cone of a finitely generated group is defined to be the asymptotic cone of

its Cayley graph. It is a fact that the asymptotic cone of a hyperbolic group is an R-tree and

is homogeneous. If the hyperbolic group is non-elementary, then its asymptotic cone is actually

isomorphic to A2ω (see [9]).

2.4.4 Proposition. ([10] or [9]) Let G be a non-elementary hyperbolic group. Let U be a

non-principal ultrafilter on N. By [9, Proposition 3.1.1] the asymptotic cone of G is a richly

branching R-tree.

2.5 The model companion of RT

In this section we to build up to the proof that the theory rbRT of richly branching R-trees is

the model companion of RT.

2.5.1 Definition. Let M |= Tp with underlying metric space (M,d, p), and a = a1, ..., ak ∈M .

For convenience assume a1 = p. Let b = b1, ..., bn ∈M . For y1, ..., yn from the appropriate sorts,

define the partial type DMb (y1, ..., yn/a) to be:

{|d(al, yj)− dM(al, bj)| = 0, |d(yi, yj)− dM(bi, bj)| = 0 | i, j = 1, ..., n; l = 1, ..., k}.

2.5.2 Lemma. Let M |= Tp with underlying metric space (M,d, p), and a = a1, ..., ak ∈ M .

For convenience assume a1 = p. Let b = b1, ..., bn ∈ M . If c1, ..., cn ∈ M are such that

(c1, ..., cn) |= DMb (y1, ..., yn/a), then for any quantifier free formula ϕ(x1, ..., xk, y1, ..., yn) we

have ϕ(a1, ..., ak, b1, ..., bn)M = ϕ(a1, ..., ak, c1, ..., cn)M.

Proof. This is proved by an easy induction on the definition of quantifer free formulas.

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The following lemma is the main tool for proving Theorem 2.5.4.

2.5.3 Lemma. Assume M is an ω1-saturated model of rbRT. Let K be a non-empty, finitely

generated R-tree with basepoint p. For any e ∈ M and any countable collection {βi | i ∈ ω}

of branches at e, there exists an isometric embedding f of K into M such that f(p) = e and

f(k) ∩ βi = {e} for all i ∈ ω.

Proof. Assume M is an ω1-saturated model of rbRT. Let K be a non-empty, finitely generated

R-tree with basepoint p. By Lemma 2.1.10, there is a minimal set of generators for K, namely,

the set of endpoints of K. We will prove this lemma by induction on the size of this minimal

generating set.

Base Case: Assume K has one generator, namely the basepoint p. So, K = {p}. Let e ∈M ,

and define f : K → M by f(p) = e. This isometric embedding clearly satisfies f(K) ∩ β = {e}

for every branch at e. So the lemma is true when K has one generator.

Inductive Step: Let K have n > 1 generators. Assume that if K ′ is an R-tree that has fewer

than n generators and basepoint p′, then for any e ∈ M and any collection {βi | i ∈ ω} of

branches at e, there exists an isometric embedding f ′ of K ′ into M sending p′ to e such that

f ′(K ′)∩ (∪i∈ωβi) = {e}. Let e ∈M and take an arbitrary collection {βi | i ∈ ω} of branches at

e. There must be generators b1, b2 of K so that p ∈ [b1, b2]. List these two, plus the rest of the

generators as b1, ..., bn.

Case I: If n = 2, then we just have 2 generators b1 and b2 and K = [b1, b2]. First, assume

p is a generator (say it is b1.) Find a branch β at e that does not intersect any branch from

{βi | i ∈ ω} (except at e). This branch exists by Lemma 2.3.7 because M is ω1-saturated and

hence has uncountably many branches at e. The branch has infinite height by Lemma 2.3.2, so

we may find a point c ∈ β such that d(e, c) = d(p, b2). Both [e, c] ⊆M and [p, b2] are isometric

to the interval [0, d(e, c)] in R. So, there is an isometry sending p to e with the properties we

wanted. Now, assume p is not a generator. Using Lemma 2.3.7, find two branches at e that

only intersect the members of {βi | i ∈ ω} at e. On one of these branches let c1 be the point

that is d(p, b1) away from e. On the other, let c2 be the point that is d(p, b2) away from e. Let

f be the isometric embedding sending K = [b1, b2] to [c1, c2], with f(b1) = c1 and f(b2) = c2.

Then clearly f(p) = e and f(K) intersects {βi | i ∈ ω} only at e.

Case II: If n ≥ 3, let K ′ be the R-tree generated by b1, ..., bn−1 and let f ′ : K ′ → M be an

isometric embedding such that f ′(p) = e and f ′(K ′)∩ (∪i∈ωβi) = {e}. Note that K ′ is a closed

subtree of K. Let a be the closest point in K ′ to bn. Look at f ′(a) ∈M . Find a branch at f ′(a)

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that only intersects f ′(K ′) at a. We know this branch has infinite height. So, we may let c be

a point on this branch with distance d(a, bn) from f ′(a), and extend f ′ by sending the segment

[a, bn] ∈ K to the segment [f ′(a), c] isometrically. Let f be this extension of f ′. Clearly f is an

isometric embedding which sends p to e and f(K) ∩ (∪i∈ωβi) = {e}.

2.5.4 Theorem. The Lp-theory rbRT is the model companion of RT.

Proof. Clearly any model of rbRT is a model of RT. If M is a model of RT, M can be extended

to a model of rbRT. For example, use Lemma 2.2.8 to glue in a copy of the 3-branching R-

tree from Lemma 2.2.9 at every point in M . By the definition of model companion (Definition

1.7.10), it remains to show that rbRT is model complete. We will show the equivalent condition

given in Proposition 1.7.9: if every M |= rbRT, is an existentially closed model of rbRT, then

rbRT is model complete. Let N |= rbRT be an extension of M. We may assume M and N

are ω1-saturated. (This is because we may consider the structure((M,d, p), (N, d, p), ι

)where

ι is the embedding from M to N, and take an ω1-saturated extension of that structure. If we

can verify the definition of existentially closed in that setting, it will be true of M and N.) Let

a1, ..., ak ∈M.

Claim: For any b1, ..., bn ∈ N , there exist c1, ..., cn ∈M with c1, ..., cn |= DNb (y1, ..., yn/a).

Proof of claim: Let b1, ..., bn ∈ N . Recall that Ea ⊆ M is the subtree generated by a1, ..., ak.

Define an equivalence relation on b1, ..., bn by bi ∼ bj if bi and bj have the same closest point in

Ea. Let A1, ..., Am be the equivalence classes of this equivalence relation, and for j = 1, ...,m

let ej be the unique closest point in Ea common to the members of Aj . For each j = 1, ...,m

let Kj be the R-tree generated by Aj ∪ {ej}, with basepoint pj = ej . Note that each Kj is

closed and for i 6= j, Ki ∩Kj = ∅. For each j = 1, ...,m, by Lemma 2.5.3, there is an isometric

embedding fj : Kj → M sending pj to ej such that fj(Kj) does not intersect Ea except at ej .

Note that ej is the unique closest point in Ea for every point in fj(Kj).

Let f be the union of the functions fj for j = 1, ...,m. If bi and bj are both in Al, then

d(bi, bj) = d(fl(bi), fl(bj)) = d(f(bi), f(bj)). If bi and bj are in Al 6= Am respectively, then

d(bi, bj) = d(bi, el) + d(el, em) + d(em, bj)

= d(fl(bi), fl(el)) + d(el, em) + d(fm(em), fm(bj))

= d(f(bi), el) + d(el, em) + d(em, f(bj))

= d(f(bi), f(bj)).

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Therefore the function f is an isometric embedding fromm⋃j=1

Kj to M . Let ci = f(bi) for all

i ∈ {1, ..., n}. Then clearly d(bi, bj) = d(f(bi), f(bj)) = d(ci, cj) for all i, j ∈ {1, ..., n}. Now let

l ∈ {1, ..., k} and i ∈ {1, ..., n}. Let ej be the closest point to bi in Ea. Then

d(al, bi) = d(al, ej) + d(ej , bi) = d(al, ej) + d(f(ej), f(bi)) = d(al, ej) + d(ej , ci) = d(al, ci).

Thus, the claim is true.

Now, it follows by Lemma 2.5.2 that for any quantifier free formula ϕ(x1, ..., xk, y1, ..., yn)

and any b1, ..., bn ∈ N there exist c1, ..., cn ∈M such that

ϕ(a1, ..., ak, b1, ..., bn)N = ϕ(a1, ..., ak, c1, ..., cn)N = ϕ(a1, ..., ak, c1, ..., cn)M.

So, if ε > 0 and b1, ..., bn ∈ N such that

|ϕ(a1, ..., ak, b1, ..., bn)N − infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N| ≤ ε

we know we can find c1, ..., cn ∈M with

ϕ(a1, ..., ak, c1, ..., cn)M = ϕ(a1, ..., ak, b1, ..., bn)N.

This implies

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M ≤ infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N.

Also, the fact that M ⊆ N implies

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M ≥ infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N.

We conclude that given any inf-formula infs1,y1

... infsn,yn

ϕ(x1, ..., xk, y1, ..., yn) and any a1, ..., ak ∈ N

we know

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M = infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N.

Therefore M is an existentially closed model of rbRT.

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2.6 Properties of the model companion

In this section various properties of rbRT are presented. The theory rbRT has quantifier elimi-

nation and is complete and stable. It is not categorical in any cardinal.

2.6.1 Lemma. The Lp-theory rbRT has quantifier elimination.

Proof. By Theorem 2.5.4, Theorem 2.2.6 and Proposition 1.7.13.

2.6.2 Corollary. The Lp-theory rbRT is complete.

Proof. Observe that we may embed the structure {p} consisting of just a basepoint in any model

of rbRT. As in first order logic, this fact along with quantifier elimination implies that rbRT is

complete.

2.6.3 Lemma. Let M |= RT, and let b, c ∈ M . Let A ⊆ M and define A′ = A ∪ {p}. Then

tpM(b/A) = tpM(c/A) if and only if b and c have the same unique closest point e ∈ EA′ and

d(b, e) = d(c, e).

Proof. Assume the situation described in the hypotheses. For the forward direction, assume

tpM(b/A) = tpM(a/A). Then we know d(b, a) = d(c, a) for all a ∈ EA′ , which implies b and c

must have the same unique closest point e ∈ EA′ . Moreover, it is clear that d(b, e) must equal

d(c, e). For the other direction, assume b and c have the same unique closest point e ∈ EA′

and that d(b, e) = d(c, e). Since rbRT has quantifier elimination, it suffices to show that the

quantifier-free types of c and b over A are the same. To show the quantifier-free types over

A are the same, it suffices to show d(a, b) = d(a, c) for all a ∈ A and d(p, b) = d(p, c). This

follows easily from our assumptions, since for any a ∈ A′, the point e must be on both [a, b] and

[a, c].

2.6.4 Theorem. The theory rbRT is stable. Indeed when κ is an infinite cardinal, rbRT is

κ-stable if and only if κ satisfies κω = κ.

Proof. Let κ be an infinite cardinal. Let M |= rbRT be κ+-saturated, with underlying R-tree

(M,d, p). First, assume κ = κω. Let |A| = κ. Then

|EA| ≤ |A×A|2ω = κ22ω ≤ κω2ω = κω = κ.

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Counting possible 1-types using Lemma 2.6.3 shows

|S1(A)| ≤ |EA × R≥0| = |EA|2ω ≤ |EA|ω2ω = κω2ω = κω = κ.

Thus the rbRT is κ-stable.

For the other direction, note that given any infinite κ we may construct, via a tree construc-

tion, a subset A of M with |A| = κ and |EA| = κω. Now assume that κω > κ, and let A be

such a subset. For each e ∈ EA, choose be on a branch out of e that intersects EA only at e

with d(e, be) = 1. We may always find such a branch provided our model is saturated enough.

The set {be | e ∈ EA} has cardinality κω, and for any e 6= f in EA it is straightforward to show

d(tp(be/A), tp(bf/A)) ≥ 2. Since κω > κ, this implies the theory is not κ-stable.

Let κ be a cardinal so that κ = κω and κ > 2ω. Let U be a κ-universal domain for rbRT. A

subset of U is small if its cardinality is < κ.

2.6.5 Definition. Let A,B and C be small subsets of U . Say A is ∗-independent from B over

C, denoted A |∗^CB, if and only if for all a ∈ A we have dist(a,EB∪C) = dist(a,EC).

2.6.6 Lemma. A |∗^CB if and only if for all a ∈ A the closest point to a in EB∪C is the same

as the closest point to a in EC .

Proof. Assume A |∗^CB. Take an arbitrary a ∈ A. Let e1 be the unique closest point to a in

EB∪C and e2 the unique closest point to a in EC . We assumed dist(a,EB∪C) = dist(a,EC),

which implies d(a, e1) = d(a, e2). Since e2 ∈ EC ⊆ EB∪C , we know e1 ∈ [a, e2] by Lemma 2.1.1.

Therefore, e1 = e2. Since a was arbitrary, we know this holds for all a ∈ A. For the other

direction, assume for all a ∈ A the closest point to a in EB∪C is the closest point to a in EC .

Then, clearly dist(a,EB∪C) = dist(a,EC) for all a ∈ A.

2.6.7 Theorem. The |∗^ independence relation is the model theoretic independence relation for

rbRT.

Proof. Note, in what follows, we will abbreviate unions such as B ∪C as BC. We will show |∗^satisfies all the properties of a stable independence relation on a universal domain of a stable

theory as given in [1, Theorem 14.12]. Then, by [1, Theorem 14.14], we know |∗^ is the model

theoretic independence relation for the stable theory rbRT.

(1) Invariance under automorphisms

Any automorphism σ satisfies σ(EA) = Eσ(A) and is distance preserving.

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(2) Symmetry: if A |∗^CB, then B |∗^C

A.

Assume A |∗^CB. This means for all a ∈ A we have that the closest point in EBC to a is

ea ∈ EC . Thus, by Lemma 2.1.2, for any a ∈ A, for any y ∈ EBC we have [a, y] ∩ EC 6= ∅. It

follows that for any x ∈ EA, for any y ∈ EBC there exists a point of EC on [x, y]. Let b ∈ B.

Then for any x ∈ EA there is a point of EC on [x, b]. It follows that the closest point in EA to

any b ∈ B is in EC .

(3) Transitivity: A |∗^CBD if and only if A |∗^C

B and A |∗^BCD.

We know

EC ⊆ EBC ⊆ EBCD

which implies

dist(a,EC) ≥ dist(a,EBC) ≥ dist(a,EBCD).

Therefore dist(a,EBCD) = dist(a,EC) if and only if

dist(a,EBC) = dist(a,EC) and dist(a,EBCD) = dist(a,EBC).

Hence

A |∗^C

BD if and only if A |∗^C

B and A |∗^BC

D.

(4) Finite character: A |∗^CB if and only if a |∗^C

B for all finite tuples a ∈ A.

This is clear from the definition.

(5) Extension: for all A,B,C we can find A′ such that tp(A/C) = tp(A′/C) and A′ |∗^CB.

By finite character and compactness, it suffices to show this statement when A is a finite

tuple. Let e ∈ EC be the unique point closest to EA = EA. Let β < κ be the cardinality of

EB . Then there are at most β branches in EB at any point of EB . Using Lemma 2.3.7, we

may modify the proof of Lemma 2.5.3 so that we avoid any given collection of branches of size

< κ. Since A is finite, we may use this modified Lemma 2.5.3 to embed a copy of EA = EA on

branches at e that do not intersect EB except at e. The image of A under this embedding gives

us A′.

(6) Local Character: if a = a1, ..., am is a finite tuple, there is a countable B0 ⊆ B such that

a |∗^B0B.

Let ei be the closest point of EB to ai for i = 1, ...,m. Let Bi be a countable subset of B

such that ei is an element of EBi. Let B0 =

⋃mi Bi.

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(7) Stationarity: if tp(A/M) = tp(A′/M), A |∗^MB, and A′ |∗^M

B, then tp(A/BM) =

tp(A′/BM), where M is a small submodel of U .

By quantifier elimination, tp(A/BM) is determined by {tp(a/BM) | a ∈ A} plus the infor-

mation {d(a1, a2) | a1, a2 ∈ A}. These distances {d(a1, a2) | a1, a2 ∈ A} are fixed by tp(A/M).

Thus, it suffices to show the conclusion in the case when A = {a} and A′ = {a′}. If a or a′ is in

M the conclusion is obvious, so assume a, a′ /∈M . The type of a (or a′) over BM is determined

by two parameters, the unique point in EBM that is closest to a, and the distance from a to

that point. Since a |∗^MB, it follows that the closest point in M = EM to a is the same as the

closest point in EBM to a, and the same is true for a′. Since tp(a/M) =tp(a′/M), we know a

and a′ have the same closest point e in M and d(a, e) = d(a′, e). Since e is also the closest point

in EBM to a and a′, we know that tp(a/BM) =tp(a′/BM) by Lemma 2.6.3.

2.6.8 Remark. The proof of stationarity above does not use the fact that M is a model, it

works for any set M .

2.6.9 Notation. Now that we know |∗^ is model theoretic independence for rbRT, we will

simply denote it by | .

2.6.10 Lemma. Let M |= rbRT with underlying R-tree (M,d, p). Let α be an infinite cardinal

and assume there exists a ∈M with degree α. Then the density character of M is at least α.

Proof. Assume the situation described in the hypotheses. Note that the density character of M

must be at least ω. Every branch at a has infinite extent, so on each branch out of a we may

find b so that d(a, b) = 1. Let (bi | i < α) be the collection of all these points. Note that for

i 6= j, a ∈ [bi, bj ] and therefore d(bi, bj) = 2. For each i < α, let Oi be an open ball of radius 12

centered at bi. Then Oi is contained in the same branch at a as b, because a /∈ [b, x] for every

point x ∈ Oa. Therefore, these Oi form a collection of disjoint open sets in M with cardinality

α. Thus, the density character of M must be at least α.

2.6.11 Theorem. The theory rbRT is not ω-categorical.

Proof. Any isomorphism g between models M and N of rbRT will be a homeomorphism on the

underlying R-trees, and therefore g must preserve branching.

Using Lemma 2.2.8, build a separable model M of rbRT so that each branch point in the

underlying R-tree M has degree three. Let N0 be the R-tree R with basepoint 0. Let N0 be

the corresponding Lp-structure, which is a model of RT. Let A0 = {a0i | i ∈ ω} be a countable

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dense subset of N0. For each i, let the R-tree Ti be R≥0 with basepoint 0, and let Ti be the

corresponding model of RT. Note that Ti is separable. Define N1 to be the extension of N0

that results from the application of Lemma 2.2.8 and let (N1, d, p) be its underlying R-tree.

There are now three branches at each point in A0 ⊆ N1. Moreover, because N1 is equal to

the countable union(⋃

i∈ω Ti)∪ N0, and each Ti is separable, we know that N1 is separable.

The points on the rays we added (with the exception of the basepoints) all still have only two

branches. So, we must iterate this construction.

Let A1 = {a1i | i ∈ ω} be a countable, dense subset of N1 \N0 and apply Lemma 2.2.8, to N1

to add the rays Ti to each point in A1. Call the resulting extension N2. In the underlying R-tree

N2 of N2 there are exactly three branches out of each point in A1 ∪ A0, which is a countable,

dense subset of N1. Note that N2 is still separable, since it is a countable union of separable

spaces. Generally, for n ≥ 1, let An = {ani | i ∈ ω} be a countable dense subset of Nn \ Nn−1

and apply Lemma 2.2.8 to get an extension Nn+1 of Nn such that in the R-tree Nn+1, all the

points in⋃ni=0Ai have exactly three branches and all the points in Nn+1 \

⋃ni=0Ai have exactly

two branches. Define N =⋃n∈ω Nn. This will be a model of RT such that there are exactly

three branches at every point in⋃∞i=0Ai, which is a countable, dense subset of the underlying

space N of N. Therefore, N is separable.

Alternatively, we may modify the construction above to get a separable model M of rbRT so

that every branch point in its underlying R-tree M has degree four. Simply let each Ti be a copy

of R with basepoint 0. The separable models M and N are not homeomorphic, and therefore

cannot be isomorphic.

2.6.12 Theorem. Let κ > ω be a cardinal. The theory rbRT is not κ-categorical.

Proof. Let κ > ω be a cardinal. Using Lemma 2.2.8, we could construct a model M so that the

set of branch points in its underlying R-tree M is dense and of size κ, and there are κ-many

branches at each branch point.

Alternatively, we could construct a model N so that in the underlying R-tree N of N the

set of branch points is dense and of size κ, there are κ branches at pN, but every other branch

point has degree three. By Lemma 2.6.10 both M and N have density character at least κ. It

follows from the construction using Lemma 2.2.8, M and N both have density character at most

κ. Clearly M and N are not homeomorphic. Thus, they cannot be isomorphic.

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CHAPTER 3

HYPERBOLIC ISOMETRIES OF R-TREES

3.1 Introduction to isometries of R-trees

In this chapter we study classes of R-trees equipped with isometries. This section presents some

necessary background (parts of which will not be used until Chapter 4). Given an R-tree M and

a geodesic segment [a, b] ⊆ M , we will use the notation (a, b] to mean [a, b] \ {a}, and likewise

for [a, b).

By an isometry of an R-tree M , we mean a surjective distance preserving function from

M → M . Isometries of R-trees fall into two categories. If an isometry f of an R-tree M has a

fixed point it is called elliptic, otherwise it is hyperbolic. The quantity ||f || := infx∈M d(x, f(x))

is called the translation distance of f .

3.1.1 Lemma. (See [7, 1.3]) Let f be an isometry of an R-tree M . If ||f || = 0, then f is

elliptic. If ||f || > 0, then f is hyperbolic and acts as a translation along an axis, which is a copy

of R in M . The points on this axis are moved by exactly distance ||f ||.

For a hyperbolic isometry of an R-tree f , let Af denote the axis of f .

3.1.2 Lemma. (See [7, 1.3]) Let M be an R-tree and let f : M →M be a hyperbolic isometry

of M . Let Af be the axis of f . Then for all x ∈M ,

dist(x,Af ) =d(x, f(x))− ||f ||

2.

Let f : M → M be an isometry of an R-tree M . For any such f and M , define f0(x) = x.

Let a ∈ M . Define the f -order of a to be the cardinality of the orbit of a under f . When it is

clear what function we are referring to, we will simply call this the order of a. For any m ∈ N,

define

fix(fm) := {x ∈M | d(x, fm(x)) = 0}.

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Note that fix(fm) a closed subtree of M (see [6, Chapter 3, Lemma 1.1].) Note also that if

a ∈ fix(fm), then the f -order of a must divide m. Recall from Section 2.2 that µ(x, y) denotes

the midpoint of [x, y], and that µ is a function uniformly 0-definable in all models of T . Define

µm(x) := µ(x, fm(x))

which is the midpoint of [x, fm(x)] for m ∈ N>0. Define µ0(x) = x.

3.1.3 Lemma. Let M be an R-tree and let f be an elliptic isometry of M . Let m ∈ N and

x ∈M . Then µm(x) is the unique point in fix(fm) that is closest to x.

Proof. This follows directly from [6, Chapter 3, Lemma 1.1], applied to the isometry fm.

3.2 The signatures Ls and Ls-structures

In this section we define the continuous signatures Ls we use when studying R-trees with an

isometry. For each s, we also give an Ls-theory Ts such that models of Ts have unique underlying

spaces and functions up to isomorphism.

Let s ∈ N>0. Let Ls be the signature Lp from page 16, plus a family of function symbols

(fn | n ∈ N), where the arity of fn is (n;n+s) and fn has modulus of uniform continuity ∆(ε) = ε,

for each n ∈ N. Let (M,d, p, f) be such that (M,d, p) is a pointed metric space, f : M →M is

a function that satisfies the modulus of uniform continuity ∆(ε) = ε and d(p, f(x)) ≤ n+ s for

all x ∈ Bn(p). We define the Ls-structure corresponding to (M,d, p, f) to be the Lp structure

corresponding to (M,d, p), together with the interpretation fMn = f � Bn(p) : M (n) →M (n+s).

Let Ts be the Ls-theory equal to the theory Tp from Section 1.8 together with the axioms

supm,x

d(fn(Im,n(x)), Im+s,n+s(fm(x))

)= 0

with m,n ∈ N such that m ≤ n.

3.2.1 Lemma. Let (M,d, p) be a pointed R-tree, and let f : M → M be a function satisfying

∆(ε) = ε and d(p, f(x)) ≤ n + s for all x ∈ Bn(p). Then the Ls-structure M corresponding to

(M,d, p, f) is a model of Ts.

Proof. By Lemma 1.8.8 we know M |= Tp. By definition, fMn = f � Bn(p), and IM

m,n is the

inclusion map between M (n) = Bn(p) and M (m) = Bm(p) for all m ≤ n ∈ N. By assumption,

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d(p, f(x)) ≤ n+ s for all x ∈ Bn(p). Therefore

M |= supm,x

d(fn(Im,n(x)), Im+s,n+s(fm(x))

)= 0.

3.2.2 Corollary. Let (M,d, p) be a pointed R-tree, and let f : M → M be an isometry such

that d(p, f(p)) ≤ s. Then the Ls-structure M corresponding to (M,d, p, f) is a model of Ts.

Proof. Because f is an isometry, it will satisfy the modulus of uniform continuity ∆(ε) = ε. Since

d(p, f(p)) ≤ s, for all x ∈ Bn(p) we have d(p, f(x)) ≤ d(p, f(p)) + d(f(p), f(x)) = d(p, f(p)) +

d(p, x) ≤ n+ s.

3.2.3 Lemma. Let M =(

(M (n), d(n)), pn, Im,n, fn)

be an Ls-structure that is a model of

Ts. Let (M,d, p) be the underlying metric space of M. Then there exists a unique function

f : M →M such that M is isomorphic to the Ls-structure corresponding to (M,d, p, f).

Proof. Let M =(

(M (n), d(n)), pn, Im,n, fn)|= Ts. Let (M,d, p) be the underlying metric space

for M. Let Bn(p) denote the closed ball of radius n centered at p in (M,d, p). By Lemma 1.8.9,

we may assume M (n) = Bn(p) and IMm,n is the inclusion of Bm(p) in Bn(p).

For all m ≤ n

M |= supm,x

d(fn(Im,n(x)), Im+s,n+s(fm(x))) = 0

implies fMn ◦ IM

m,n(x) = IMm+s,n+s ◦ fM

m (x) for all x ∈ M (m). This implies fMm = fM

n � Bm(p).

Take the union of the functions fMn to get f : M →M such that f restricted to Bn(p) is equal

to fMn for all n ∈ N.

For M |= Ts, we call the unique (M,d, p, f) guaranteed by Lemma 3.2.3 the underlying metric

space and function of M. If the function is an isometry, we call (M,d, p, f) the underlying metric

space and isometry of M.

3.3 Theories of hyperbolic isometries of R-trees

In this section we axiomatize certain classes of R-trees with a hyperbolic isometry and find

model companions for those theories. Let s ∈ N>0 and Ls as defined on page 51. Let r ∈ R>0

with s ≥ r. We will use the abbreviation ||fn|| for the Ls-formula infn,x

d(x, fn(x)).

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3.3.1 Definition. Let Kr,s be the class of M |= Ts with underlying metric space and function

(M,d, p, f) so that (M,d, p) is a pointed R-tree, and f : M →M is a hyperbolic isometry such

that d(p, f(p)) ≤ s and r ≤ ||f ||.

3.3.2 Definition. Let HRTr,s be the Ls-theory consisting of the following Ls-conditions:

1. the axioms of Ts and RT;

2. for n ∈ N the axiom

supn,x

supn,y|d(x, y)− d(f(x), f(y))| = 0;

3. for n ∈ N the axiom

supn+s,y

min{n−· d(f(p), y), infn,x

d(f(x), y)} = 0;

4. for n ∈ N the axiom

supn,x

(r −· d(x, f(x))) = 0;

5. the axiom

d(p, f1(p))−· s = 0.

3.3.3 Lemma. The class Kr,s is exactly the class of Ls-structures that are models of HRTr,s.

Proof. Let M =(

(M (n), d(n)), pn, Im,n, fn)|= HRTr,s and let (M,d, p, f) be its underlying

metric space and function. Recall that fMn = f � Bn(p), and M (n) = Bn(p) for each n ∈ N.

We know that (M,d, p) is a pointed R-tree by Theorem 2.2.3. The axioms in item (2) above

guarantee that f is an isometry, and the axioms in (3) guarantee that the image of fMn is dense in

M (n+s) for every n ∈ N. Then since M (n) and M (n+s) are complete, we conclude that each fMn

is surjective. Therefore, f is a surjective isometry from M to M . The axioms in (4) guarantee

that r ≤ ||fn|| for each n ∈ N, which implies that r ≤ ||f ||. The axiom in (5) implies that

d(p, f(p)) ≤ s.

Let M =(

(M (n), d(n)), pn, Im,n, fn)∈ Kr,s. The axioms in (1) are satisfied in M by Corol-

lary 3.2.2 and Theorem 2.2.3. The axioms in (2) are true in M because f is an isometry, and

the axioms in (3) are true in M because f is onto. The axiom in (4) is true in M because f is

hyperbolic with ||f || ≥ r. That d(m)(p, f1(p)) −· s = 0 follows directly from d(p, f(p)) ≤ s, so

axiom (5) is true in M. Therefore, M |= HRTr,s.

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For a model M of HRTr,s, we refer to the underlying metric space and function (M,d, p, f)

of M as the underlying R-tree and (hyperbolic) isometry of M.

3.3.4 Theorem. The Ls-theory HRTr,s has the amalgamation property over substructures of

models. That is, if M0, M1 and M2 are substructures of models of HRTr,s and ϕ1 : M0 →M1,

ϕ2 : M0 →M2 are embeddings, then there exists a model N of HRTr,s and embeddings gi : Mi →

N such that g1 ◦ ϕ1 = g2 ◦ ϕ2.

Proof. Let (M0, d0, p0, f0) be the underlying R-tree and isometry of M0 and likewise for

(M1, d1, p1, f1) and (M2, d2, p2, f2). As in the proof of Lemma 2.2.6, we may assume that

M0 ⊆ M1, and M0 ⊆ M2 with M0 = M1 ∩M2 and the ϕi are inclusion maps for i = 1, 2.

In particular, this means M0 ⊆ M1 and the isometry f0 = f1 � M0. Likewise, M0 ⊆ M2 and

f0 = f2 � M0. Therefore, f1(x) = f0(x) = f2(x) for all x ∈M0.

Let the Lp-structure N′ |= RT with underlying metric space (N, d, p), and the embeddings

gi : Mi → N′ be the result of applying Lemma 2.2.6 to the reducts of M0, M1 and M2 to Lp.

Recall from the proof of Lemma 2.2.6 that N = M1 ∪ M2 where M1 ∩ M2 = M0, and the

embeddings gi are the inclusion maps of Mi into N for i = 1, 2. Also, p0 = p1 = p2 = p. Recall

that d(x, y) = di(x, y) if x, y ∈Mi, and d(x, y) = di(x, ey) + dj(y, ey) for x ∈Mi and y ∈Mj if

i 6= j, where ey is the closest point in M0 to y. Moreover, di is equal to the restriction of d to

Mi for i = 0, 1, 2.

Define f on N by: f = f1 on M1 ⊆ N , and f = f2 on M2 ⊆ N . This is well defined because

f1 and f2 are equal on M1 ∩M2 = M0. We know f ◦ g1 = g1 ◦ f1 and f ◦ g2 = g2 ◦ f2, because

gi is the inclusion of Mi into N for i = 1, 2 .

It remains to show that f is an isometry of N . Let x, y ∈ N . If x, y ∈ Mi for one of

i = 1, 2, then d(f(x), f(y)) = d(fi(x), fi(y)) = d(x, y). If x ∈ M1 \M0 and y ∈ M2 \M0, then

f(x) ∈M1 \M0 and f(y) ∈M2 \M0. Let ey be the closest point to y in M0. Then the closest

point to f(y) = f2(y) in M0 ⊆M2 ⊆ N is f2(ey) = f(ey) = f1(ey). This implies,

d(f(x), f(y)) = d(f(x), f(ey)) + d(f(y), f(ey))

= d1(f1(x), f1(ey)) + d2(f2(y), f2(ey))

= d1(x, ey) + d2(y, ey)

= d(x, ey) + d(y, ey).

Therefore f is an isometry.

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Note that since p and f(p) are both in M0, we have d(p, f(p)) = d0(p0, f0(p0)) ≤ s. Note

also that for every x ∈M , if x ∈Mi, then f(x) ∈Mi for i = 1, 2. This implies that d(x, f(x)) =

di(x, fi(x)) ≥ r, and thus, ||f || ≥ r > 0. So, (N, d, p) is a pointed R-tree and f : N → N is

a hyperbolic isometry such that d(p, f(p)) ≤ s and r ≤ ||f ||. Therefore, the Ls-structure N

corresponding to (N, d, p, f) is in the class Kr,s by Definition 3.3.1. By Lemma 3.3.3, we know

that N |= HRTr,s. Since f ◦ g1 = g1 ◦ f1 and f ◦ g2 = g2 ◦ f2, we know the gi give embeddings

g1 : M1 → N and g2 : M2 → N of Ls-structures . Lastly, it is clear that g1 ◦ϕ1 = g2 ◦ϕ2, because

on the level of the underlying R-trees, these functions are inclusion maps.

In picturing this situation, it may help to recall that M0 is closed under f0 and f−10 , which

implies that the axis A0 of f0 is contained in M0. Since f0 = f1 � M0, the axis A1 of f1 must

equal A0, and likewise the axis A2 of f2 must equal A0.

3.4 Model companions: hyperbolic case

For this section, let s ∈ N>0 and let r ∈ R>0 such that r ≤ s. In this section, we build up to the

definition of the theory rbHRTr,s, and show it is the model companion of the theory HRTr,s.

3.4.1 Definition. Let M |= Ts with underlying metric space and function (M,d, p, f). Let

a = a1, ..., ak ∈ M and b = b1, ..., bn ∈ M , and for convenience let a1 = p. For y1, ..., yn from

the appropriate sorts, define the partial type DMf,b(y1, ..., yn/a) to consist of:

• for m, l ∈ N and i = 1, ..., k, j = 1, ..., n the condition

|d(fm(ai), f l(yj))−(d(fm(ai), f l(bj))

)M| = 0;

• for m, l ∈ N and i, j = 1, ..., n the condition

|d(fm(yi), f l(yj))−(d(fm(bi), f l(bj))

)M| = 0.

3.4.2 Lemma. Let M |= HRTr,s with underlying metric space and function (M,d, p, f). Let

a = a1, ..., ak ∈ M , b = b1, ..., bn ∈ M , and c1, ..., cn ∈ M , and let a1 = p. If (c1, ..., cn) |=

DMf,b(y1, ..., yn/a), then for any quantifier free Ls-formula ϕ(x1, ..., xk, y1, ..., yn)

ϕ(a1, ..., ak, b1, ..., bn)M = ϕ(a1, ..., ak, c1, ..., cn)M.

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Proof. The proof is analogous to that of Lemma 2.5.2.

3.4.3 Lemma. Assume (M,d, p) is a pointed R-tree and let f be a hyperbolic isometry of M .

Let a1, ..., ak ∈M for k ≥ 1, and assume p = a1. Let A be the set

⋃m∈Z{fm(ai) | i = 1, ..., k}

Then,

1. EA is closed under f and f−1.

2. the axis Af of f is contained in EA;

3. EA = EA;

4. for all x ∈ EA, the degree of x in EA (the number of branches at a in EA) is at most

countable;

Proof. Let M , the points a1, ..., ak and the set A be as above.

Proof of (1): If x ∈ EA, then x ∈ [fm(ai), fn(aj)] for some m,n ∈ Z and some i, j ∈ {1, ..., k}.

Then f(x) ∈ [fm+1(ai), fn+1(aj)] and f−1(x) ∈ [fm−1(ai), fn−1(aj)], which are also clearly

contained in EA.

Proof of (2): The segment [a1, f(a1)] ⊆ EA must contain the segment [e, f(e)] where e is the

point on the axis Af closest to a1. Also, Af =⋃m∈Z f

m([e, f(e)]). It follows by part (1) that

Af ⊆ EA.

Proof of (3): Assume there is a sequence (xn)∞n=1 of points in EA converging to a point y in

M . For all i = 1, ..., k, let ei be the point on the axis closest to ai. Then, fm(ei) is the closest

point on the axis to fm(ai) for any m ∈ Z. So for any fm(ai) and any f l(aj) with l,m ∈ Z and

i, j ∈ {1, ..., k} we know

[fm(ai), f l(aj)] = [fm(ai), fm(ei)] ∪ [fm(ei), fn(ej)] ∪ [f l(aj), f l(ej)].

Therefore, EA = Af ∪⋃

i=1,..,k

⋃m∈Z

[fm(ai), fm(ei)], since each [fm(ei), fn(ej)] is contained in Af .

If y ∈ Af , then y ∈ EA. If y /∈ Af , then eventually the sequence (xn)∞n=1 is bounded away

from Af , that is, there exists δ > 0 so that dist(xn, Af ) > δ for all n ∈ N>0. Therefore, there

is a tail of (xn)∞n=1 contained in⋃i=1,..,k

⋃m∈Z[fm(ai), fm(ei)]. We may replace the original

sequence with this subsequence. Now, because there are only finitely many choices for i, some

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subsequence of (xn)∞n=1 must be contained in⋃m∈Z[fm(ai), fm(ei)] for some fixed i ∈ {1, ..., k}.

Again, replace the sequence with this subsequence. Now, for m 6= l ∈ Z the sets [fm(ai), fm(ei)]

and [f l(ai), f l(ei)] have distance at least ||f || from one another. This implies that eventually,

the elements of the sequence (xn)∞n=1 must be in one of them. Each [fm(ai), fm(ei)] is closed,

and therefore y ∈ EA.

Proof of (4): This follows from the fact that EA is generated by a countable set of points.

3.4.4 Lemma. Assume N |= HRTr,s. Let (N, d, p, f) be the underlying metric space and

hyperbolic isometry of N. Let T be a non-empty closed subtree of N such that T is closed under

f and f−1. Let b1, ..., bn ∈ N \ T. Let B be the set

⋃m∈Z{fm(bi) | i = 1, ..., n}.

Define an equivalence relation on B by: x ∼ y if and only if x and y have the same unique

closest point in T. Let {Gh}∞h=1 be the equivalence classes of ∼. For convenience and without

loss of generality, assume b1, ..., bn ∈⋃nh=1Gh. For each h, let eh be the unique closest point in

T common to all members of Gh. Then,

1. fm(bi) and f l(bi) are in different equivalence classes for any m 6= l and any i = 1, ..., n;

2. for all h ∈ N the equivalence class Gh has cardinality at most n;

3. the set {Gh}∞h=1 of equivalence classes of ∼ is closed under f and f−1;

4. for h ≥ n+ 1, there is some j ∈ {1, ..., n} and some m ∈ N such that fm(Gj) = Gh.

Proof. Assume the situation described in the hypotheses. For any m ∈ Z, if x ∈ Gh, then

fm(eh) is the closest point to fm(x) in T, because T is closed under the isometries f and f−1.

Proof of (1): Towards contradiction, assume there exist l,m ∈ Z and i ∈ {1, ..., n} so that

f l(bi) and fm(bi) are in the same equivalence class Gh. Then fm−l(eh) is the closest point to

fm−l(f l(bi)) = fm(bi) in T. So fm−l(eh) = eh, which contradicts that f is hyperbolic.

Proof of (2): Towards contradiction, assume there is an h ∈ N so that Gh has cardinality

≥ n+ 1. Each point in Gh is of the form fm(bi) for i = 1, ..., n and m ∈ Z. By the pigeonhole

principle, since there are > n distinct points in Gh there must be two that are different images

of the same bi. This contradicts (1).

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Proof of (3): Let Gh be an equivalence class and let x ∈ Gh. Let Gj be the equivalence class

of f(x). Then, ej = f(eh). This implies Gj = f(Gh). Therefore, f(Gh) is also an equivalence

class of ∼. The argument for the closure of {Gh}∞h=1 under f−1 is analogous.

Proof of (4): Let h ≥ n + 1. Let c the minimum of {i | ∃m ∈ Z, fm(bi) ∈ Gh}. Let Gj be

the equivalence class of bc. Since b1, ..., bn ∈⋃nh=1Gh, we conclude j ∈ {1, ..., n}. The fact that

fm(bc) ∈ Gh implies that fm(ej) = eh and thus fm(Gj) = Gh.

3.4.5 Theorem. Let M |= HRTr,s. Then M is an existentially closed model of HRTr,s if and

only the underlying R-tree of M is richly branching.

Proof. Let M |= HRTr,s and let (M,d, p, f) be the underlying R-tree and isometry of M . For

the forward direction, assume M is an existentially closed model. We know that (M,d, p) is

a pointed R-tree. We will show the reduct of M to Lp is a model of rbRT, which implies

M is richly branching. It suffices to verify that the conditions ϕn = 0 defined in 2.3.4 are

true in M. Let n ∈ N and a ∈ M (n). Then, ϕn(a) is an inf-formula with a parameter from

M. Let N be an extension of M with at least 3 branches of infinite height at every point in⋃{fm(a) | m ∈ Z}. To find such an extension, use Lemma 2.2.8 to add 3 new rays Rb1,Rb2, and

Rb3 at each b ∈⋃{fm(a) | m ∈ Z} so that Rbi ∩Rbj = {b} for i 6= j. This gives us an Lp-structure

N extending the reduct of M to Lp. Let (N, d, p) be the underlying R-tree of N. Extend f to a

hyperbolic isometry of N by defining f(Rbi ) = Rf(b)i isometrically for i = 1, 2, 3. Let N be the

Ls-structure corresponding to (N, d, p, f). Because M is existentially closed, we know

ϕMn (a) = ϕN

n (a).

We also know ϕNn (a) = 0 because there are at least 3 branches of infinite height at a. Therefore

ϕMn (a) = 0. The number n and a ∈ M (n) were arbitrary. Thus for all n ∈ N we know

ϕn = supn,x

ϕ(x) = 0, and therefore the Lp-structure corresponding to the underlying metric

space of M is a model of rbRT. So, M must be a richly branching R-tree.

Now, for the other direction assume that the underlying R-tree (M,d, p) of M is richly

branching. As in the proof of Theorem 2.5.4, we may assume M is ω1-saturated without loss of

generality. Since M is ω1-saturated, its reduct to Lp is also ω1-saturated. Also as in the proof

of Theorem 2.5.4, to show M is existentially closed, by Lemma 3.4.2 it suffices to show: for any

a1, ..., ak ∈ M , any extension N |= HRTr,s with underlying R-tree (N, d, p) and any b1, ..., bn ∈

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N , there exist c1, ..., cn ∈M such that c1, ..., cn satisfy the partial type DNf,b(x1, ..., xn/a).

Let a1, ..., ak ∈ M , and let N |= HRTr,s be an extension of M. Let b1, ..., bn ∈ N . We may

assume that none of the bi are in M , because if so, we may let ci = bi for that i and then add

it and its images under f and f−1 to our list of parameters. So, assume b1, ..., bn ∈ N \M . Let

A be the set⋃m∈Z{fm(ai) | i = 1, ..., k}. Let T=EA. By Lemma 3.4.3, T is a non-empty closed

subtree of N which is closed under f and f−1. Define

B =⋃m∈Z{fm(bi) | i = 1, ..., n},

and define the equivalence relation ∼ on B with equivalence classes Gh as in Lemma 3.4.4. For

h ∈ N, let Kh be the subtree of N generated by Gh ∪ {eh}, with basepoint ph = eh. Each Kh

is finitely generated, since |Gh| ≤ n by Lemma 3.4.4, part (2). Every point in Kh has eh as its

closest point in EA. Also, if Kh ∩Kl 6= ∅, then h = l. If x ∈ Kh and y ∈ Kl for h 6= l, then

d(x, y) = d(x, eh) + d(eh, el) + d(y, el) by Lemma 2.1.3.

Next, we build an isometric embedding

g :∞⋃h=1

Kh →M.

Step 1: By Lemma 3.4.3, any point in EA has at most countable degree. So, we use Lemma

2.5.3 to find an isometric embedding g1 : K1 → M sending p1 to e1 whose image does not

intersect EA except at e1. Define the partial function g on⋃l∈Z f

l(K1) by:

• g = g1 on K1;

• for all x ∈ K1, for all l ∈ Z, define g(f l(x)) = f l(g1(x)) = f l(g(x)). This defines g on

f l(K1) for every l ∈ Z.

Clearly g is isometric embedding onK1, and g fixes e1. Letm, l ∈ Z and y ∈ fm(K1), z ∈ f l(K1).

Let y′ = f−m(y) ∈ K1 and z′ = f−l(z) ∈ K1. Then,

d(y, z) = d(y, fm(e1)) + d(fm(e1), f l(e1)) + d(f l(e1), z) (3.1)

= d(y′, e1) + d(fm(e1), f l(e1)) + d(e1, z′) (3.2)

= d(g(y′), e1) + d(fm(e1), f l(e1)) + d(e1, g(z′)) (3.3)

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= d(fm(g(y′)), fm(e1)) + d(fm(e1), f l(e1)) + d(f l(e1), f l(g(z′))) (3.4)

= d(fm(g(y′)), f l(g(z′))) (3.5)

= d(g(fm(y′)), g(f l(z′))) = d(g(y), g(z)) (3.6)

Line (3.1) is true because fm(e1) and f l(e1) are the closest points in EA to y and z respectively.

Line (3.2) follows because f−m and f−l are isometries. Line (3.3) is true because g is an isometry

when restricted to K1 and line (3.4) follows because fm and f l are isometries. The function g

fixes e1, so e1 is the closest point in EA to g(y), and therefore fm(e1) is the closest point in EA

to fm(g(y)). Thus, (3.5) follows from (3.4). Lastly, (3.6) is true by the definitions of g and the

points y′ and z′. So g is an isometry on⋃l∈Z f

l(K1). Also, it follows from the fact that EA is

closed under f and f−1 that the image of g intersects EA only at the points f l(e1) for l ∈ Z.

The isometry g commutes with f and f−1 by definition.

Step 2: Extend g to(⋃

l∈Z fl(K1)

)⋃ (⋃l∈Z f

l(K2)). There are two cases.

Case I: If there are any points in K2 at which g has already been defined, then K2 = f l(K1)

for some l ∈ Z. In this case g is already defined on K2 and on all of its images under f and f−1.

Case II: If there are no points in K2 at which g is defined, use Lemma 2.5.3 to find an

isometric embedding g2 : K2 → M sending p2 to e2 whose image does not intersect EA except

at e2 and whose image does not intersect the image of g we have constructed so far. We extend

our definition of g as follows:

• on K2 we set g = g2;

• for all x ∈ K2, for all l ∈ Z, define g(f l(x)) = f l(g2(x)).

To check g is still an isometry, let m, l ∈ Z and y ∈ fm(K1), z ∈ f l(K2). Let y′ = f−m(y) ∈ K1

and z′ = f−l(z) ∈ K2. Then, fm(e1) is the closest point in EA to y and f l(e2) is the closest

point in EA to z. Therefore

d(y, z) = d(y, fm(e1)) + d(fm(e1), f l(e2)) + d(f l(e2), z) (3.7)

= d(y′, e1) + d(fm(e1), f l(e2)) + d(e2, z′) (3.8)

= d(g(y′), e1) + d(fm(e1), f l(e1)) + d(e2, g(z′)) (3.9)

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= d(fm(g(y′)), fm(e1)) + d(fm(e1), f l(e2)) + d(f l(e2), f l(g(z′))) (3.10)

= d(fm(g(y′)), f l(g(z′))) (3.11)

= d(g(fm(y′)), g(f l(z′))) = d(g(y), g(z)) (3.12)

Line (3.7) follows from Lemma 2.1.3, and line (3.8) comes from applying the isometries f−m

and f−l. Then line (3.9) is true because g is an isometric embedding on K2 and g(e2) = e2.

Then fm and f l are applied, resulting in (3.10), and Lemma 2.1.3 gives (3.11). The last line

comes from the definition of g. Therefore, the function g on(⋃

l∈Z fl(K1)

)⋃ (⋃l∈Z f

l(K2))

is

an isometric embedding which commutes with f and f−1. Also, the image of g intersects EA

only at the points f l(e1) or f l(e2) for l ∈ Z.

Step j: For j ≥ 3, when g has been defined on⋃j−1i=1

⋃l∈Z f

l(Ki), we extend the domain to

include⋃l∈Z f

l(Kj). If g has already been defined at any point of Kj , then g is already defined

on⋃l∈Z f

l(Kj) as described above in Case I. If not, we find an isometry gj defined on Kj and

extend g as described in Case II. At each step, the new points from⋃l∈Z f

l(Kj) in the domain of

g all have different closest points in EA as those from⋃j−1i=1

⋃l∈Z f

l(Ki) already in the domain.

Using Lemma 2.1.3, we are able to show our extension is an isometric embedding as in Step 2.

Therefore, after every step the partial function g is an isometric embedding and g commutes with

f . Also, the image of g intersects EA only at the points {f l(eh) | l ∈ {1, ..., j}, h ∈ {1, ..., n}}.

By part (4) of Lemma 3.4.4, for h ≥ n+ 1 there is some j ∈ {1, ..., n} and some m ∈ N such

that fm(Kj) = Kh, and fm(ej) = eh. Therefore, g is fully defined on⋃∞h=1Kh after n steps.

Moreover, g is an isometric embedding and g commutes with f . Also, the image of g intersects

EA only at the points

{f l(eh) | l ∈ Z, h ∈ {1, ..., n}}.

Let ci = g(bi). Then since f commutes with g for all x ∈⋃∞h=1Kh it follows that

fm(ci) = fm(g(bi)) = g(fm(bi)).

Therefore, for any m, l ∈ N and any i, j ∈ {1, ..., n},

d(fm(bi), f l(bj)) = d(g(fm(bi)), g(f l(bj))) = d(fm(ci), f l(cj)).

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In addition, if ej is the closest point in EA to fm(bi), then for any point x ∈ EA,

d(fm(bi), x) = d(fm(bi), ej) + d(ej , x) (3.13)

= d(g(fm(bi)), g(ej)) + d(ej , x) (3.14)

= d(fm(ci), ej) + d(ej , x) (3.15)

= d(fm(ci), x). (3.16)

Lines (3.13) and (3.16) are true by Lemma 2.1.2 because ej is the closest point in EA to fm(bi)

and fm(ci). Line (3.14) follows from (3.13) by an application of the isometry g and line (3.15)

comes from (3.14) by the definition of ci and the fact that g(ej) = ej . Therefore, we have found

c1, ..., cn ∈M that satisfy the partial type Dfb (x1, ..., xn), and we are done.

Note that once we defined the subtrees Kh, the key to building g was the fact that given

Kh for any h = 1, ..., n, there is an isometric embedding gh : Kh → M sending ph to eh whose

image does not intersect EA except at eh. Once we know that, the rest follows from Lemma

3.4.4 and from the fact that if l 6= k, then Kh and Kl have different closest points in EA.

3.4.6 Definition. Let the Ls-theory rbHRTr,s consist of the axioms of HRTr,s together with

the axioms of rbRT.

3.4.7 Theorem. The Ls-theory rbHRTr,s is the model companion of HRTr,s.

Proof. Clearly any model of rbHRTr,s is a model of HRTr,s. Any model M of HRTr,s may be

extended to a model N of rbHRTr,s in the following manner. Let M |= HRTr,s with underlying

R-tree and isometry (M,d, p, f). Let X be the R-tree we constructed in Lemma 2.2.9 where

every point has degree 3, and choose a basepoint q ∈ X. Use Lemma 2.2.8 to glue in a copy of

X at each point a ∈M , so that q gets identified with a. Let (N, d, p) be the resulting extension

of (M,d, p). Denote the copy of X at a by Xa. Then Xa ∩M = {a}, and Xb ∩Xa = ∅ for b 6= a

in M . Extend the isometry f from M to N by sending Xa to Xf(a) in the obvious way, using

the identity map on (X, q). Let N be the Ls-structure corresponding to (N, d, p, f). It follows

that N |= rbHRTr,s.

It remains to show that rbHRTr,s is model complete. Let M |= rbHRTr,s. Then, M is an

existentially closed model of HRTr,s by Theorem 3.4.5. Let N |= rbHRTr,s be an extension of M.

Clearly N is also a model of HRTr,s. Then, since M is an existentially closed model of HRTr,s

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we know for any inf-formula infs1,y1 ... infsn,yn ϕ(x1, ..., xk, y1, ..., yn) and any a1, ..., ak ∈M

infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)N = infs1,y1

... infsn,yn

ϕ(a1, ..., ak, y1, ..., yn)M.

Therefore, M is an existentially closed model of rbHRTr,s. Thus, any model M of rbHRTr,s is

an existentially closed model of rbHRTr,s, and therefore rbHRTr,s is model complete.

3.5 Properties of the model companions

For this section, fix s ∈ N>0 and r ∈ R such that r ≤ s. In this section, we show that rbHRTr,s

has quantifier elimination, characterize the completions of rbHRTr,s, and discuss stability.

3.5.1 Lemma. The Ls-theory rbHRTr,s has quantifier elimination.

Proof. By Theorem 3.4.7, Theorem 3.3.4 and Proposition 1.7.13.

3.5.2 Theorem. Let t ∈ R≥0 such that r ≤ t ≤ s, and let q ∈ R>0 such that r ≤ q ≤ t. Let the

Ls-theory rbHRTq,tr,s be the theory rbHRTr,s, together with the axioms

∣∣||fs|| − q∣∣ = 0

and

|d(p, fs(p))− t| = 0.

The theory rbHRTq,tr,s is a completion of rbHRTr,s.

Proof. Since rbHRTr,s has quantifier elimination by Lemma 3.5.1, rbHRTq,tr,s has quantifier

elimination by [1, Remark 13.4]. Let M |= rbHRTq,tr,s with underlying R-tree and isometry

(M,dM , pM, fM) and N |= rbHRTq,tr,s with underlying R-tree and isometry (N, dN , pN, fN). The

first new axiom guarantees that

||fM|| = q = ||fN||

because [pM, fM(pM)] ⊆ M (s) must intersect the axis of fM, so ||fs||M = ||fM||, and likewise

for fN. The second new axiom guarantees that

dM (pM, fM(pM)) = t = dN (pN, fN(pN)).

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Then, the distance dist(pM, AfM) of pM to the axis AfM of fM is determined, since

dist(pM, AfM) =12

(dM (pM, fM(pM))− ||fM||

).

Likewise, the distance dist(pN, AfN) of pN to the axis of fN is determined and must be equal

to dist(pM, AfM). Then, for all m ∈ Z

dM (pM, (fM)m(pM)) = 2 dist(pM, AfM) + |m| · ||fM||

= 2 dist(pN, AfN) + |m| · ||fN||

= dN (pN, (fN)m(pN)).

This implies that there is an isometry from the fM-orbit of pM in M to the fN-orbit of pN

in N taking pM to pN. Therefore, the substructure of M generated by pM embeds in any

model of rbHRTq,tr,s. Then, because rbHRTq,tr,s has quantifier elimination, we know rbHRTq,tr,s is

complete.

It is not hard to see that any completion of rbHRTr,s has an axiomatization of the form

rbHRTq,tr,s. The completions of rbHRTr,s are exactly the Ls-theories Th(M) for M |= rbHRTr,s.

Given M |= rbHRTr,s, choose q = ||fM|| and t = d(pM, fM(p)). Then, M |= rbHRTq,tr,s and by

Theorem 3.5.2 rbHRTq,tr,s = Th(M). For the rest of this section, fix t ∈ R≥0 such that r ≤ t ≤ s,

and q ∈ R>0 such that r ≤ q ≤ t.

Next, we turn to the question of the stability for each completion of rbHRTr,s. Given

M |= rbHRTq,tr,s and A ⊆ M , let TA be the closed subtree of M generated by the set {fm(a) |

a ∈ A ∪ {p}, m ∈ Z}.

3.5.3 Lemma. Assume M |= rbHRTr,s, and let b, c ∈ M and A ⊆ M . Then tpM(b/A) =

tpM(c/A) if and only if b and c have the same unique closest point e ∈ TA and d(b, e) = d(c, e).

Proof. Assume the situation described in the hypotheses. The forward direction is the same as in

Lemma 2.6.3. For the other direction, assume b and c have the same unique closest point e ∈ TA

and that d(b, e) = d(c, e). Since rbHRTq,tr,s has quantifier elimination, it suffices to show that the

quantifier-free types of c and b over A are the same. To show the quantifier-free types are the

same, by Lemma 3.4.2 it suffices to show d(a, fn(b)) = d(a, fn(c)) and d(b, fn(b)) = d(c, fn(c))

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for all n ∈ N, for all a ∈ TA. Let a ∈ TA and n ∈ N. Then,

d(a, fn(b)) = d(a, fn(e)) + d(fn(e), fn(b)) (3.17)

= d(a, fn(e)) + d(e, b) (3.18)

= d(a, fn(e)) + d(e, c) (3.19)

= d(a, fn(e)) + d(fn(e), fn(c)) (3.20)

= d(a, fn(c)) (3.21)

for any n ∈ N. Line (3.17) is true because fn(e) must be the closest point in TA to fn(b). The

next line follows because fn is an isometry, and line (3.19) follows because d(e, c) = d(e, b).

Then, reverse those two steps with c instead of b. Also,

d(b, fn(b)) = d(b, e) + d(e, fn(e)) + d(fn(e), fn(b)) (3.22)

= 2d(b, e) + d(e, fn(e)) (3.23)

= 2d(c, e) + d(e, fn(e)) (3.24)

= d(c, e) + d(e, fn(e)) + d(fn(e), fn(c)) (3.25)

= d(c, fn(c)) (3.26)

Line (3.22) is true by Lemma 2.1.3, and line (3.26) follows from (3.25) for the same reason.

Therefore, tpM(b/A) = tpM(c/A).

3.5.4 Theorem. The theory rbHRTq,tr,s is stable. Indeed when κ is an infinite cardinal, rbHRTq,tr,s

is κ-stable if and only if κ satisfies κω = κ.

Proof. The proof is the same as in Theorem 2.6.4, using Lemma 3.5.3 instead of Lemma 2.6.3.

Let κ be a cardinal so that κ = κω and κ > 2ω. Let U be a κ-universal domain for rbHRTq,tr,s.

3.5.5 Definition. Let A,B and C be small subsets of U . Let A := {fm(a) | a ∈ A∪ {p}, m ∈

Z}, and let B and C be defined analogously. Say A is H -independent from B over C, denoted

A |H^CB, if and only if A | bC B. That is, A |H^C

B if and only if for all a ∈ A we have

dist(a,EdBC) = dist(a,E bC).

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3.5.6 Theorem. The |H^ independence relation is the model theoretic independence relation for

rbHRTq,tr,s.

Proof. The general line of argument is close to that of proof of Theorem 2.6.7, but in places we

need lemmas from this chapter instead of those used in the proof of 2.6.7. As in that proof we

use [1, Theorem 14.12] and show |H^ has all seven of the properties required by that theorem.

Invariance under automorphisms is clear because any automorphism of a model of rbHRTq,tr,s

preserves | and satisfies σ(A) = σ(A). Symmetry and transitivity follow from the definition

of |H^ , and from the fact that | is symmetric and transitive. Finite character follows from

the fact that | has finite character, and from the fact that A =⋃a∈A∪{p}{fm(a) | m ∈ Z}.

Extension has much the same proof as in the proof of Theorem 2.6.7, but using Lemma 2.5.3

as in the proof of Theorem 3.4.5. For local character we use the same line of reasoning as in

Theorem 2.6.7. The argument for stationarity uses Lemma 3.5.3 instead of Lemma 2.6.3.

3.5.7 Theorem. The theory rbHRTq,tr,s is not ω-categorical.

Proof. Since Lp ⊆ Ls are both countable signatures, and rbRT is the reduct of rbHRTq,tr,s to Lp,

this follows from the fact that rbRT is not ω-categorical by [1, Proposition 12.13].

3.5.8 Theorem. Let κ > ω be a cardinal. The theory rbHRTq,tr,s is not κ-categorical.

Proof. Let κ > ω be a cardinal. We construct non-isomorphic models of rbHRTq,tr,s, each with

density character κ. First, use Lemma 2.2.8 to construct a model N of rbRT with underlying

space (N, d, pN ) such that there are κ-many branches at pN , the set of branch points in N is

of size ≤ κ, and at each branch point x 6= pN there are at most ω-many branches. This N will

have density character equal to κ.

Next, let M |= rbHRTq,tr,s be separable with underlying R-tree and isometry (M,d, p, f). The

degree of any point in a separable R-tree is at most ω by Lemma 2.6.10. Build an extension

W1 of M by using Lemma 2.2.8 to add a copy of N at each point in {fm(p) | m ∈ Z} ⊂ M ,

identifying the basepoint pN ∈ N with the point. For each point fm(p) for m ∈ Z, label the

new copy of N by Nm. The R-tree W1 has density character at least κ by Lemma 2.6.10. It

follows from the construction that W1 has density character at most κ, therefore the density of

W1 is κ. Extend the isometry f to all of W1 by defining f(Nm) = Nm+1 isometrically. Then,

the Ls-structure W1 corresponding to (W1, d, p, f) is a model of rbHRTq,tr,s.

To find the other non-isomorphic model, let W2 |= rbHRTq,tr,s be κ-saturated with underlying

R-tree and isometry (W2, d2, p2, g). Then by Lemma 2.3.7 there are at least κ-many branches

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at every point in W2. Choose a ∈ W2 outside the orbit of p2, and choose κ-many distinct rays

at a. Choose κ-many distinct rays at p2. Let A2 ⊂ W2 be the subspace consisting of a, p2 and

the chosen rays. Note that A2 has density equal to κ. Apply the Downward Lowenheim-Skolem

Theorem ([1, Proposition 7.3]) to W2 to get an elementary substructure W2 of density character

≤ κ which contains A2. Then, W2 |= rbHRTq,tr,s and has density character equal to κ by Lemma

2.6.10.

In W1 only the points {fm(p) | m ∈ Z} have degree κ, while the rest have at most degree ω.

In W2, the basepoint p2 has degree κ and there is at least one point a outside the orbit of p2

which has degree κ. Thus, W1 and W2 cannot be isomorphic.

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CHAPTER 4

ELLIPTIC ISOMETRIES OF R-TREES

For this chapter, we will need to refer to the background given in Section 3.1 and Section 3.2.

4.1 Theories of elliptic isometries on R-trees

For this section, fix s ∈ N>0. In this section, we axiomatize classes of R-trees with an elliptic

isometry and find model companions for those theories. Let Ls be the signature defined on page

51.

4.1.1 Definition. Let K0,s be the class of M |= Ts with underlying metric space and function

(M,d, p, f) so that (M,d, p) is a pointed R-tree, and f : M → M is an elliptic isometry such

that d(p, f(p)) ≤ s.

4.1.2 Definition. Let ERTs be the Ls-theory consisting of the following Ls-conditions:

1. the axioms of Ts and RT;

2. for n ∈ N the axiom

supn,x

supn,y|d(x, y)− d(fn(x), fn(y))| = 0;

3. for n ∈ N the axiom

supn+s,y

min{d(f(p), y)−· n, infn,x

d(f(y), x)} = 0;

4. for n ∈ N such that n ≥ s

2, the axiom

||fn|| = 0;

5. the axiom d(s)(p, f1(p))−· s = 0.

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4.1.3 Lemma. The class K0,s is exactly the class of Ls-structures that are models of ERTs.

Proof. This proof is the same as the proof of Lemma 3.3.3, except for the axioms in (4). Assume

M ∈ K0,s. Then µ1(p) is the closest fixed point to p. Since µ1(p) is the midpoint of [p, f(p)]

and d(p, f(p)) ≤ s, the axioms in (4) are true in M. Conversely, the axioms in (4) imply that

there must be a point in M fixed by f by Lemma 3.1.1.

For a model M of ERTs, we will refer to the underlying metric space and function (M,d, p, f)

of M as the underlying R-tree and (elliptic) isometry of M.

4.1.4 Theorem. The Ls-theory ERTs has the amalgamation property over substructures of

models.

Proof. Let (M0, d0, p0, f0) be the underlying R-tree and isometry of M0 and likewise for

(M1, d1, p1, f1) and (M2, d2, p2, f2). As in the proof of Lemma 2.2.6, we may assume that

M0 ⊆ M1, and M0 ⊆ M2 with M0 = M1 ∩M2 and the ϕi are inclusion maps for i = 1, 2.

In particular, this means M0 ⊆ M1 and the isometry f0 = f1 � M0. Likewise, M0 ⊆ M2 and

f0 = f2 � M0. Therefore, f1(x) = f0(x) = f2(x) for all x ∈M0.

Define (N, d, p) and f just as in the proof of Lemma 3.3.4. The isometry f of N is elliptic,

since f � M1 ⊆M = f1, and f1 had a fixed point. Therefore, the Ls-structure N corresponding

to (N, d, p, f) is a model of ERTs that has the properties we want.

4.2 Orbits under an elliptic isometry

In this section, we provide lemmas which characterize the possible orbits of points in an R-tree

with an elliptic isometry.

Given an R-tree M , an elliptic isometry f of M , and a ∈M , the following lemmas describe

the structure of the geodesic segment [µ1(a), a] between a and the closest fixed point to a.

(When we just say a point is fixed we mean fixed by f , that is, of f -order one.) That is, they

describe how [µ1(a), a] may be split up into subsegments of constant f -order, and how those

segments must be arranged. This in turn determines the structure of the f -orbit of a ∈M , that

is, it determines all distances d(fk(a), f l(a)) for k < l ∈ N.

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4.2.1 Lemma. Let f : M →M be an elliptic isometry of an R-tree M . Let a ∈M . Then

1. if m,n ∈ N>0 and m|n, then µn(a) is on [µm(a), a];

2. if a has finite f -order, b ∈ [µ1(a), a] and b has f -order m ∈ N>0, then for all x ∈ [µ1(a), b],

the f -order of x divides m and for all y ∈ [b, a], the f -order of y is a multiple of m;

3. if b ∈ [µ1(a), a] and b has f -order ∞, then every x ∈ [b, a] has f -order ∞;

4. if µn(a) has f -order n, then anything in (µn(a), a] has f -order ∞ or f -order kn for some

k ∈ N with k > 1.

Proof. Let f : M →M be an elliptic isometry of an R-tree M . Let a ∈M .

Proof of (1): Assume m,n ∈ N>0 and m|n. Then, fix(fm) ⊆ fix(fn), which implies µm(a) ∈

fix(fn). By Lemma 3.1.3, µn(a) is the closest point in fix(fn) to a. Therefore by Lemma 2.1.1,

µn(a) ∈ [µm(a), a].

Proof of (2): Assume a has finite f -order, b ∈ [µ1(a), a] and b has f -order m ∈ N>0. Both

µ1(a) and b are in fix(fm), which implies [µ1(a), b] ⊆ fix(fm). Therefore, for any x ∈ [µ1(a), b],

the f -order of x divides m. Next, let y ∈ [b, a]. Then, µ1(a) = µ1(y), because otherwise there

would be a point of f -order one closer to a than µ1(a). This implies b ∈ [µ1(a), y] = [µ1(y), y].

Then, if n is the f -order of y, we know b ∈ [µ1(y), y] ⊆ fix(fn), which implies n is a multiple of

m.

Proof of (3): Assume b ∈ [µ1(a), a] and b has f -order ∞. Towards contradiction, assume

there exists x ∈ [b, a] with finite f -order n. Then, [µ1(a), x] ⊆ fix(fn) which implies that

b ∈ fix(fn) and hence the f -order of b divides n. This is our contradiction.

Proof of (4): This follows from (2) and (3). Nothing in (µn(a), a] can be in fix(fn) because

µn(a) is the closest point in fix(fn) to a. Therefore, if the order of the point is kn, we know

k > 1.

4.2.2 Definition. Let M be an R-tree, f : M → M an elliptic isometry of M , and a ∈ M .

Define

O(a) := {t ∈ N | ∃x ∈ [µ1(a), a] such that x has f -order t}.

4.2.3 Lemma. Let M be an R-tree, f : M → M an elliptic isometry of M , and a ∈ M . Let

1 < t1 < t2 < ... be a list of the members of O(a) in ascending order. Then, µt(a) ∈ [µ1(a), a]

for all t ∈ O(a), and for j < l we have µtj (a) ∈ [µ1(a), µtl(a)] and tj divides tl. Moreover, every

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point in (µ1(a), µt1 ] has order t1, and for j = 1, 2, ... every point in (µtj (a), µtj+1(a)] has order

tj+1. Additionally,

1. if the f -order of a is finite and > 1, then

• O(a) is finite and the largest element tk of O(a) is the order of a (which implies

µtk(a) = a);

• [µ1(a), a] = [µ1(a), µt1(a)]∪ (µt1(a), µt2(a)]∪ ...∪ (µtk−1(a), µtk(a)], where this union

is disjoint;

2. if the order of a is infinite, then either O(a) is finite with largest element tk and

• [µ1(a), a] = [µ1(a), µt1(a)] ∪ (µt1(a), µt2(a)] ∪ ... ∪ (µtk−1(a), µtk(a)] ∪ (µtk(a), a]

where this union is disjoint and every point in (µtk(a), a] has infinite order;

or O(a) is infinite and

• {µtj (a)}∞j=1 converges to a point µ∞(a) so that every point in [µ∞(a), a] has infinite

order;

• [µ1(a), a] = [µ1(a), µt1 ] ∪( ∞⋃j=1

(µtj (a), µtj+1(a)])∪ [µ∞(a), a] where this union is dis-

joint.

Proof. Let M be an R-tree, f : M → M an elliptic isometry of M , and a ∈ M . Let 1 < t1 <

t2 < ... be a list of the members of O(a) in ascending order. By Lemma 4.2.1 part (1) we

have µt(a) ∈ [µ1(a), a] for all t ∈ O(a). Then, it follows from part (4) of Lemma 4.2.1 that

µ1(a) = µ1(µt(a)) for each t ∈ O(a). If t ∈ O(a), then µt(a) actually has order t. Then, by

the fact that 1 < t1 < t2 < ... and Lemma 4.2.1 part (2) it follows that for j < l we have

µtj (a) ∈ [µ1(µtl(a)), µtl(a)] = [µ1(a), µtl(a)] and tj divides tl. The t ∈ O(a) represent all of the

possible finite orders of points on [µ1(a), a], therefore, every point in (µ1(a), µt1 ] has order t1,

and for all j, every point in (µtj (a), µtj+1(a)] has order tj+1.

Proof of (1): Assume the order of a is finite and > 1. By Lemma 4.2.1 part (2) any t ∈ O(a)

must divide the order of a. Then clearly O(a) is finite and its largest element tk is the order

of a. That [µ1(a), a] = [µ1(a), µt1(a)] ∪ (µt1(a), µt2(a)] ∪ ... ∪ (µtk−1(a), µtk(a)] follows from the

facts that µtj (a) ∈ [µ1(a), µtl(a)] for j < l, and µtk(a) = a. The fact that this is a disjoint union

is clear from an examination of the orders of the points in each piece.

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Proof of (2): Assume the order of a is infinite.

Case I: Assume the set O(a) is finite. Let tk be the largest element of O(a). Then, every point

in (µtk(a), a] must have infinite order. That [µ1(a), µtk(a)] = [µ1(a), µt1(a)] ∪ (µt1(a), µt2(a)] ∪

... ∪ (µtk−1(a), µtk(a)] is clear by part (1). Then, it follows that [µ1(a), a] = [µ1(a), µt1(a)] ∪

(µt1(a), µt2(a)]∪ ...∪ (µtk(a), a] by the fact that µtj (a) ∈ [µ1(a), µtl(a)] for j < l. The fact that

this is a disjoint union is clear from an examination of the orders of the points in each piece.

Case II: Assume O(a) is infinite. For j < l, we know d(µ1(a), µtj (a)) < d(µ1(a), µtl(a))

because µtj (a) ∈ [µ1(a), µtl(a)]. So, {d(µ1(a), µtj (a))}∞j=1 is a strictly increasing sequence of

positive real numbers, bounded above by d(µ1(a), a). Therefore this sequence converges to some

real number R ≤ d(µ1(a), a). Let µ∞(a) be the point on [µ1(a), a] with distance R from µ1(a).

Then, {µtj (a)}j∈N must converge to µ∞(a) ∈ [µ1(a), a] (since [µ1(a), a] is an isometric copy of

a real interval).

Claim: Every point in [µ∞(a), a] must have infinite order.

Proof of claim: Towards contradiction assume some point in [µ∞(a), a] has finite order. Then,

µ∞(a) must have finite order, say it has order m. Then, m ∈ O(a). But, O(a) is an infinite

collection of positive integers, and thus there must exist tj ∈ O(a) such that tj > m. Then,

d(µ1(a), µtj (a)) > d(µ1(a), µ∞(a)) = R. But, by how we found R, we know d(µ1(a), µtj (a)) < R.

This is a contradiction. Thus, every point in [µ∞(a), a] has infinite order. Note that µ∞(a) is

the closest point of infinite order to µ1(a) on [µ1(a), a].

Since µtj (a) ∈ [µ1(a), a] for all j = 1, ..., k and µ∞(a) ∈ [µ1(a), a] it is clear that

[µ1(a), µt1(a)] ∪( ∞⋃j=1

(µtj (a), µtj+1(a)])∪ [µ∞(a), a] ⊆ [µ1(a), a].

If z ∈ [µ1(a), a], then if z has finite order it will be in

[µ1(a), µt1(a)] ∪( ∞⋃j=1

(µtj (a), µtj+1(a)])

If z has infinite order, it will be in [µ∞(a), a], since µ∞(a) is the closest point to µ1(a) on

[µ1(a), a] of infinite order. Thus,

[µ1(a), a] = [µ1(a), µt1 ] ∪( ∞⋃j=1

(µtj (a), µtj+1(a)])∪ [µ∞(a), a].

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4.2.4 Lemma. Let M be an R-tree, f : M → M an elliptic isometry of M , and a, b ∈ M . If

O(a) = O(b) and d(a, µt(a)) = d(b, µt(b)) for all t ∈ O(a), then d(a, µn(a)) = d(b, µn(b)) for all

n ∈ N.

Proof. Let M be an R-tree, f : M → M an elliptic isometry of M , and a, b ∈ M . Assume

O(a) = O(b) and d(a, µt(a)) = d(b, µt(b)) for all t ∈ O(a). Let n ∈ N. By Lemma 4.2.1 part

(1) we know µn(a) ∈ [µ1(a), a]. Let tk ∈ O(a) be the largest number in O(a) that divides n.

Then, µtk(a) = µn(a), because µtk(a) will be the point in fix(fn) that is closest to a. Since

O(a) = O(b), we conclude µtk(b) = µn(b). Now, d(a, µt(a)) = d(b, µt(b)) for all t ∈ O(a) implies

d(a, µn(a)) = d(b, µn(b)).

4.2.5 Lemma. Let M be an R-tree, f : M → M an elliptic isometry of M , and a, b ∈ M . If

d(a, µn(a)) = d(b, µn(b)) for all n ∈ N, then for all k, l ∈ Z

d(fk(a), f l(a)) = d(fk(b), f l(b)).

Proof. Let M be an R-tree, f : M → M an elliptic isometry of M , and a, b ∈ M . Assume

d(a, µn(a)) = d(b, µn(b)) for all n ∈ N. Let k, l ∈ Z with k ≤ l. Then, l − k ∈ N and

d(fk(a), f l(a)) = d(a, f l−k(a)) = 2d(a, µl−k(a))

because f is an isometry and by the definition of µl−k. By our assumption, 2d(a, µl−k(a)) =

2d(b, µl−k(b)). Therefore, d(fk(a), f l(a)) = 2d(b, µl−k(b)) = d(b, f l−k(b)) = d(fk(b), f l(b)).

4.2.6 Remark. In the elliptic case, we need an analog of Lemma 3.4.3. Here, (M,d, p) is a

pointed R-tree and f an elliptic isometry of M , a ∈M , and A = {fm(a) | m ∈ Z}. However, in

the elliptic case the subtree EA =⋃m∈Z f

m([µ1(a), a]) is not necessarily closed.

4.2.7 Lemma. Assume (M,d, p) is a pointed R-tree and f an elliptic isometry of M . Let

a1, ..., ak ∈M for k ≥ 1, and assume p = a1. Let A :=⋃m∈Z{fm(ai) | i = 1, ..., k}. Then,

1. EA is closed under f and f−1;

2. for all x ∈ EA, the degree of x in EA is at most 2ω.

Proof. Assume (M,d, p) is a pointed R-tree and f an elliptic isometry of M . Let a1, ..., ak ∈M

for k ≥ 1, and assume p = a1.

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Proof of (1): It is clear that EA must be closed under f and f−1. If y ∈ EA \EA, then y is a

limit point of a sequence (xn)∞n=1 of points from EA. Then, because f is an isometry, f(y) must

be the limit of (f(xn))∞n=1. These points are all in EA since EA is closed under f , therefore

f(y) ∈ EA. The proof for f−1 is the same.

Proof of (2): This follows from the fact that the cardinality of EA is at most 2ω.

4.3 Model companions: elliptic case

For this section, fix s ∈ N. In this section, we characterize the model companion of the theory

ERTs.

Next, we define an Ls-theory rbERTs extending ERTs which will turn out to be the model

companion of ERTs. Informally, we want to add the following conditions when (M,d, p, f) is

the underlying R-tree and elliptic isometry of a model M of ERTs, and r ∈ R>0, h, l,m ∈ N>0:

If x ∈ M has order m, then there exist y1, ..., yl ∈ M such that for all i, j = 1, ..., l we have

d(x, yi) = r, d(yi, yj) = 2r (if i 6= j), and every element of (x, yi] has order hm.

In order to express suitable forms of these conditions using Ls-conditions, we need to do

some preliminary work.

To begin, we introduce an extension by definitions of ERTs in which the midpoint function

µ(x1, x2) is available. Using it we also have at hand the functions µm(x) = µ(x, fm(x)), in

terms of which we can express the orders of elements. Recall from Lemma 2.2.4 that in models

of RT the distance d(y, µ(x1, x2)) is given by the interpretation of certain Lp-formulas. Let

M |= RT and let (M,d, p) be the underlying metric space of M. Specifically, by Lemma 2.2.4,

for x1, x2, y ∈M (n) the distance d(y, µ(x1, x2)) is given by the interpretation of the formula

ψn(x1, x2, y) = max{d(x1, y)−· d(x1, x2)2

, d(x2, y)−· d(x1, x2)2

}

where all the variables are over M (n).

With this background, we introduce an extension by definitions ERTs(µ) of ERTs in which

the midpoint function is available. We denote the extended continuous signature by Ls(µ). In

addition to Ls, it contains for each n ∈ N a binary function symbol µ(n) of arity (n, n;n) whose

interpretation will be the restriction of the midpoint function to the sort M (n). The assigned

modulus of uniform continuity for µ(n) is ∆(ε) =ε

2.

The theory ERTs(µ) consists of ERTs together with the following Ls(µ)-conditions that

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serve to define µ(n): for each n ∈ N

supn,x1

supn,x2

supn,y

∣∣∣d(µ(n)(x1, x2), y)− ψn(x1, x2, y)∣∣∣ = 0.

Using Lemma 2.2.4, it is clear that each model M |= ERTs has a unique expansion to a

model of ERTs(µ). We denote this expansion of M by M(µ). If M |= ERTs and (M,d, p, f) is

the underlying R-tree and elliptic isometry of M, then for each n ∈ N, the interpretation of µ(n)

in M(µ) is the restriction of the midpoint function of M to the sort M (n) = Bn(p).

For each m ∈ N>0 and n ∈ N we define

µ(n)m (x) = µ(n+ms) (In,n+ms(x), fm(x))

where x is of sort n. This is an Ls(µ)-term of arity (n;n+ms). For M and (M,d, p, f) as above,

the interpretation of µ(n)m in M(µ) is the restriction of the function µm (giving the nearest fixed

point of fm) to M (n). As we have done in other similar situations, we will often write µ instead

of µ(n) and µm instead of µ(n)m in Ls(µ)-formulas, leaving it up to the reader to assign the needed

sort indices.

Our next result shows explicitly how certain Ls(µ)-conditions can be simplified by eliminating

occurrences of the function symbols µm. By applying this result several times, we can explicitly

produce equivalent Ls-conditions for the axioms we add to ERTs(µ).

4.3.1 Lemma. Assume M |= ERTs with underlying R-tree and isometry (M,d, p, f). Let

ϕ(y, z1, ..., zk) = 0 be an Ls(µ)-condition where y is of sort n. Let a ∈M (n), and b1, ..., bk ∈M

be from appropriate sorts. Let m ∈ N. Then

(inf

n+ms,ymax{ψn+ms(a, fm(a), y), ϕ(y, b1, ..., bk)}

)M(µ)

= 0

if and only if

ϕM(µ)(µm(a), b1, ..., bk) = 0.

Proof. Assume the hypotheses of the lemma. For the left to right direction, assume

(inf

n+ms,ymax{ψn+ms(a, fm(a), y), ϕ(y, b1, ..., b)}

)M(µ)

= 0.

Then, for all ε > 0 there exists c ∈M (n) so that ϕM(µ)(c, b1, ..., bk) ≤ ε and ψM(µ)n+ms(a, f

m(a), c) ≤

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ε, hence, d(c, µm(a)) ≤ ε. Then by the uniform continuity of the function ϕM(µ), we conclude

ϕM(µ)(µm(a), b1, ..., bk) = 0. In the other direction, the infn,y

is clearly witnessed by µm(a).

In the Ls(µ)-conditions we will add to ERTs(µ), we need to use the sup quantifier over the

set fix(fm) ∩M (n). The next results show how to eliminate such quantifiers using formulas of

Ls(µ).

4.3.2 Lemma. For any m ∈ N, the sets fix(fm) are uniformly definable in models of ERTs.

Proof. Let M |= ERTs with underlying R-tree and isometry (M,d, p, f). In M , the distance

dist(x,fix(fm)) is equal to the distance from x to µm(x) by Lemma 3.1.3. By definition, µm(x)

is the midpoint of [x, fm(x)]. Therefore, for x ∈ M (n) the predicate dist(x,fix(fm) ∩M (n)) =

d(x, µm(x)) is given by interpretation of the formula 12d(x, fm(x)) in M. The same formula

works for any model M, therefore we have uniform definability.

Now, consider any Ls(µ)-formula ψ(x). Using Lemma 4.3.2 and the proof of [1, Proposition

9.17] we can show that

supx∈fix(fm)∩M(n)

ψ(x)

is the interpretation of an Ls(µ)-formula in all models of ERTs(µ). Specifically, the proof of [1,

Proposition 9.17] uses the fact that given a formula ψ(x) with range interval Iψ = [0, b] and

n ∈ N>0, there exists an increasing, continuous function α : [0, 2n] → [0, b] with α(0) = 0 such

that |ψ(y) − ψ(z)| ≤ α(d(y, z)) holds in any M(µ) |= ERTs(µ) and any y, z ∈ M (n). Then,

it is shown that in any M(µ) |= ERTs(µ), the predicate infx∈fix(fm)∩M(n)

ψ(x) is equal to the

interpretation of the Ls(µ)-formula

infn,x

(ψ(x) + α

(12d(x, fm(x))

).

Lastly, the sup quantification is handled by noting

supx∈fix(fm)∩M(n)

ψ(x) = b−·(

infx∈fix(fm)∩M(n)

(b−· ψ(x)

)).

Finally, we are ready to extend ERTs to the L-theory rbERTs, which turns out to be its

model companion.

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4.3.3 Definition. For r ∈ Q, h, l,m, n ∈ N>0, define ϕr,h,l,m,n(x) to be the Ls(µ)-formula

infn+r,y1,...,yl

max{|d(x, yi)− r|, |2r − d(yi, yj)|, d(µkm(yi), x), d(µhm(yi), yi)}

where the maximum is over all 1 ≤ k ≤ h− 1 and all 1 ≤ i 6= j ≤ l, and x is a variable of sort

M (n). Further, define ϕr,h,l,m,n to be

supx∈fix(fm)∩M(n)

min({d(x, fk(x)) : k < m} ∪ {ϕr,h,l,m,n(x)}

)

which we regard as an Ls(µ)-formula as discussed above.

4.3.4 Definition. Let rbERTs(µ) be the Ls(µ)-theory whose axioms are the axioms of ERTs(µ)

and rbRT together with all the conditions ϕr,h,l,m,n = 0 where r ∈ Q; h, l,m, n ∈ N>0. We also

let rbERTs be the restriction of rbERTs(µ) to Ls.

Note that the preceding discussion allows us to write explicit Ls-conditions that axiomatize

rbERTs, and yields the following description of its models:

4.3.5 Proposition. 1. Every model M of rbERTs has a unique expansion M(µ) to a model

of rbERTs(µ), and every model of rbERTs(µ) is of this form.

2. Let M be an ω1-saturated model of ERTs, with underlying R-tree and elliptic isometry

(M,d, p, f). Then M |= rbERTs if and only if M is a richly branching R-tree and for all

x ∈ M of order m ≥ 1, all r ∈ R>0, and all h, l ∈ N, there exist y1, ..., yl ∈ M such that

for all i, j = 1, ..., l we have d(x, yi) = r, d(yi, yj) = 2r (if i 6= j) and every element of

(x, yi] has order hm.

For more explanation of statement (2) in the proposition above, see the proof of 4.3.6 below.

Next, we develop some technical machinery needed for the proof that rbERTs is the model

companion of ERTs.

4.3.6 Lemma. Let κ ≥ ω1 be a cardinal. Assume M is a κ-saturated model of rbERTs with

underlying R-tree and isometry (M,d, p, f).

1. For any a ∈ M with order m ∈ N>0, any r ∈ R>0 and any h ∈ N there are at least κ

many points b on distinct branches at a such that d(a, b) = r and all the points in (a, b]

have order hm.

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2. For any a ∈ M with order m ∈ N>0, and any r ∈ R>0 there are at least κ-many points b

on distinct branches at a whose orbits are disjoint such that d(a, b) = r, and all the points

in (a, b] have infinite order.

3. For any a ∈ M with order ∞, and any r ∈ R>0 there are at least κ many points b on

distinct branches at a whose orbits are disjoint such that d(a, b) = r, and all the points in

[a, b] have infinite order.

Proof. Assume the hypotheses of the lemma. Let a ∈M with order m ∈ N>0. Take r ∈ R>0 and

h, l ∈ N. Let n be large enough so that a ∈ Bn(p) = M (n). Then because M |= ϕr,h,l,m,n = 0,

and a ∈ fix(fm) ∩M (n) we conclude

min({dM(a, fk(a)) : k < m} ∪ {ϕM

r,h,l,m,n(a)})

= 0.

Since the order of a is actually m, we know that min{dM(a, fk(a)) : k < m} > 0. Therefore,

ϕMr,h,l,m,n(a) must equal 0.

Because M is κ-saturated for κ ≥ ω1, we know that the inf in ϕMr,h,l,m,n(a) is satisfied exactly

by Lemma 1.7.3. Therefore, there exist b1, ..., bl ∈ M (n+r) such that for all i, j ∈ {1, ..., l} and

all k ∈ {1, ..., h− 1} :

|d(a, bi)− r| = 0 (4.1)

|2r − d(bi, bj)| = 0 (4.2)

d(µkm(bi), a) = 0 (4.3)

d(µhm(bi), bi) = 0. (4.4)

The statements in (4.1) give that d(a, bi) = r for each 1 ≤ i ≤ l. The statements in (4.1) and

(4.2) imply by Lemma 2.1.2 that for i 6= j the points bi and bj are on different branches a. For

each i = 1, ..., l, statement (4.4) gives that bi ∈ fix(fhm), while the statements in (4.3) say that

for all 1 ≤ k ≤ h− 1 the closest point to bi in fix(fkm) is a. This implies that the order of every

point in (a, bi] is greater than km for all 1 ≤ k ≤ h − 1. Therefore the order of bi is exactly

hm for i = 1, ..., l, and the order of every point in (a, bi] must be hm. Note that this gives us

l-many such points bi on distinct branches out of a, but it does not guarantee that their orbits

do not intersect. To get l′-many such points bi so that their orbits do not intersect, we can use

this argument with l ≥ l′mh.

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Proof of (1): Let h ∈ N. Towards contradiction, assume there are < κ many points b on

distinct branches at a such that d(a, b) = r and all the points in (a, b] have order hm. List

them as {bi}i∈I where card(I) < κ. Let A be the set {a} ∪ {bi}i∈I . Note that this set also has

cardinality < κ. The set of Ls(A)-conditions

{|d(a, y)− r| = 0; |d(bi, y)− 2r| = 0; d(µkm(y), a) = 0; d(µhm(y), y) = 0 | i ∈ I, 1 ≤ k ≤ h− 1}

is finitely satisfiable in M. Therefore, by κ-saturation there is a point b that satisfies all these

conditions at once. Since d(µkm(b), a) = 0 for all 1 ≤ k ≤ h− 1 and d(µhm(b), b) = 0, all points

in (a, b] must have order hm. Since |d(a, b)− r| = 0 and |d(bi, b)− 2r| = 0, the point b must be

on a different branch out of b than all the bi. This is a contradiction.

Proof of (2): To see that there is at least one point b on a branch at a such that d(a, b) = r

and all the points in (a, b] have order ∞, note that the set of Ls(a) conditions

{|d(a, y)− r| = 0; d(µkm(y), a) = 0; | k ∈ N>0}

is finitely satisfiable in M. So, by κ-saturation there is some b satisfying all of the Ls-conditions

in the set at once. This b is on a branch at a such that d(a, b) = r and all the points in (a, b]

have order ∞.

Now, towards contradiction, assume there are < κ many points b on distinct branches at a

with disjoint orbits, such that d(a, b) = r, and all the points in (a, b] have order ∞. List them

as {bi}i∈I for card(I) < κ. Let A be the set {a} ∪⋃l∈N{f l(bi)}i∈I . Note that A has cardinality

< κ The set of Ls(A)-conditions

{|d(a, y)− r| = 0; |d(f l(bi), y)− 2r| = 0; d(µkm(y), a) = 0 | i ∈ I, k, l ∈ N>0}

is finitely satisfiable in M. Therefore, by κ-saturation there is a point b that satisfies all these

conditions at once. This point b must be on a different branch at a from all the bi, and it must

be such that µkm(b) = a for all k ∈ N>0. Therefore b has infinite order, which is a contradiction.

Proof of (3): Let a ∈M have order ∞. Because rbRT ⊆ rbERTs, we may use Lemma 2.3.7

to conclude that there are at least κ-many distinct branches at a, and each of these branches has

infinite extent. So, we have κ-many branches at a different from the branch that contains µ1(a).

Then, for any r ∈ R>0, there exist κ-many points b on distinct branches at a with d(a, b) = r

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and a ∈ [µ1(a), b]. It follows that µ1(a) = µ1(b), and by Lemma 4.2.1 b must have infinite order.

Moreover, these b must all have disjoint orbits. This is because if there are distinct b and b′

meeting the criteria above so that fm(b) = b′ for some m ∈ Z, then since a is the midpoint of

[b, b′], a must be fixed by fm by Lemma 3.1.3. But, a had order ∞, so this is impossible.

In the next three lemmas, we build up the tools for showing that every highly saturated

model of rbERTs is existentially closed as a model of rbERTs. (From this fact it follows that

rbERTs is the model companion of ERTs; see Theorem 4.3.13 below).

4.3.7 Lemma. Assume N |= ERTs, and let (N, d, p, f) be the underlying R-tree and elliptic

isometry of N. Let b1, ..., bn ∈ N , and T a non-empty closed subtree of N such that T is closed

under f and f−1. Let B be the set

⋃m∈Z{fm(bi) | i = 1, ..., n}.

Define an equivalence relation on B by: x ∼ y if and only if x and y have the same unique

closest point in T. Let {Gh}∞h=1 be the equivalence classes of ∼. Without loss of generality,

assume b1, ..., bn ∈⋃nh=1Gh. For each h, let eh be the unique closest point in T common to all

members of Gh. Then,

1. the set of equivalence classes of ∼ is closed under f and f−1;

2. for each h ≥ n+ 1, there is some j ∈ {1, ..., n} and some m ∈ Z such that fm(Gj) = Gh

and fm(ej) = eh.

Proof. The proofs are the same as for Lemma 3.4.4.

Let N |= rbERTs be an extension of M |= rbERTs where M is highly saturated. Let T be a

closed subtree of M. In the next lemma, given a point b ∈ N we see how to embed the segment

from b to T into M so that the structure of the orbit of b is preserved.

4.3.8 Lemma. Let M |= rbERTs and N |= ERTs an extension of M with (N, d, p, f) the

underlying R-tree and elliptic isometry of N and (M,d, p) the underlying R-tree of M. Assume

M is κ-saturated for κ ≥ ω1. Let T be a non-empty closed subtree of M that is closed under f

and f−1. Let b ∈ N \M and e ∈ T be the closest point in T to b. Take α < κ and {βi | i ∈ α}

a family of branches at e in M . Then, there is an isometric embedding g of [e, b] into M so

that g(e) = e and d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N, and the image of g does not

intersect⋃{βi | i ∈ α} except at e.

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Proof. Let the situation be as described in the statement of the lemma. Recall that µ1(e) is

the midpoint of [e, f(e)]. The R-tree T is closed under f and convex in N , so e ∈ T implies

µ1(e) ∈ T. Therefore, [µ1(e), e] ⊆ T, and so by Lemma 4.2.1 part (1), µm(e) ∈ T for all m ∈ N.

Assume e has infinite order. Then b must also have infinite order. To embed [e, b] in M , apply

Lemma 4.3.6 to find a point y on a branch out of e in M that does not intersect⋃{βi | i ∈ α}

except at e, such that d(e, y) = d(e, b) and every point in [e, y] has infinite order. We may

avoid these α-many branches because there are at least κ-many branches out of e with the

properties we need. Then, define g(e) = e and g(b) = y and extend g to [e, b] isometrically.

Every point on [e, b] has infinite order. It follows that µm(e) = µm(b) for all m ∈ N>0. Every

point in [g(e), g(b)] = [e, g(b)] has infinite order, so we know µm(g(b)) = µm(g(e)) for all m ∈

N>0. We conclude µm(g(b)) = µm(g(e)) = µm(e) = µm(b) for all m ∈ N>0. Therefore,

d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N>0.

Next assume both e and b have order one. Then all of [e, b] has order one. Use Lemma 4.3.6

to find y ∈ M on a branch out of e that does not intersect⋃{βi | i ∈ α} except at e, with

d(e, y) = d(e, b) and so that every point in [e, y] has order one. Define g(e) = e and g(b) = y and

extend to [e, b] isometrically. Note that since b and g(b) both have order one, µm(b) = b and

µm(g(b)) = g(b) for all m ∈ N>0. Therefore, d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N>0.

Now, assume m ∈ N>0 is the order of e and b has finite order tk > 1. Then by Lemma 4.2.3

we know

[µ1(b), b] = [µ1(b), µt1(b)] ∪ (µt1(b), µt2(b)] ∪ ... ∪ (µtk−1(b), µtk(b)]

where 1 < t1 < t2 < ... < tk are the elements of O(b), every point in (µ1(b), µt1 ] has order t1,

and every point in (µtj (b), µtj+1(b)] has order tj+1 for j = 1, ..., k − 1.

Case I: Assume there is t ∈ O(b) so that m = t and e = µt(b). If t = 1, begin by defining

g on [µ1(b), µt1(b)] = [e, µt1(b)]. Use Lemma 4.3.6 to find y1 ∈M on a branch out of e = µ1(b)

that does not intersect⋃{βi | i ∈ α} except at e, with d(e, y1) = d(e, µt1(b)) and so that every

point in (e, y1] is has order t1. Define g(e) = e and g(µ1(b)) = y1 and extend g to all of [e, µt1(b)]

isometrically. Next, we extend the domain of g to include (µt1(b), µt2(b)]. Use Lemma 4.3.6 to

find y2 ∈ M on a branch out of y1 = g(µt1(b)) that does not intersect the previously defined

image of g, with d(y1, y2) = d(µt1(b), µt2(b)) and so that every point in (y1, y2] is has order t2.

Define g(µt2(b)) = y2 and extend g to all of [e, µt1(b)] isometrically. Repeat this process until

you have embedded all of [e, b]. Note that, µt(b) ∈ [e, b] for all t ∈ O(b), and by our construction,

d(e, µt(b)) = d(g(e), µt(g(b))) for all t ∈ O(b). This implies d(b, µt(b)) = d(g(b), µt(g(b))) for all

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t ∈ O(b). So by Lemma 4.2.4 we know that d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N>0.

If t > 1, then t = tj for some j = 1, ..., k. Begin by defining g on [e, µtj+1(b)], which is equal

to [µtj (b), µtj+1(b)]. Use Lemma 4.3.6 to find yj+1 ∈M on a branch out of e = µtj (b) that does

not intersect⋃{βi | i ∈ α} except at e, with d(e, yj+1) = d(e, µtj+1(b)) and so that every point

in (e, yj+1] is has order tj+1. Define g(e) = e and g(µtj+1(b)) = yj+1 and extend to [e, µtj+1(b)]

isometrically. Then continue as above to define g on (µtj+2(b), µtj+3(b)] ∪ ... ∪ (µtk−1(b), µtk(b)],

which is the rest of [e, b]. Note that µt(b) ∈ [e, b] for all t ∈ O(b) such that t > tj , and by

our construction, d(e, µt(b)) = d(g(e), µt(g(b))) for all t ∈ O(b) such that t > tj . This implies

d(b, µt(b)) = d(g(b), µt(g(b))) for all t ∈ O(b) such that t > tj . For t ∈ O(a) such that t ≤ tj ,

we know that µt(b) = µt(e) ∈ M , and e ∈ [µt(b), b] and d(e, g(b)) = d(e, b). It follows that

d(b, µt(b)) = d(g(b), µt(g(b))) for t ∈ O(a) such that t ≤ tj . So by Lemma 4.2.4 we know that

d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N>0.

Case II: Assume m = 1, but e 6= µ1(b). Begin by defining g on [e, µ1(b)]. Use Lemma

4.3.6 to find y1 ∈ M on a branch out of e that does not intersect⋃{βi | i ∈ α} except

at e, with d(e, y1) = d(e, µ1(b)) and so that every point in (e, y1] is has order one. Define

g(e) = e and g(µ1(b)) = y1 and extend to [e, µ1(b)] isometrically. Then, proceed as in Case I

to embed the rest of [e, b]. Note that µt(b) ∈ [e, b] for all t ∈ O(a), and by our construction,

d(e, µt(b)) = d(g(e), µt(g(b))) for all t ∈ O(a). It follows that d(b, µt(b)) = d(g(b), µt(g(b))) for

t ∈ O(a) such that t ≤ tj . So by Lemma 4.2.4 we know that d(g(b), µm(g(b))) = d(b, µm(b)) for

all m ∈ N>0.

Case III: Assume m > 1 but e 6= µm(b). Say m = tj . Begin by defining g on [e, µtj (b)]. Use

Lemma 4.3.6 to find yj ∈M on a branch out of e that does not intersect⋃{βi | i ∈ α} except at

e, with d(e, yj) = d(e, µtj (b)) and so that every point in (e, yj ] is has order tj . Define g(e) = e

and g(µtj (b)) = yj and extend to [e, µtj (b)] isometrically. Then, proceed as in Case I to embed

the rest of [e, b]. In this case, we also may conclude that d(g(b), µm(g(b))) = d(b, µm(b)) for all

m ∈ N>0.

Now, assume m ∈ N>0 is the order of e and that b has order ∞. There are two more cases

under this assumption.

Case IV: Assume O(a) is finite with largest element tk. Then, by Lemma 4.2.3, [µ1(b), b] =

[µ1(b), µt1(b)] ∪ (µt1(b), µt2(b)] ∪ ... ∪ (µtk(b), b] where every point in (µ1(b), µt1 ] has order t1,

every point in (µtj (b), µtj+1(b)] has order tj+1 for j = 1, ..., k − 1, and every point in (µtk(b), b]

has infinite order. Because µtk(b) has finite order, and µm(µtk(b)) = µm(b) for all m ∈ N we

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may use the cases above to define g with the desired properties on [e, µtk(b)]. Then, by Lemma

4.3.6 part (3) we can find y∞ such that d(yk, y∞) = d(µtk(b), b) and every point in (yk, y∞] has

infinite order, on a branch in M out of g(µtk(b)) = yk that does not intersect⋃{βi | i ∈ α}\{e}

and does not intersect the image of g built previously except at yk. Define g(b) = y∞ and extend

g to [µtk(b), b] isometrically. This defines g on [e, b] such that µm(g(b)) = g(µm(b))

Case V: Assume the order of b is infinite and O(a) is infinite. Then by Lemma 4.2.3 we have

{tj}∞j=1 with 1 < tj for all j so that every point in (µ1(b), µt1 ] has order t1, and for all j, every

point in (µtj (b), µtj+1(b)] has order tj+1. Also, {µtj (b)}∞j=1 converges to a point µ∞(b) so that

every point in [µ∞(b), b] has infinite order, and

[µ1(b), b] = [µ1(b), µt1(b)] ∪( ∞⋃j=1

(µtj (b), µtj+1(b)])∪ [µ∞(b), b].

For each j ∈ N, proceed as in Case II to find isometric embeddings gj : [e, µtj (b)]→M such that

gj ⊆ gj+1 and so that d(e, gj(µtj (b)) = d(e, µtj (b)). Let g be the union of those embeddings.

Then, g is an isometric embedding of [e, µ∞(b)) into M .

The distances d(e, g(µtj (b))) are all ≤ d(e, b) and hence form a bounded, increasing sequence

of real numbers. Since

d(g(µtj (b)), g(µtl(b))

)=∣∣d(e, g(µtj (b))

)− d(e, g(µtl(b))

)∣∣it follows that {g(µtj (b))}j∈N form a Cauchy sequence in M . Since M is complete, {g(µtj (b))}j∈N

converges to z ∈ M . Define g(µ∞(b)) = z. We now have an isometric embedding of [e, µ∞(b)]

into M , and the last step is to extend to all of [e, b]. Use Lemma 4.3.6 to find y ∈ M with

d(µ∞(b), b) = d(z, y) so that every point in [z, y] has infinite order. Find this y on a branch

out of z = g(µ∞(b)) that does not intersect⋃{βi | i ∈ α} \ {e} and does not intersect the

previously defined image of g except at z. Set g(b) = y and extend to [z, y] isometrically. We

may show that in this last case d(g(b), µm(g(b))) = d(b, µm(b)) is true for all m ∈ N, since the

corresponding statement was true for each gj .

In this next lemma, we extend our embedding from Lemma 4.3.8 to the subtree defined by

the orbit of the segment.

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4.3.9 Lemma. Let M |= rbERTs and N |= ERTs an extension of M, with (N, d, p, f) the

underlying R-tree and elliptic isometry of N, and (M,d, p) the underlying R-tree of M . Assume

M is κ-saturated for some cardinal κ ≥ ω1. Let T be a non-empty closed subtree of M closed

under f and f−1. Let b ∈ N \M and e ∈ M the closest point in T to b. Take α < κ be a

cardinal, and {βi | i ∈ α} a family of branches at e in M . Then, there is an isometric embedding

g of⋃m∈Z f

m([e, b]) into M so that g(e) = e and g commutes with f and so that the image of

g does not intersect⋃i∈α{fm(βi) | m ∈ Z} except at the points fm(e) for m ∈ Z.

Proof. Assume the situation described in the hypotheses. Then, by Lemma 4.3.8 there exists

an isometric embedding g : [e, b] → M so that g(e) = e, the image of g intersects⋃i∈α βi only

at e, and d(g(b), µm(g(b))) = d(b, µm(b)) for all m ∈ N. Then, by Lemma 4.2.5, for all k, l ∈ Z

d(fk(b), f l(b)) = d(fk(g(b)), f l(g(b))).

Therefore we may extend g to an isometric embedding from⋃m∈Z f

m([e, b]) =⋃m∈Z[e, fm(b)])

into M by defining g(fm(b)) = fm(g(b)) for each m ∈ Z and extending isometrically to the

segments between. It is immediate from the definition of g that g commutes with f . The image

of g does not intersect⋃i∈α{fm(βi) | m ∈ Z} except at the points fm(e) for m ∈ Z, because

the image of g can only intersect⋃i∈α βi at e and g commutes with f .

4.3.10 Lemma. Let M |= rbERTs and N |= ERTs an extension of M with (N, d, p, f) the

underlying R-tree and elliptic isometry of N and (M,d, p) the underlying R-tree of M. Assume

M is κ-saturated for κ > 2ω. Let a1, ..., ak ∈M , b1, ..., bn ∈ N \M . Let the set B and {Gh}∞h=1

and eh be as in the hypotheses of Lemma 4.3.7, with the tree T = EA. Define Kh to be the

subtree of N generated by Gh ∪ {eh}. Let K = Kh for some h ∈ N>0. Then, there exists an

isometric embedding

g :⋃m∈Z

fm(K)→M

with g(eh) = eh such that g commutes with f , and so that the image of g intersects EA only at

the points fm(eh) for m ∈ Z.

Proof. Assume the situation described in the hypotheses of the lemma. We may assume without

loss of generality that b1, ..., bk are exactly the members of B that are in K, where k ≤ n. Let

e = eh be the closest point in EA common to all points in Gh. Then e is the closest point in

EA to any point from K.

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Define A1 ⊆ A2 ⊆ ... ⊆ Ak ⊆⋃m∈Z f

m(K) as follows:

• A1 =⋃m∈Z

fm([e, b1])

• A2 = A1 ∪⋃m∈Z

fm([z2, b2]) where z2 is the closest point to b2 in [e, b1];

• Ai+1 = Ai∪⋃m∈Z

fm([zi+1, bi+1]) where zi+1 is the closest point to bi+1 in [e1, b1]∪ [z2, b2]∪

... ∪ [zi, bi] ⊆ K.

Then, Ak =⋃m∈Z f

m(K), since K is generated by b1, ..., bk. We begin by defining g on [e, b1].

By Lemma 4.3.9 there exists an isometric embedding g1 of A1 =⋃m∈Z f

m([e, b1]) into M so

that g1(e) = e and g1 commutes with f . Moreover, we may find g1 so that the image of g1 only

intersects EA at e. This is because the degree of e in EA is at most 2ω by Lemma 4.2.7, so

when we set T = EA, we may let {βi | i ∈ 2ω} be the collection of branches at e that intersect

EA at points other than e. Define g = g1 on A1. Since we have just embedded A1 in M , we

now assume, for notational convenience, that A1 ⊆ M and that g1 was an inclusion map. Let

T1 = EA ∪A1.

Claim: T1 is a closed subtree of M . Assume towards contradiction that there is a Cauchy

sequence (xj)∞j=1 in T1 whose limit y is not in T1. Then, the Cauchy sequence must be bounded

away from EA ⊆ T1, which means there exists a δ > 0 and a point c in [e, b1] with distance δ/2

from e so that eventually, the sequence (xj)∞j=1 is contained in fm([c, b1]) ⊆ A1. But, these sets

fm([c, b1]) for m ∈ Z all have distance at least δ from one another. Therefore, (xj)∞j=1 must

eventually be in one of them. Each fm([c, b1]) is closed and hence y is in A1 ⊆ T1, which is

a contradiction. The sets EA and A1 are both closed under f and f−1, thus the subtree T1 is

closed under f and f−1.

It is straightforward to see that z2 is the closest point to b2 in T1 (because the b1, ..., bk were

in K, and hence had the same closest point in EA). If b2 ∈ [e, b1], then g is already defined

on A2, so we may assume b2 /∈ A1. We extend the domain of g to A2 = A1

⋃m∈Z f

m([z2, b2])

as follows. Use Lemma 4.3.9 with the tree T1 and {βi | i ∈ 2ω} equal to the collection of

branches at z2 that intersect T1. Because z2 ∈ A1, which is generated by {b1}, we know this

collection is has size at most 2ω. (If z2 = e, then the size of the collection is 2ω. If z2 6= e, then

there are only two branches at z2 that intersect T1.) We conclude that there exists an isometric

embedding g2 of⋃m∈Z f

m([z2, b2]) into M sending z2 to z2 so that g commutes with f . Since

[e, b1] ∩ EA = {e}, and z2 is on a branch at e that does not intersect EA except at e, we know

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that the image of g2 does not intersect EA, except possibly at fm(e) for m ∈ Z. Because the

domains of g1 and g2 were disjoint except at points fm(z2) for m ∈ Z and g1(z2) = z2 = g2(z2)

it follows that g1 ∪ g2 is an isometric embedding of A2 into M whose range intersects EA only

at fm(e). As before, since we have embedded A2 in M , we may assume A2 ⊆ M and that g2

was inclusion.

Let T2 = T1 ∪ A2. Then T2 is a closed subtree of M that is closed under f and f−1. Then,

z3 is the closest point to b3 in T2. We extend the domain of g to A3 = A2

⋃m∈Z f

m([z3, b3]) in

the same way we extended from A1 to A2. Proceed to extend g in the same manner until you

have extended it to Ak =⋃m∈Z f

m(K). This g has the desired properties.

In this next lemma, given M |= ERTs, for any h ∈ N we add new rays each point a ∈M of

relative order h. That is, if a has order m, the new rays will have order hm, and if a has infinite

order the new rays will also have infinite order.

4.3.11 Lemma. Let M |= ERTs with underlying R-tree and isometry (M,d, p, f). Let h ∈ N>0.

Then there exists an extension N |= ERTs of M with underlying R-tree and isometry (N, d, p, f)

such that:

• if a ∈ M , and m ∈ N>0 is the order of a, then there is a new ray Ra at a contained in(N \M

)∪ {a}, such that the f -order of every point in Ra \ {a} is equal to hm;

• if a ∈M has infinite order, there exists a new ray R at a such that every point on R has

infinite order.

Proof. Assume M |= ERTs with underlying R-tree and isometry (M,d, p, f). Let h ∈ N>0.

Enumerate M by {ai | i ∈ |M |}. If ai has finite f -order, denote the f -order of ai by mi. Let

Ti be an R-tree made up of h copies of R≥0, which all intersect at 0 but are otherwise disjoint.

(We could easily build such an R-tree using Theorem 2.2.6.) Let qi = 0 be the basepoint of Ti.

Label the h rays of Ti by Rai1 , R

ai2 , ..., R

ai

h . Use Lemma 2.2.8 to find an extension N of M that

glues in a new copy of Ti at ai, identifying qi and ai. Let (N, d, p) be the underlying R-tree of

N. Extend the isometry f on M to these new rays by setting f(Raj ) = Rf(a)j+1 for j = 1, ..., h− 1,

and f(Rah) = Rf(a)1 for each a ∈ M (mapping Raj isometrically to Rf(a)

j+1 by using the fact that

they are both isometric to R≥0). It is straightforward to check that this extension of f is a

well-defined isometry of N .

Let a ∈ M have order m. To show that every point (except the basepoint) in each “new”

ray in N has order hm, it suffices to show that every point in Ra1 has order hm. If h = 1, then

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fm(Ra1) = Rfm(a)1 = Ra1 , but for k < m fk(Ra1) = R

fk(a)1 6= Ra1 . Therefore, every point on the

Ra1 has order m. If h = 2, then fm(Ra1) = Rfm(a)2 is a ray out of fm(a) = a distinct from Ra1 ,

and

f2m(Ra1) = fm(fm(Ra1)) = fm(Rfm(a)

2 ) = fm(Ra2) = Rfm(a)1 = Ra1 .

For any k < 2m, with k 6= m, we know fm(a) 6= a, so clearly fk(Ra1) 6= Ra1 . Therefore, the order

of each point but the basepoint on Ra1 is 2m. For arbitrary h ∈ N>0 the argument is analogous.

Note that in this process, points of infinite order in M also get new rays added to them, so that

every point on those new rays has infinite order.

4.3.12 Lemma. Let M |= ERTr,s with underlying R-tree and isometry (M,d, p, f), a =

a1, ..., ak ∈ M and b = b1, ..., bn ∈ M with a1 = p. If c1, ..., cn ∈ M satisfy the par-

tial type DMf,b(y1, ..., yn/a) given in Definition 3.4.1, then for any quantifier free Ls-formula

ϕ(x1, ..., xk, y1, ..., yn) we have ϕ(a1, ..., ak, b1, ..., bn)M = ϕ(a1, ..., ak, c1, ..., cn)M.

Proof. The proof is analogous to that of Lemma 2.5.2.

4.3.13 Theorem. The Ls-theory rbERTs is the model companion of ERTs.

Proof. Clearly any model of rbERTs is also a model of ERTs. Assume M |= ERTs has un-

derlying R-tree and isometry (M,d, p, f). By iterating Lemma 4.3.11, we find an extension of

M that is a model of rbERTs. Let (ni)∞i=1 be a sequence of positive integers such that every

positive integer appears infinitely many times in the sequence. Let M0 = M. For i ≥ 1, let

Mi be the result of applying Lemma 4.3.11 to Mi−1 with h = ni. This process creates an

increasing chain (Mi | i ∈ ω) of models of ERTs. Let the Ls-structure W be the union of this

chain. Then the underlying space W of W is an R-tree by Lemma 2.2.7, and the isometry fW

on W is the union of the isometries fMi . Therefore, fW is elliptic, since it extends fM0 , and

we know d(pW, fW(pW)) = d(pM0 , fM0(pM0)). Let a ∈ W . Let m be the order of a and let

h, l ∈ N>0. For large enough i, the number h has appeared at least l times in our sequence

(ni)∞i=1. Therefore, at a with order m ∈ N>0, there are at least l-many rays so that every point

but a along the ray is of order hm. So, for any r ∈ R>0 we may find b1, ..., bl on these rays with

d(a, bi) = r for i = 1, ..., l so that each point in (a, bi] has order hm. In addition, at any point

a ∈M of order ∞, there are arbitrarily many rays at a such that every point on those rays has

order ∞. This implies that W |= rbERTs.

It remains to show rbERTs is model complete. It suffices by Proposition 1.7.9 to show

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every model of rbERTs is an existentially closed model of rbERTs. Let M |= rbERTs. Let

N |= rbERTs be an extension of M with underlying R-tree and isometry (N, d, p, f). We may

assume that M is κ-saturated for κ > 2ω. (The reason here is the same as the reason that we

may assume ω1-saturation in the proof of Theorem 2.5.4.) Since M is κ-saturated, its reduct to

Lp is also κ-saturated. Also as in the proof of Theorem 2.5.4, to show M is existentially closed

it suffices, by Lemma 3.4.2, to show: for any a1, ..., ak ∈ M , for any extension N |= HRTr,s,

for any b1, ..., bn ∈ N there exist c1, ..., cn ∈ M such that c1, ..., cn satisfy the partial type

DNf,b(x1, ..., xn/a).

Let a1, ..., ak ∈ M , and let b1, ..., bn ∈ N . We may assume that none of the bi are in M ,

because if so, we may let ci = bi for that i and then add it and its images under f and f−1 to

our set of parameters A. So, assume b1, ..., bn ∈ N \M . Let T = EA, and note by Lemma 4.2.7

that this makes T a closed subtree of N that is closed under f and f−1. Define

B =⋃m∈Z{fm(bi) | i = 1, ..., n},

the equivalence relation ∼ on B and the classes Gh as in Lemma 4.3.7. For h ∈ N, let Kh be

the subtree of N generated by Gh ∪ {eh}. Every point in Kh has eh as its closest point in EA.

Also, if Kh ∩Kl 6= ∅, then every point in Kh and every point in Kl must have the same closest

point in EA. This implies that if Kh ∩Kl 6= ∅, then h = l.

Lemma 4.3.10 implies that given Kh for any h = 1, ..., n, there is an isometric embedding

gh : Kh → M sending ph to eh whose image does not intersect EA except at eh and so that gh

commutes with f . Therefore, we may build an isometric embedding

g :∞⋃i=1

Ki →M

exactly as in the proof of Theorem 3.4.5.

Let ci = g(bi). Then since g commutes with f , for all x ∈⋃∞i=1Ki we know

fm(ci) = fm(g(bi)) = g(fm(bi)).

Therefore, for any m, l ∈ Z and any i, j ∈ {1, ..., n},

d(fm(bi), f l(bj)) = d(g(fm(bi)), g(f l(bj))) = d(fm(ci), f l(cj)).

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In addition, for any point x ∈ EA,

d(fm(bi), x) = d(fm(bi), ej) + d(ej , x)

= d(g(fm(bi)), g(ej)) + d(ej , x)

= d(fm(ci), ej) + d(ej , x) = d(fm(ci), x).

Therefore, we have found c1, ..., cn ∈M that satisfy the partial type Dfb (x1, ..., xn).

4.4 Properties of the model companions

4.4.1 Lemma. The Ls-theory rbERTs has quantifier elimination.

Proof. By Theorem 4.3.13, Theorem 4.1.4 and Proposition 1.7.13.

4.4.2 Theorem. Let π be the partial type

{d(p, µm(p)) = rm | m ∈ N}

where rm ∈ R≥0. Assume π is consistent with rbERTs and let rbERTπs be the Ls-theory con-

sisting of the conditions in rbERTs ∪ π. Then the Ls-theory rbERTπs is complete.

Proof. Assume the situation given in the hypotheses. Let M and N be models of rbERTs.

Consider the substructures generated by pM and pN in M and N respectively. If follows from

Lemma 4.2.5 and the axioms in π that these substructures are isomorphic. Therefore, the

substructure of M generated by pM embeds in any model of rbERTπs . Then, because rbERTπs

has quantifier elimination, we know rbERTπs is complete.

It is not hard to see that any completion of rbERTs has an axiomatization of the form

rbERTπs . The completions of rbERTs are exactly the Ls-theories Th(M) for M |= rbERTs. To

see this, given M |= rbERTs, let

π = {d(p, µm(p)) = dM(pM, µm(pM)) | m ∈ N}.

Then, M |= rbERTπs and by Theorem 4.4.2 we conclude rbERTπs is equivalent to Th(M).

In the rest of this section, fix a completion rbERTπs of rbERTs. Next, we turn to the question

of the stability of rbERTπs . Given M |= rbERTπs and A ⊆M , let TA be the closed subtree of M

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generated by the set {fm(a) | a ∈ A ∪ {p}, m ∈ Z}. That is, TA = E{fm(a)|a∈A∪{p}, m∈Z}.

4.4.3 Lemma. Let M |= rbERTπs , and let b, c ∈ M and A ⊆ M . Then tpM(b/A) = tpM(c/A)

if and only if

• b and c have the same unique closest point in e ∈ TA and d(e, b) = d(e, c);

• d(b, µm(b)) = d(c, µm(c)) for all m ∈ N.

Proof. Assume the situation described in the hypotheses. For the forward direction assume

tpM(b/A) = tpM(c/A). Then for all x ∈ TA, d(x, b) = d(x, c). (The distance to a point in

the closure of a set is determined by distances to points in the set, and the type tpM(b/A)

determines all distances to points in E{fm(a)|a∈A∪{p}, m∈Z}.) Therefore, b and c must have the

same closest point e in TA and it must be that d(e, b) = d(e, c). Also, for any m ∈ N we know

d(b, µm(b)) =12d(b, fm(b)) =

12d(c, fm(c)) = d(c, µm(c)).

For the other direction, assume b and c have the same unique closest point in e ∈ TA

and d(e, b) = d(e, c), and assume d(b, µm(b)) = d(c, µm(c)) for all m ∈ N. Since rbERTπs has

quantifier elimination, it suffices to show that the quantifier-free types of c and b over A are

the same. To show the quantifier-free types are the same, by Lemma 4.3.12 it suffices to show

d(x, fn(b)) = d(x, fn(c)) and d(b, fn(b)) = d(c, fn(c)) for all n ∈ N, for all x ∈ TA. Since

d(b, µm(b)) = d(c, µm(c)) for all m ∈ N, Lemma 4.2.5 implies d(b, fn(b)) = d(c, fn(c)) for all

n ∈ N. If x ∈ TA and n ∈ N, then fn(e) ∈ [x, fn(b)] and fn(e) ∈ [x, fn(c)] because fn(e) must

be the closest point in TA to fn(b) and fn(c) (since TA is closed under f and f−1. Therefore,

d(x, fn(b)) = d(x, fn(e)) + d(fn(e), fn(b)) = d(x, fn(e)) + d(fn(e), fn(c)) = d(x, fn(c)). This

implies that d(x, fn(b)) = d(x, fn(c)) for all x ∈ TA and all n ∈ N.

4.4.4 Theorem. The Ls-theory rbERTπs is stable. Indeed, when κ is an infinite cardinal,

rbERTπs is κ-stable if and only if κ satisfies κω = κ.

Proof. By a counting argument as in Theorem 2.6.4, using Lemma 4.4.3 instead of Lemma

2.6.3.

Let κ be a cardinal so that κ = κω and κ > 2ω. Let U be a κ-universal domain for rbERTπs .

4.4.5 Definition. Let A,B and C be small subsets of U . Let A := {fm(a) | a ∈ A∪ {p}, m ∈

Z}, and let B and C be defined analogously. Say A is E-independent from B over C, denoted

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A |E^CB, if and only if A | bC B in the sense of models of rbRT. That is, A |E^C

B if and only

if for all a ∈ A we have dist(a,EdBC) = dist(a,E bC).

4.4.6 Theorem. The |E^ independence relation is the model theoretic independence relation for

rbERTπs .

Proof. Again here as in Theorem 3.5.6, the general line of argument is close to that of proof

of Theorem 2.6.7, but in places we need lemmas from this chapter instead of those used in the

proof of 2.6.7. As in that proof we use [1, Theorem 14.12], and show |E^ has all seven of the

properties required by that theorem.

Invariance under automorphisms is clear because any automorphism of a model M of rbERTπs

preserves | and satisfies σ(A) = σ(A). Symmetry and transitivity follow from the definition of

|E^, and from the fact that | is symmetric and transitive. Finite character follows from the fact

that | has finite character, and from the fact that A =⋃a∈A∪{p}{fm(a) | m ∈ Z}. Extension

has much the same proof as in the proof of 2.6.7, but using Lemma 4.3.10 instead of Lemma

2.5.3. For local character we use the same line of reasoning as in Theorem 2.6.7. The argument

for stationarity uses Lemma 4.4.3 instead of Lemma 2.6.3.

4.4.7 Theorem. The theory rbERTπs is not ω-categorical.

Proof. Since Lp ⊆ Ls are both countable signatures, and rbRT is the reduct of rbERTπs to Lp,

this follows from the fact that rbRT is not ω-categorical by [1, Proposition 12.13].

4.4.8 Theorem. Let κ > ω be a cardinal. The theory rbERTπs is not κ-categorical.

Proof. Let κ > ω be a cardinal. We construct non-isomorphic models of rbERTπs , each with

density character κ. First, let M be a separable model of rbERTπs with underlying R-tree and

isometry (M,d, p, f). The degree of any point in a separable R-tree is at most ω by Lemma

2.6.10. Pick a point q ∈M fixed by f . Use Lemma 2.2.8 to add κ-many new distinct rays at q,

all of which are fixed pointwise by f , and let W1 be this extension of M.

Next, we use the procedure in the beginning of the proof of Theorem 4.3.13 to extend W1 to

a model W′1 of rbERTπs . Let (ni)∞i=1 be a sequence of positive integers such that every positive

integer appears infinitely many times in the sequence. Let M0 = W1. For i ≥ 1, let Mi be

the result of applying Lemma 4.3.11 to Mi−1 with h = ni. This process creates an increasing

chain (Mi | i ∈ ω) of models of ERTs. Let the Ls-structure W′1 be the union of this chain,

with underlying R-tree and isometry (W ′1, d, p, f). As in the proof of Theorem 4.3.13, it is

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straightforward to see that W′1 |= rbERTπs . Moreover, at each of the countably many steps in

the construction above, we added at most finitely many branches at any given point. Therefore,

the degree of every point x 6= q in W ′1 is at most ω.

Let A1 ⊂ W ′1 be a subspace consisting of q and the κ-many rays we added at q. Note that

A1 has density equal to κ. Apply the Downward Lowenheim-Skolem Theorem ([1, Proposition

7.3]) to W′1 to get an elementary substructure W1 of density character ≤ κ which contains A1.

Because W1 contains a point of degree κ, we know W1 has density character equal to κ by

Lemma 2.6.10. Also, W1 |= rbERTπs , and q is the only point in the underlying R-tree of degree

κ, every other point has degree at most ω.

To find the other non-isomorphic model, let W2 |= rbERTπs be κ-saturated with underlying

R-tree and isometry (W2, d2, p2, g). Then by Lemma 2.3.7 there are at least κ-many branches

at every point in W2. Choose a ∈ W2 different from the basepoint, and choose κ-many distinct

rays at a. Choose κ-many distinct rays at the basepoint p2. Let A2 ⊂ W2 be the subspace

consisting of a, p2 and the chosen rays. Apply the Downward Lowenheim-Skolem Theorem ([1,

Proposition 7.3]) to W2 to get an elementary substructure W2 of density character ≤ κ which

contains A2. Then, W2 |= rbERTπs and has density character equal to κ by Lemma 2.6.10.

There are at least two distinct points (p2 and a) with degree κ in the underlying R-tree W2 of

W2. In W1 only the point q has degree κ, while the rest of the points have at most degree ω.

Thus, W1 and W2 cannot be isomorphic.

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REFERENCES

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[2] Ben Yaacov I., Usvyatsov A., Continuous first order logic and local stability. To appear inTrans. Amer. Math. Soc.

[3] Ben Yaacov I., Continuous first order logic for unbounded metric structures, submitted.

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[5] Bridson M.R., Haefliger A., Metric Spaces of Non-positive Curvature. Grundlehren dermathematischen Wissenschaften 319, Springer, 1999.

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[7] Culler, M., Morgan, J.W., Group actions on R-trees. Proc. London Math. Soc. 55 (1987),571–604.

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[10] Dyubina, A., Polterovich, I. Explicit constructions of universal R-trees and asymptoticgeometry of hyperbolic spaces. Bull. London Math. Soc. 33 (2001), 727–734.

[11] Hodges, W., A Shorter Model Theory. Cambridge University Press, Cambridge, 1997.

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[13] Roe, J., Lectures on Coarse Geometry. University Lecture Series, 31. American Mathe-matical Society, Providence, RI, 2003.

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AUTHOR’S BIOGRAPHY

Sylvia Ellen Booth Carlisle was born February 13th, 1980 in St. Louis, Missouri. She grew up

mainly in Iowa City, Iowa, where in her freshman year her geometry teacher encouraged her to

stop sitting at the back of the classroom and join the high school math club. Sylvia graduated

from Carleton College in 2002 and moved on to the doctoral program in mathematics at the

University of Illinois, Urbana-Champaign. After graduating, she will be a tenure-track assistant

professor at Eastern Illinois University in Charleston, Illinois.

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