Generalizations of Coarse Properties in Large Scale Spaces

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Doctoral Dissertations Graduate School

8-2017

Kevin Michael Sinclair University of Tennessee, Knoxville, [email protected]

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Part of the Geometry and Topology Commons

Recommended Citation Sinclair, Kevin Michael, "Generalizations of Coarse Properties in Large Scale Spaces. " PhD diss., University of Tennessee, 2017. https://trace.tennessee.edu/utk_graddiss/4651

This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a dissertation written by Kevin Michael Sinclair entitled "Generalizations of Coarse Properties in Large Scale Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics.

Jerzy Dydak, Major Professor

We have read this dissertation and recommend its acceptance:

Nikolay Brodskiy, Morwen Thistlethwaite, Michael Berry

Accepted for the Council:

Dixie L. Thompson

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) Generalizations of Coarse Properties in Large Scale Spaces

A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville

Kevin Michael Sinclair August 2017 c by Kevin Michael Sinclair, 2017

ii To my parents, John and Vernice Sinclair, who taught me the importance of education, and

to my girlfriend, Danna Sharp, who kept me sane up to this point.

iii Acknowledgments

I would like to thank Dr. Dydak for guiding and teaching me throughout my graduate career. His creativeness showed me new ideas and his patience gave me time enough time to understand them. I would like secondly to thank Kyle Austin and Michael Holloway for helping me to get oﬀ the ground in large scale geometry. I also want to thank Dr. Brodskiy,

Dr. Thistlethwaite and Dr. Berry for serving on my committee. Finally, I would like to thank Pam Armentrout for all the things she did to ensure graduate school went smoothly.

iv Abstract

Many results in large scale geometry are proven for a metric space. However, there exists many large scale spaces that are not metrizable. We generalize several concepts to general large scale spaces and prove relationships between them. First we look into the concept of coarse amenability and other variations of amenability on large scale spaces. This leads into the definition of coarse sparsification and connections with coarse amenability. From there, we look into an equivalence of Sako’s definition of property A on uniformly locally finite spaces and prove that finite coarse asymptotic definition implies it. As well, we define large scale exactness and prove implications with large scale property A and coarse amenability.

We ﬁnally look into a stronger concept of bounded geometry on large scale spaces that is a coarse invariant and leads to a way to decompose large scale spaces.

v Table of Contents

1 Introduction1

1.1 Scales...... 2

1.2 Coarse Structures...... 4

1.3 Large Scale Structures...... 5

2 Generalizations of Coarse Properties 10

2.1 Coarse Amenability...... 10

2.2 Coarse Uniform Local Amenability with respect to all probability measures. 14

2.3 Coarse Sparsiﬁcation...... 20

3 Property A 25

3.1 Property A on Large Scale Structures...... 25

3.2 Coarse Exactness...... 34

4 Coarse Generalization of Bounded Geometry 42

4.1 Bounded Scale Measure...... 42

4.2 Decomposition of ls-spaces...... 46

vi Bibliography 50

Vita 53

vii Chapter 1

Introduction

When introducing concepts in coarse geometry one normally starts by assuming X to be a metric space. On the other hand, it was quickly proved that there exist coarse structures that are not metrizable. Coarse structures were defined by Higson and Roe as a collection of subsets of X × X called entourages. The original motivation for studying coarse structures was to try to approach the Novikov conjecture which states that the higher signatures of smooth manifolds are homotopy invariants. Since then coarse structures have been utilized in many other fields, and a number of new properties and ideas have arisen. Many of these applications are given in a metric space with bounded geometry. From there, J. Dydak gave an alternate definition of coarse structures denoted as large scale structures. It was shown that coarse structures and large scale structures are in one to one correspondence with each other. Therefore, rather than working with coarse structures, we will prove results for large scale structures. In this dissertation, we will approach such properties only defined for a metric space in order to generalize them to general large scale spaces. In particular we will begin by generalizing the idea of uniform local amenability presented in [BNS+12]

1 and then move onto deﬁnitions that are connected to uniform local amenability. We will

then prove whether the relations between these concepts still hold in a general coarse space.

Additionally, we will look into generalizing the idea of bounded geometry to one that is a

coarse invariant. To begin we ﬁrst need to establish several deﬁnitions and concepts used

throughout this dissertation.

1.1 Scales

Before deﬁning a large scale structure we must ﬁrst introduce the concept of scales and how

to compare two scales against each other.

Deﬁnition 1. Given a space X, we say U is a scale of X if U is a cover of X. In other

words U is a collection of subsets of X such that ∪U∈U U = X.

Note that for any family B of X, not necessarily a scale, we can extend it to a scale via the trivial extension e(B) deﬁned as B ∪ {{x} : x ∈ X}. Therefore, we can think of any family as a scale even if it does not cover the entirety of X.

When defining large scale structures we need a way to compare two scales. This is done through the idea of coarsenings, refinements, and star-refinements.

Deﬁnition 2. Given two scales U and V, we say U coarsens V, denoted V ≺ U if for every

U ∈ U there exists V ∈ V such that V ⊂ U. Similarly, we say that V reﬁnes U.

In Coarse Geometry we study large scale properties of a space X. As such we need a way

to create larger scales from pre-existing one. This is achieved through the act of starring

scales.

2 Deﬁnition 3. Given a set X, V ⊆ X, and a collection of subsets U we deﬁne the star of V with respect to U, denoted st(V, U), as the union of all sets in U that have nonempty intersection with V , i.e. st(V, U) = ∪{U ∈ U : U ∩ V 6= ∅}.

Given two collections of subsets V and U we deﬁne st(V, U) = {st(V, U),V ∈ V}.

Given two collections of subsets V and U we say V star reﬁnes U if st(V, V) ≺ U.

Stars and star reﬁnement satisfy the following basic properties:

1. If U ⊆ X and V is a scale of X, then U ⊆ st(U, V).

2. If U is a collection of subsets of X and V is a scale of X, then U ≺ st(U, V).

3. If U ⊆ V and W is any collection of subsets of X, then st(U, W) ⊆ st(V, W).

4. If U ≺ V and W is any collection of subsets of X, then st(U, W) ≺ st(V, W).

5. If U ⊂ X and V1, V2 are collections of subsets of X with V1 ≺ V2, then st(U, V1) ≺

st(U, V2).

6. If U is any collection of subsets of X and V1, V2 are collections of subsets of X with

V1 ≺ V2, then st(U, V1) ≺ st(U, V2).

7. If U is a scale and V is any collection of subsets of X, then V ≺ st(U, V).

The motivating example for understanding scales and stars of scales is the cover of a metric space by r-balls for any r > 0. Let Br be a cover of X by r-balls and let U be any other cover

of X. Notice that st(U, Br) = {B(U, 2r): U ∈ U where B(U, 2r) is the 2r neighborhood of

any set U. Therefore, we can think of the act of starring as fattening up any scale to create

a new scale that reﬁnes the original. Clearly V ≺ st(V, U). This allows us to think of the

3 sets of st(V, U) as neighborhoods to the sets of V. In other words, we can consider V ∈ V as points of X with neighborhoods contained in st(V, U).

While the motivating example comes from a metric we will see the deﬁnition of a Large

Scale Structure is more general in nature [DH06]. However, before we deﬁne a Large Scale

Structure, or an ls-structure for short, we should ﬁrst introduce the idea of a coarse structure on X.

1.2 Coarse Structures

Coarse structures were originally introduced by Higson and Roe for use in index theory and signature theory. In particular, coarse structures were defined to provide an approach to the Novikov conjecture, a still open conjecture stating that the higher signatures of smooth manifolds are homotopy invariant [Roe03]. Like scales, metric spaces naturally give rise to coarse structures. However, coarse structures can easily be defined for more general spaces without a metric. Before giving the definition of a coarse structure, we must first introduce some notation.

Deﬁnition 4. Let X be a set and consider the product space X × X .

1. The diagonal of X to be ∆ = {(x, x): x ∈ X}.

2. For U ⊂ X × X, deﬁne the inverse of U to be U −1 = {(y, x):(x, y) ∈ U}.

3. Given two subsets U, V ∈ X × X, deﬁne the product of U and V to be U ◦ V = {(x, z):

(x, y) ∈ U, (y, z) ∈ V for some y ∈ X}.

4. For U ⊆ X × X and for x ∈ X, deﬁne U[x] = {y ∈ X :(y, x) ∈ U}.

4 Deﬁnition 5. (Roe, 2003) A Coarse Structure on X is a family U of subsets of X × X that satisfy the following:

• ∆ ∈ U.

• U ∈ U implies U −1 ∈ U

• If U, V ∈ U then U ◦ V ∈ U

• If U ∈ U and V ⊂ U, then V ∈ U.

• If U, V ∈ U then U ∪ V ∈ U.

The elements of a uniform structure are called either controlled sets or entourages.

1.3 Large Scale Structures

The deﬁnition of coarse structures is analytical in nature, but is not very useful for topologists and geometers. With that in mind large scale structures were deﬁned in [DH06] as a more topological way of looking at coarse properties of a space.

Deﬁnition 6. (J Dydak, 2005) Given a set X.A Large Scale Structure, LSSX is a nonempty set of families B of subsets of X (called uniformly bounded or uniformly LSSX - bounded one LSSX is ﬁxed) satisfying the following conditions:

•B 1 ∈ LSSX implies B2 ∈ LSSX if each element of B2 consisting of more than one

point is contained in some element of B1.

•B 1, B2 ∈ LSSX implies st(B1, B2) ∈ LSSX .

5 Alternatively a large scale structure is a nonempty filter of scales on X under reverse star refinement that is closed under refinements of scales.

Elements of the ls-structure are called either uniformly bounded covers or uniformly bounded families. Note that the first property of our definition gives us that the trivial scale, {{x} : x ∈ X} is always in our large scale structure. Therefore if B ∈ LSSX then all refinements of e(B) also belong to LSSX . The second condition can be thought of

as a generalization of the triangle inequality.

The idea of a uniformly bounded cover is very closely related to the idea of a controlled set

in a coarse structure. In [DH06] it was shown that coarse structures and ls-structures are

equivalent by the following propositions.

Proposition 1.0.1. Every ls-structure LSSX on X induces a coarse structure C on X as

follows: A subset E of X × X is declared controlled if and only if there is B ∈ LSSX such

that E ⊂ ∪B∈BB × B.

Proposition 1.0.2. Every coarse structure C on X induces a large scale structure LSSX

on X as follows: B is declared uniformly bounded if and only if there is a controlled set E

such that ∪B∈BB × B ⊂ E.

Therefore properties of coarse structures can also be deﬁned and proved on large scale structures. As well, we can deﬁne functions between large scale structures in a way similar to functions between coarse structures.

Let f : X → Y be a function between two metric spaces (X, dX ) and (Y, dY ). We say that

f is uniformly continuous if for every > 0 there exists a δ > 0 such that given x, y ∈ X,

dX (x, y) < δ implies dY (f(x), f(y)) < . In other words, if two points are close in the domain

6 they will stay close in the codomain. This idea has been generalized to the idea of closeness

between two large scales spaces.

Deﬁnition 7. Let X and Y be large scales spaces. A map f : X → Y is Large Scale continuous,

or ls-continuous, if for every uniformly bounded cover U in X, f(U) is a uniformly bounded cover in Y .

For a large scale continuous function f, we say f is a coarse embedding if for every

uniformly bounded family U ∈ Y , f −1(U) = {f −1(U)|U ∈ U} is uniformly bounded in X.

We say f is coarsely surjective if there exists a uniformly bounded family U in Y such

that Y ⊆ st(f(X), U).

Note that if f is not surjective, then f(U) will not be a cover of Y . However, recall that

any uniformly bounded family can be trivially extended to a uniformly bounded cover. From

the deﬁnition it is clear that the composition of two large scale continuous maps is still large

scale continuous. In a large scale space we can consider points in a uniformly bounded set

as equal. Therefore, we will deﬁne a coarse equivalence on the uniformly bounded sets.

Deﬁnition 8. Let X and Y be large scale space. Given maps f, g : X → Y , we say f and g

are close if there exists a uniformly bounded cover V of Y such that for every x ∈ X there

is V ∈ V with f(x), g(x) ∈ V .

A large scale continuous map f : X → Y between two large spaces is a coarse equivalence

if there exists a large scale continuous map g : Y → X such that f ◦ g is close to the function

idY and g ◦ f is close to the function idX . If f : X → Y is a coarse equivalence then we say

X and Y are coarsely equivalent.

7 This is equivalent to saying that {{g ◦ f(x), x}|x ∈ X} and {{f ◦ g(y), y}|y ∈ Y } are uniformly bounded families.

Theorem 1.1. A ls-continuous map f : X → Y is a large scale equivalence if and only if f is a coarse embedding and is coarsely surjective.

Proof. Assume that f : X → Y is a large scale equivalence. First we will show that f is a coarse embedding. Let U be a uniformly bounded cover of Y and let g : Y → X be the function such that g ◦ f : X → X is close to the identity idX . Therefore, there exists a uniformly bounded family V of X such that for every x ∈ X there exists a V ∈ V with x ∈ V and g ◦ f(x) ∈ V . Let U be a uniformly bounded family in Y . Since g is large scale continuous, then g(U) = {g(U): U ∈ U} is a uniformly bounded family in X. We claim the uniformly bounded family W = st(g(U), V) coarsens f −1(U).

Let U ∈ U. Consider any x ∈ X such that f(x) = y ∈ U. From the deﬁnition of coarse equivalence, there exists a Vy ∈ V such that {g ◦ f(x), x} ⊆ Vy. Therefore, {g(y), x} ⊆ Vy

−1 for all x ∈ X with f(x) = y, or {g(y), f (y)} ⊆ Vy.

−1 −1 Now, for U ∈ U, f (U) = ∪y∈U f (y) ⊆ ∪y∈U Vy for Vy deﬁned above. However, for y ∈ U we know {g(y), x} ∈ Vy for all x ∈ X with f(x) = y, so Vy ∩ g(U) 6= ∅. Thus for all y ∈ U

−1 Vy ⊆ st(g(U), V). Therefore f (U) ⊆ st(g(U), V), and from the deﬁnition of a large scale structure we have f −1(U) is a uniformly bounded family, and f is a coarse embedding.

Next we will show that f is coarsely surjective. From the deﬁnition of coarse equivalence, there exists a uniformly bounded cover U of Y such that for every y ∈ Y there exists a

U ∈ U with {f ◦ g(y), y} ⊆ U. Arbitrarily take y ∈ Y . Then there exists a U ∈ U with

{f ◦ g(y), y} ⊆ U. Therefore y ∈ st(f ◦ g(Y ), U). Therefore Y ⊆ st(f ◦ g(Y ), U), and since

8 f ◦ g(Y ) ⊆ f(X) we have that Y ⊆ st(f ◦ g(X), U), and f is coarsely surjective.

Now, suppose f is a large scale continuous map that is a coarse embedding and is coarsely

surjective. First we will construct a large scale continuous function g : Y → X. From

coarse surjectivity, we know there exists a uniformly bounded family U in Y such that

Y ⊆ st(f(X), U). Thus, for every y ∈ Y there exists a Uy ∈ U with Uy ∩ f(X) 6= ∅. Choose an xy ∈ X such that f(xy) ∈ Uy. Therefore deﬁne g : Y → X with g(y) = xy.

Claim: f ◦ g is close to idY . Let y inY . From the construction of g, there exists Uy ∈ U with y ∈ Uy, and f ◦ g(y) ∈ Uy. Thus {f ◦ g(y), y} ⊆ Uy and thus f ◦ g is close to idY .

−1 Let y ∈ Y . Claim: g ◦ f is close to idX . From coarse embeddability we know f (U) is

a uniformly bounded family in X. Choose x ∈ X. We know from coarse surjectivity that

there exists Uf(x) ∈ U with both f(x) ∈ Uf(x) and f ◦ g ◦ f(x) ∈ Uf(x) from the construction

−1 of the function g. Therefore {g ◦ f(x), x} ⊆ f (Uf (x)) = {z ∈ X|f(z) ∈ Uf(x)}, and g ◦ f is close to idX .

Finally we will show that g is large scale continuous. Let U be a uniformly bounded family of

Y . Since f ◦ g is close to idY , then there exists a uniformly bounded family V of Y such that for every y ∈ Y there exists V ∈ V with {f ◦g(y), y} ⊆ V . Let U ∈ U. For every y ∈ U, there

exists V ∈ V such that {f ◦ g(y), y} ⊆ V . Therefore, for every U ∈ U, f ◦ g(U) ⊂ st(U, V),

and we have that f ◦ g(U) is uniformly bounded in Y . Finally, since f is a coarse embedding

f −1 ◦ f ◦ g(U) is a uniformly bounded family and for all U ∈ U, g(U) ⊆ f −1 ◦ f ◦ g(U). This

means that g(U) is uniformly bounded in X from the deﬁnition of large scale structure.

This result will be used many times when showing a property is a coarse invariant.

9 Chapter 2

Generalizations of Coarse Properties

2.1 Coarse Amenability

Amenability of groups was introduced by Von Neumann who showed that the Banach-Tarski paradox is caused by the lack of amenability of the group of isometries of the 3D-space

R3. Amenability now is used in many areas of mathematics and admits many different definitions. One of the most common definitions of amenability is a geometric interpretation due to Følner for finitely generated groups. Let G be a finitely generated group with a word length metric and let a subset A ⊆ G be given. We define the R-boundary of A in the following way

∂RA = {g ∈ G \ A | d(g, A) ≤ R}

Følner’s deﬁnition of amenability compares the cardinality of ﬁnite sets against their boundaries.

10 Definition 9. A finitely generated group G is amenable if for every R > 0 and > 0 there exists a finite set F ⊆ G such that |∂ F | R ≤ . |F |

Such ﬁnite sets are referred to as Følner sets.

From there analogs of amenability were deﬁned for various coarse spaces. Cencelj, Dydak,

and Vavpetic [CDV12] came up with a deﬁnition of coarsely amenable spaces by dualizing

the concept of paracompactness.

Deﬁnition 10. X is large scale weakly paracompact if for all r, s > 0 there is a

uniformly bounded cover U of X of Lebesgue number at least s such that every r-ball B(x, r)

is contained in only ﬁnitely many elements of U.

The motivation for the deﬁnition of Coarse Amenability came as a result of analyzing

expanders

Deﬁnition 11. The horizon hor(A, U) = {Ui | A ∩ Ui 6= ∅,Ui ∈ U}.

Deﬁnition 12. X is coarsely amenable if for each s > r > 0 and each > 0 there is

a uniformly bounded cover U of X such that for each x ∈ X the horizon hor(B(x, s), U) is

ﬁnite and |hor(B(x, r), U)| > 1 − . |hor(B(x, s), U)|

Since not all large scale spaces have a metric we need a deﬁnition for Coarse Amenability

that relies solely on covers rather than r-balls. The generalization turns out to be simpler

in nature and will imply the above deﬁnition in the metric space case.

11 Deﬁnition 13. Let X have a large scale structure. X is coarsely amenable if for each

uniformly bounded cover W of X and for each > 0, there exists a uniformly bounded cover

U of X such that for all x ∈ X hor(st(x, W), U) is ﬁnite and

|hor(x, U)| > 1 − |hor(st(x, W), U)|

First we will show this deﬁnition is a coarse invariant.

Theorem 2.1. Let X and Y be large scale spaces, and let f : X → Y be a coarse embedding.

Then if Y is coarsely amenable, then X is also coarsely amenable.

Proof. Let W be a uniformly bounded cover of X and let > 0 be given. Then, f(W) is a

uniformly bounded cover of Y and there exists a uniformly bounded cover U of Y such that

for any x ∈ X |hor(st(f(x), f(W)), U)| is ﬁnite and

|hor(f(x), U)| > 1 − . |hor(st(f(x), f(W)), U)|

Since f is a coarse embedding we know f −1(U) is a uniformly bounded family of X.

First suppose U ∈ hor(f(x), U). Thus, f(x) ∈ U and therefore x ∈ f −1(U) and

f −1(U) ∈ hor(x, f −1(U)). Therefore |hor(x, f −1(U))| ≥ |hor(f(x), U)|.

Secondly, notice that if f −1(U) ∈ hor(st(x, W), f −1(U)) then there exists a W ∈ W and

a z ∈ X with x ∈ W and z ∈ W ∩ f −1(U). Therefore, f(x) ∈ f(W ) and f(y) ∈

f(W ) ∩ U. This gives us that U ∈ hor(st(f(x), f(W)), U), and |hor(st(x, W), f −1(U))| ≤

12 |hor(st(f(x), f(W)), U)|. Putting this together we get

and X is coarsely amenable.

This general deﬁnition of coarse amenability relies only on points in X. We can however

deﬁne a general coarse amenability in a very similar way to the metric case.

Theorem 2.2. X is coarsely amenable if and only if for all > 0 and uniformly bounded

covers V, W with V reﬁning W then there exists a uniformly bounded U of X such that for

all x ∈ X, the horizon, hor(st(x, W), U) is ﬁnite and

|hor(st(x, V), U)| > 1 − |hor(st(x, W), U)|

Proof. Suppose X is coarsely amenable. Let V be a uniformly bounded cover and let > 0.

Let V be the trivial cover of X. Thus V reﬁnes W trivially, and by assumption there exists a uniformly bounded cover U such that, for all x ∈ X hor(st(x, W), U) is ﬁnite, and

Now, assume the second condition. Let > 0 and let V and W be two uniformly bounded

covers with V reﬁning W. By assumption, there exists a uniformly bounded cover U such

13 that, for all x ∈ X, hor(st(x, W), U) is ﬁnite and

|hor(x, U)| > 1 − |hor(st(x, W), U)|

If U ∈ hor(x, U) then x ∈ U. Since V is a cover of X, then x ∈ st(x, V) so U∩st(x, V) 6= ∅ and

thus U ∈ hor(st(x, V), U) and hor(x, U) ⊂ hor(st(x, V), U). Next, if U ∈ hor(st(x, V), U),

then there exists a V ∈ V such that U ∩ V 6= ∅. Since V reﬁnes W, there exists a W ∈ W

such that V ⊂ W . Therefore, W ∩ U 6= ∅, U ∈ hor(st(x, W), U), and hor(st(x, V, U) ⊂

hor(st(x, W, U). Putting this together we get

2.2 Coarse Uniform Local Amenability with respect to

all probability measures

A version of local uniform amenability was deﬁned and generalized by Brodzki, Niblo,

Spakula, Willett, and Wright [BNS+12]. The inspiration for this deﬁnition was to create a form of amenability that is preserved by passing to subspaces.

14 Deﬁnition 14. Let X be a metric space. X is said to be uniformly locally amenable if for all R, > 0, there exists S > 0 such that for any ﬁnite subset F of X there exists E ⊆ X such that diam(E) ≤ S and

|∂RE ∩ F | < |E ∩ F |.

Uniform local amenability, or ULA, means that all ﬁnite subsets of a space X must be amenable with Følner sets of uniform size. This deﬁnition of local uniform amenability is equivalent to the following more analytical characterization:

For all R, and for all > 0 there exists S > 0 such that for all ﬁnite subsets F of X there exists a function φ ∈ l1(X) such that

• diam(supp(φ)) ≤ S

• the following inequality holds

X X X |φ(x) − φ(y)| < |φ(x). x∈F y∈F x∈F d(x,y)≤R

These deﬁnitions can be used to prove equivalences of several coarse properties. However, in order to accomplish this a stronger version of local uniform amenability must be introduced where probability measures are used to localize.

Deﬁnition 15. Let X be a space. X is said to have uniform local amenability with respect to probability measures, or ULAµ if for all R, > 0 there exists S > 0 such that for all probability measures µ on X there exists a ﬁnite subset E ⊆ X such that

• diam(E) ≤ S

15 • the following inequality holds

µ(∂RE) < µ(E).

Again, the deﬁnition for ULAµ is equivalent to the following more analytical approach:

For all R, > 0 there exists S > 0 such that for all probability measures µ on X there exists a function φ ∈ l1(X) such that

• diam(supp(φ)) ≤ S

• the following inequality holds

X X X µ(x) |φ(x) − φ(y)| < µ(x)|φ(x)|. x∈supp(µ) y∈supp(µ) x∈supp(µ) d(x,y)≤R

Both ULA and ULAµ are coarse invariants and it has been shown that ULAµ implies ULA.

However, both definitions require both a metric and the subsets E to be finite subsets. Finite subsets can be considered as simply points in large scale spaces. Therefore, these concepts are not completely clear in large scale spaces, but can be generalized to definitions that fit the idea behind large scale spaces.

Deﬁnition 16. If U is an ls-cover of X, then ∂U (A) = (st(A, U) \ A)

Deﬁnition 17. A set E is U-bounded if E ⊂ U for some U ∈ U.

Deﬁnition 18. Let X be a large scale space. We say X has coarse uniform local amenability with respect to all probability measures, or coarse ULAµ, if for each U, > 0 there exists V such that for all probability measures µ on X there exists a set E ⊂ X such that:

16 • E is V-bounded

• the following inequality hold

µ(∂U (E)) < · µ(E)

The ﬁrst thing to notice in the large scale generalization is that the subset E need not

be finite. We will first show that this definition still passes to subsets as originally intended

and is still a coarse invariant.

Theorem 2.3. Let X be a set with coarse uniform local amenability with respect to all

probability measures. If Y ⊆ X then Y also has coarse ULAµ.

Proof. Suppose X an ls-space has coarse ULAµ and let Y ⊂ X. Let U be a uniformly bounded cover of Y, and trivially extend U to a uniformly bounded cover U 0 on X. Thus there exists a V0 on X such that for all probability measures φ on X, there exists E0 ⊂ X

0 0 0 0 such that E is V -bounded and φ(∂U 0 (E )) < · φ(E )

Now, deﬁne V = V0 ∩ Y to be a uniformly bounded cover on Y . Let µ be a probability

measure on Y and extend it trivially to a probability measure φ on X. Therefore, there

0 0 0 0 0 0 exists E ⊂ X such that E is V -bounded and φ(∂U 0 (E )) < · φ(E ). If E ∩ Y = ∅ then

0 0 0 0 φ(E ) = 0 ≤ φ(∂U 0 (E )) which is a contradiction, so E ∩Y 6= ∅ and we can deﬁne E = E ∩Y .

Clearly, E is (V )-bounded by construction. Next, note that E0 = E ∪ {{x}|x ∈ E0 \ Y }, and

0 0 0 0 0 U = U ∪ {{x}|x ∈ X \ Y }. Therefore ∂U 0 (E ) = st(U ,E ) \ E = st(U,E) \ E = ∂U (E) So,

0 0 0 µ(∂U (E)) = φ(∂U (E)) = φ(∂U 0 (E )) < φ(E ) = φ(E ∩ Y ) = µ(E)

17 Since µ was chosen arbitrarily, V meets our condition for all probability measures and Y has

ls − ULAµ.

Before showing this is a coarse property, we ﬁrst need to prove a basic property of stars

of sets.

Lemma 2.3.1. Let X be a set, E ⊆ X and U family in X. Then st(E, U) ⊂ f −1[st(f(E), f(U))]

Proof. Let x ∈ st(E, U). Therefore x ∈ U with U ∈ U,U ∩ E 6= ∅. These conditions give us that f(x) ∈ f(U), f(U) ∈ f(U), and f(U) ∩ f(E) 6= ∅. By deﬁnition, f(x) ∈ st(f(E), f(U)). and thus x ∈ f −1(st(f(E), f(U)))

Let > 0, U an ls-cover of X. Since f is a coarse equivalence then f(U) is an ls-cover of

Y , and thus we have an ls-cover V0 of Y such that for all probability measures φ on Y there

0 0 0 0 −1 0 exists an E that is V -bounded and φ(∂f(U)E ) < φ(E ). Consider V = f (V ) which is an ls-cover since f is a coarse equivalence.

Theorem 2.4. Let X be a large scale space, Y be a large scale space with coarse ULAµ and

let f : X → Y be a coarse equivalence. Then X has coarse ULAµ

Proof. Now let µ be a probability measure on X. Let φ be the push-forward measure

for µ, a probability measure on Y . Thus there exists an E0 that is V0-bounded and

0 0 −1 0 φ(∂f(U)E ) < φ(E ). Deﬁne E to be f (E ).

Since E0 is V0-bounded, then by construction E is V-bounded Using our lemma, we can show that µ(∂U (E)) < µ(E) directly:

−1 0 −1 0 −1 0 −1 0 µ(∂U (E)) = µ(st(f (E ), U) \ f (E )) ≤ µ(f (st(E , f(U))) \ f (E )) =

18 −1 0 0 0 0 −1 0 = µ(f [st(E , f(U)) \ E ]) = φ(∂f(U)(E )) < φ(E ) = µ(f (E )) = µ(E)

Since µ was chosen arbitrarily, V satisﬁes our properties for all probability measures on X

and thus X has ls − ULAµ.

Theorem 2.5. Let X be a space with a large scale structure. If X is coarsely amenable,

then X has coarse ULAµ.

Proof. Suppose X is coarsely amenable. Let V be a large scale cover on X and > 0. Not

pick a large scale cover U = {Us}s∈S such that for all x ∈ X:

|hor(x, U)| 1 > |hor(st(x, V), U)| 1 +

By rearranging, we get the following inequality:

Let µ be a probability measure on X with ﬁnite support. Note that if we ﬁx x ∈ X, then

Us ∈ hor(st(x, V), U) \ hor(x, U) if and only if ∪x∈Vi Vi ∩ Us 6= ∅ and x∈ / Us which is the

condition for x ∈ ∂V (Us) P Now Consider s∈S µ(∂V (Us)). By summing over x ∈ X, the previous statement gives us

19 Thus there must exist a Ut ∈ U such that

µ(∂V (Ut)) < µ(Ut)

Clearly Ut is U-bounded, and therefore X has coarse ULAµ.

2.3 Coarse Sparsiﬁcation

Metric sparsiﬁcation was introduced by Guoliang Yu to study the operator localization

property on metric spaces.

Deﬁnition 19. Let X be a space. Then X has the metric sparsiﬁcation property

(MSP) if there exists c > 0 such that for all R > 0 there exists S > 0 such that for all

probability measures µ on X there exists a subset Ω ⊆ X equipped with a decomposition

Ω = tiΩi

such that

• µ(Ω) ≥ c

• for all i, diam(Ωi) ≤ S

• for all i 6= j, d(Ωi, Ωj) > R.

In [CTWY07] it is proved that if X has MSP for some constant c > 0, then it has MSP

for any 0 < c < 1. Again, this deﬁnition depends on a metric being deﬁned. Therefore, we

20 will deﬁne coarse sparsiﬁcation for general large scale spaces that only depends on properties

of the large scale structure.

Deﬁnition 20. W is U-disjoint in X if each element of V intersects at most one element

of W. In other words, for W ∈ W, st(W, V) ∩ Wi = ∅ for all W 6= Wi ∈ W

Deﬁnition 21. X is an ls-space. X has coarse sparsiﬁcation if there is c > 0 such that for any ls-cover U of X, there exists an ls-cover V such that for all probability measures µ on X there exists a set Ω ⊂ X with Ω = tiΩi such that:

• µ(Ω) ≥ c

• each Ωi is V-bounded

• st(Ωi, U) ∩ Ωj = ∅ when i 6= j ({Ωi}i≥1 is U-disjoint, an ls-family).

We will show this deﬁnition also passes to subsets and is a coarse invariant, but ﬁrst we must establish a property of U-disjointness.

Proposition 2.5.1. Suppose f : X → Y , U is a cover of X, V coarsens f(U). If W is

V-disjoint in Y , then f −1(W) is U-disjoint.

−1 Proof. Suppose by contradiction that f (W) is not U-disjoint. If not, then there exists W1

−1 and W2 ∈ W such that f (Wi) ∩ U 6= ∅.

−1 Thus Wi = f(f (Wi)) ∩ f(U) 6= ∅. but V coarsens f(U), so ∃V ∈ V such that f(U) ⊂ V and thus W1 ∩ V 6= ∅ and W2 ∩ V 6= ∅ which contradicts the fact that W is V-disjoint in

Theorem 2.6. Let X and Y be large scale spaces and let f : X → Y be a coarse embedding.

If Y satisfies coarse sparsification then X also satisfies coarse sparsification.

21 Proof. Let Y be a space satisfying coarse sparsiﬁcation, and let f : X → Y be a coarse

equivalence. Let U be an ls-cover, µ be a probability measure. Since f is ls-continuous,

then f(U) is uniformly bounded. Now, let V be an ls-cover that coarsens U and deﬁne a

probability measure ψ to be ψ(B) = µ(f −1(B)), the ”pushforward measure”. Now, since Y

0 S∞ 0 has coarse sparsiﬁcation, there is a c > 0 and a family Ω = i=1 Ωi such that

•{ Ωi}i≥0 is V-disjoint, an ls-family

• ψ(Ω) > c

−1 0 Deﬁne Ωi = f (Ωi).

By the lemma, Ωi is U-disjoint, an ls-family, and

µ(Ω) = µ(f −1(Ω0)) = ψ(Ω0) ≥ c.

Therefore, X has coarse sparsiﬁcation.

Theorem 2.7. Let X be an ls-space satisfying the coarse sparsiﬁcation property and let

Y ⊆ X. Then Y also satisﬁes the coarse sparsiﬁcation property.

Proof. Let X satisfy coarse sparsiﬁcation and let Y ⊆ X. Let U an ls-cover of Y , µ be a

probability measure on Y, and let > 0

S∞ Extend U trivially to V by adding one point sets. Thus we have Ω = i=1 Ωi

{Ωi}i≥i is U-disjoint, an ls-family,

µ(Ω) > c. The probability for all points outside of Y is 0, and any element of the cover of

X outside of U is just a point, so Ω ∩ Y satisﬁes our properties for coarse sparsiﬁcation.

22 + In [BNS 12] it was shown that ULAµ is equivalent to metric sparsiﬁcation. Therefore,

for the large scale versions of these properties we also expect the concepts to be equivalent

for the deﬁnitions to be useful.

Theorem 2.8. Let X be a large scale space. X has coarse uniform local amenability with

respect to all probability measures if and only if X has coarse sparsiﬁcation.

Proof. Let X be an ls-space with Coarse Sparsiﬁcation Property. Let U be an ls-cover, > 0,

1 and µ be a probability measure on X. Deﬁne c = 1+ . Thus there exists an ls-cover V for µ

and a set Ω with a decomposition like in the deﬁnition. Since each Ωi are U disjoint, then

X 1 1 X µ(st(Ω , U) ≤ µ(X) = 1 ≤ µ(Ω) = µ(Ω ). i c c i

Thus by a counting argument, there exists some i such that

1 µ(∂ (Ω )) + µ(Ω ) = µ(st(Ω , U)) ≤ µ(Ω ) U i i i c i

which gives us 1 µ(∂ (Ω )) ≤ ( − 1)µ(Ω ) = µ(Ω ) U i c i i

Since each Ωi is V-bounded, we can set E = Ωi to show that X has coarse ULAµ.

Now, assume that X has coarse ULAµ. We will first show that X satisfies the conditions for coarse sparsification property on all probability measures with finite support. Let U be an ls- cover of X. Let µ be a probability measure with finite support. Fix ≥ 0. Thus there exists

a set E and an ls-cover V such that E is V-bounded and µ(∂U (E)) < · µ(E). Deﬁne µ1 = µ

23 and deﬁne F1=supp(µ1). Deﬁne E1 = E ∩ F1 which is V-bounded and µ(∂U (E1)) < · µ(E1)

Deﬁne F2 = F1 \ st(E1, U) and rescale µ1 to µ2 to be a probability measure on F2. Thus there exists some set E2 with µ2(∂U (E2)) < · µ2(E2)

Note that µ2 is just a rescaling of µ, so the following holds

µ(∂U (E2) ∩ F2) < · µ(E2).

Deﬁne F3 = F1 \(st(E1, U)∪st(E2, U)), and continue on in this process until it terminates as

supp(µ) is ﬁnite. Thus we have sets E1, ..., En that is U disjoint, and each Ei is V-bounded.

As well, µ(∂U (Ei) ∩ Fi) < · µ(Ei).

Thus deﬁne Ωi = Ei and ∪Ωi = Ω we have that

X X 1 = µ(X) = µ(F1) = µ(Ωi) + µ(∂U (Ωi) ∩ Fi) < (1 + )µ(Ωi) = (1 + )µ(Ω)

1 Thus we have that µ(Ω) ≥ 1+ . Since works for all probability measures with ﬁnite support,

1 then setting c = 1+ shows that c < µ(Ω).

Now, let ψ be any probability measure on X. Thus for any δ > 0,there exists a probability

measure µ with ﬁnite support such that |ψ(Ω) − µ(Ω)| ≤ δ. Therefore by letting δ tend

towards 0 the same Ω that works for probability measures with ﬁnite support will result

in µ(Ω) ≥ c for all probability measures µ on X. Therefore X has coarse sparsiﬁcation

property.

24 Chapter 3

Property A

3.1 Property A on Large Scale Structures

Property A was deﬁned by Guoliang Yu in [Yu00] to approach the Baum-Connes conjecture.

It can be viewed as a weaker version of amenability and is a suﬃcient condition for many

properties such as coarse embeddability into a Hilbert Space.

Deﬁnition 22. Let X be a uniformly discrete metric space. X has property A if for every

> 0 and R > 0 there exists a collection of ﬁnite subsets {Ax}x∈X ,Ax ⊆ X × N for every x ∈ X, and a constant S > 0 such that

• |Ax4Ay| ≤ when d(x, y) ≤ R |Ax∩Ay|

• Ax ⊆ B(x, S) × N

These conditions make the sets Ax and Ay be almost equal if d(x, y) ≤ R and disjoint

if d(x, y) ≥ 2S. From there Hiroki Sako deﬁned property A in [Sak13] for uniformly locally

ﬁnite coarse spaces.

25 Definition 23. A coarse space (X, C) is said to be uniformly locally finite if for every controlled set T ∈ C satisfies the inequality supx∈X |T [x]| < ∞.

When giving this deﬁnition, Sako states that a metric space with a uniformly locally

ﬁnite coarse structure is called a metric space with bounded geometry.

Definition 24. (Sako Definition) A uniformly locally finite coarse space (X, C) is said to have Property A if for every positive number and every controlled set T ∈ C, there exists a controlled set S ∈ C and a subset AC ⊂ S × N such that

C N C • For x ∈ X, Ax = {(y, n) ∈ X × ;(x, y, n) ∈ A } is ﬁnite

C 2 • ∆X × {1} ⊂ A , where ∆X is the diagonal subset of X .

C C C C •| Ax∆Ay | < |Ax ∩ Ay | if (x, y) ∈ T .

Since there is a one to one correspondence between Coarse structures and Large Scale structures, it would beneﬁt us to have an equivalent deﬁnition of property A for Large Scale structures with bounded geometry.

Deﬁnition 25. Let X be a space and let LS be a Large Scale structure on X with bounded geometry. (X, LS) is said to have Property A if for every > 0 and every uniformly bounded

LS N family U ∈ LS, there exists V ∈ LS and a family of ﬁnite subsets {Ax } of X × such that

LS N • Ax ⊂ st(x, V) ×

LS • (x, 1) ∈ Ax

LS LS LS LS •| Ax ∆Ay | < |Ax ∩ Ay | if y ∈ st(x, U).

26 From an earlier proposition we have that coarse structures and large scale structures are equivalent constructions. As such for our deﬁnition to be useful it should be that if a uniformly locally ﬁnite coarse space (X, C) has Property A, then the induced large scale structure should also have Property A and the other implication should hold as well. We will prove that this is indeed the case.

Theorem 3.1. Let X be a large scale space with bounded geometry. A uniformly locally

ﬁnite coarse space (X, C) has Property A, then the induced large scale structure (X, LS) has

Property A. As well, if (X, LS) has Property A, then the induced coarse structure (X, C) also has Property A.

Proof. First, let X be a large scale space, and suppose there exists a large scale structure, LS such that (X, LS) has Property A. Let C be the coarse structure induced by LS. Let > 0 and T ∈ C. Thus, there exists a uniformly bounded family U ∈ LS with T ⊂ ∪U∈U U ×U. Let

V be the uniformly bounded family satisfying the deﬁnition of Property A for the above and

C LS U. Deﬁne S := ∪V ∈V V ×V which is obviously a controlled set. Deﬁne A := ∪x∈X {x}×Ax .

C N C LS LS Show A ⊂ S × : Given (x, y, n) ∈ A then (x, y) ∈ {x} × Ax or y ∈ Ax ⊂ st(x, V)

Thus, there exists a V ∈ V such that x, y ∈ V . Therefore (x, y) ∈ V × V , and AC ⊂ S × N.

C N LS Show for x ∈ X, Ax = {(y, n) ∈ X × ;(x, y, n) ∈ A} = Ax : Let x ∈ X and consider

C LS LS C (y, n) ∈ Ax. By deﬁnition (y, n) ∈ Ax . Similarly, if (z, m) ∈ Ax , then (x, z, m) ∈ A , and

C C LS therefore (z, m) ∈ Ax, so Ax = Ax

LS C By deﬁnition of Ax , and A we easily get

C N C • For x ∈ X, Ax = {(y, n) ∈ X × ;(x, y, n) ∈ A } is ﬁnite

C 2 • ∆X × {1} ⊂ A , where ∆X is the diagonal subset of X .

27 The third property also follows easily as if (x, y) ∈ T , then there exists U ∈ U with x, y ∈ U.

Thus, y ∈ st(x, U), so we have

and (X, C) has Property A.

Next, assume that (X, C) has Property A then (X, LS) has Property A, where LS is the

large scale structure induced by C. Let > 0 and U be a uniformly bounded cover. Thus

there exists a controlled set T ∈ C such that ∪U∈U U × U ⊂ T . Let S be the controlled set

and A ⊂ S × N satisfying Property A for and T .

Deﬁne V = {Vx}x∈X where Vx = {y ∈ X;(y, n) ∈ Ax, n ∈ N}. It needs to be shown that V is uniformly bounded: given (y, z) ∈ ∪V ∈V V × V . Then, for some x ∈ X, (x, y, n), (x, z, m) ∈

AC ⊂ S × N for some n, m ∈ N. More importantly, (x, y), (x, z) ∈ S meaning (y, x) ∈ S−1

and thus (y, z) ∈ S−1 ◦ S which is a controlled set. Since (y, z) was chosen arbitrarily, we

have that V is uniformly bounded.

C C C Note that st(x, V) = {y ∈ X;(y, n) ∈ Az and (x, m) ∈ Az for z ∈ X}. since (x, 1) ∈ Ax,

C LS C then Ax ⊂ st(x, V. Setting Ax = Ax it is easy to show:

LS N • Ax ⊂ st(x, V) ×

LS • (x, 1) ∈ Ax

28 are true. Finally, if y ∈ st(x, U), then there exists U ∈ U such that x, y ∈ U which means

(x, y) ∈ ∪U∈U U × U ⊂ T and the following holds:

and (X, LS) has Property A.

Theorem 3.2. Let X and Y be large scale spaces with bounded geometry with large scale

structures LSX and LSY respectively. Let f : X → Y be a large scale equivalence. If

(Y, LSY ) has property A, then (X, LSX ) also has property A.

Proof. Let U be a uniformly bounded family of X and let > 0 be given. Let N =

−1 supy∈Y |f (y)|. Let W be the uniformly bounded family of Y and {Bx} the ﬁnite subsets

N −1 of Y × from the deﬁnition of property A for f(U) and N . Finally, let V = f (W) and

for each x ∈ X deﬁne

Ax = {(z, n) ∈ X × N|(f(z), n) ∈ Bf(x)}.

From the deﬁnition of V it holds trivially that Ax ⊂ st(x, V) × N. As well, it is clear that

(x, 1) ∈ Ax.

Finally, we check that if y ∈ st(x, U) then |Ax4Ay| < |Ax ∩ Ay|. If y ∈ st(x, U), then f(y) ∈ st(f(x), f(U)), and

|B 4B < |B ∩ B |. f(x) f(y) N f(x) f(y)

29 From the construction of Ax’s it follows that |Ax4Ay| ≤ N|Bf(x)4Bf(y)| and |Ax ∩ Ay| ≥

|Bf(x) ∩ Bf(y)|. Putting this all together we get

|A 4A | ≤ N|B 4B | ≤ N |B ∩ B | ≤ |A ∩ A |. x y f(x) f(y) N f(x) f(y) x y

and therefore (X, LSX ) has property A.

Since these two deﬁnitions are equivalent, we will drop the superscripts from the

collections {Ax}x∈X . The large scale deﬁnition of property A allows us to approach

relationships with other coarse properties in a more Topological way. As such, we will

use this definition to show that finite asymptotic dimension implies the large scale definition

of property A.

Asymptotic Dimension can be thought of as a large scale geometric version of he covering

dimension. One of the ﬁrst useful deﬁnitions involved the concept of R-multiplicity of a scale

on a metric space. For a scale U, the R-multiplicity of U on a metric space is the smallest

positive integer n such that for every x ∈ X the ball B(x, R) intersects at most n elements of U.

Deﬁnition 26. Suppose that X is a metric space. The asymptotic dimension of X is the

smallest positive integer n such that for every R > 0 there exists a uniformly bounded cover

U with R-multiplicity n + 1. It is denoted by asdimX = n.

From there, asymptotic dimension was generalized for large scale structures.

30 Deﬁnition 27. Let X be a large scale space with large scale structure LS. We say

asdim(X, LS) ≤ n if for every uniformly bounded cover U in LS there exists a uniformly

bounded cover V such that V is a coarsening of U with multiplicity at most n + 1.

Since the trivial cover of X is uniformly bounded, this deﬁnition gives us that there exists a uniformly bounded cover B such that each x ∈ X is contained in at most n + 1 elements of

B. It can be shown that asdim(X, LS) ≤ n if LS can be generated by a uniformly bounded family B such that the multiplicity of B is at most n + 1 [DH06]. In [CDV15] a diﬀerent characterization of asymptotic dimension was given.

Theorem 3.3. The following conditions are equivalent for any coarse space X and any integer N ≥ 0:

1. asdimX ≤ n

2. for every > 0 and every uniformly bounded cover U of X there is a (U, )-partition

of unity f : X → K(n).

3. for every uniformly bounded cover U of X there is a (U, ∞)-partition of unity f : X →

K(n).

As for the metric case, we will show that ﬁnite asymptotic dimension for coarse spaces

implies the general large scale deﬁnition of property A. First we will introduce some

terminology and a lemma. Given a uniformly bounded cover U for all positive integers

n, deﬁne stn(U) in the following way: st0(U) = U stn(U, stn−1(U))

31 Lemma 3.3.1. If y ∈ st(x, U), then st(y, stn−1(U)) ⊆ st(x, stn(U)).

Proof. Let z ∈ st(y, stn−1(U)). Then there exists U ∈ U such that z.y ∈ st(U, stn−2(U)).

Since y ∈ st(x, U) there exists U 0 ∈ U with x, y ∈ U 0. Therefore we have

U 0 ∩ st(U, stn−2(U)) 6= ∅

and thus z ∈ st(U, stn−2(U)) ⊂ st(U 0, st(U, stn−2(U))).

Putting this all together we then have

x, z ∈ st(U 0, st(U, stn−2)) = st(U 0, stn−1(U)) ⊆ stn(U)

which shows z ∈ st(x, stn(U)).

Theorem 3.4. Let (X, LS) be a large scale space. If asdim(X, LS) is ﬁnite, then (X, LS) has property A.

Proof. Let asdim(X, LS) = k. Let > 0 be given and U an arbitrary uniformly bounded

n family. Then st({x}x∈X , st (U)) is a uniformly bounded family. From the deﬁnition of ﬁnite

n asymptotic dimension, let V be a uniformly bounded family such that st({x}x∈X , st (U)) ≺ V with multiplicity at most k + 1. For each V ∈ V arbitrarily pickz ∈ V and denote it zV . and

n deﬁne a uniformly bounded family W = st(st({x}x∈X , st (U)), V)

For each x ∈ X deﬁne Ax in the following way:

m m Ax := {(x, 1)} ∪ {(zV , m) | st(x, st (U)) ∩ V 6= ∅,V 6⊂ st(x, st (U), 1 ≤ m ≤ n}.

32 n Then each Ax is ﬁnite due to the multiplicity of V against st(x, st (U)) is at most k + 1 for

all x ∈ X. As well, it follows simply that Ax ⊆ st(x, W) × N and (x, 1) ∈ Ax. All that

remains is to check what happens when y ∈ st(x, U).

n n Since st({x}x∈X , st (U)) ≺ V there exists V ∈ V with st(x, st (U)) ⊆ V . From the

construction of Ax for this ﬁxed V ∈ mathcalV we have (zV , m) ∈ Ax for 1 ≤ m ≤ n.

Additionally, we have y ∈ V and from our lemma we have (zV , m) ∈ Ay for 1 ≤ m ≤ n − 1.

Therefore, |Ax ∩ Ay| ≥ n − 1.

On the other hand, consider (zV , l) ∈ Ax4Ay. Without loss of generality suppose

l l (zV , l) ∈ Ax \ Ay. Therefore we know that V ∩ st(x, st (U)) 6= ∅ and V 6⊂ st(x, st (U)).

On the other hand, we know that either V ∩ st(y, stl(U)) = ∅ or V ⊂ st(y, stl(U)).

First suppose that V ∩ st(y, stl(U)) = ∅. From the previous lemma we have that

st(x, stl(U)) ⊂ st(y, stl+1(U)), so V ∩ st(y, stl+1(U)) 6= ∅. On the other hand, suppose

V ⊂ st(y, stl(U)). Again, by the above lemma we have V ⊂ st(y, stl(U)) ⊆ st(x, stl+1(U)).

Therefore any (zV , l) can only be contained in Ax \Ay for at most 2 distinct values of l. Each

Ax can only have at most k + 1 distinct zV ’s so |Ax4Ay| ≤ 2(2(k + 1) + 1). Putting both

inequalities together we get that

|A 4A | 4k + 6 x y ≤ |Ax ∩ Ay| n − 1

the numerators is independent of our choice of n, so to all we have to do is choose an n great

4k+6 enough so that n−1 < to satisfy the inequality. Therefore the collection {Ax}x∈X satisﬁes

the requirements and (X, LS) has property A.

33 3.2 Coarse Exactness

For metric space, exactness was introduced by Dardalet and Guentner to replace property

A. It has been shown that for metric spaces with bounded geometry, exactness and property

A are equivalent [HMS14].

Deﬁnition 28. (Exact Metric Space) A metric space X is exact if ∀R > 0 and > 0∃ a partition of unity (φi)i∈I on X subordinated to a cover V = (Vi)i∈I such that:

P •∀ x, y ∈ X with d(x, y) ≤ R, i∈I |φi(x) − φi(y)| ≤ .

• The cover V = (Vi)i∈I is a uniformly bounded family.

Since a Large Scale Space need not be a metric space, we need a new deﬁnition of exact that only relies on uniformly bounded families.

Deﬁnition 29. A space X with a large scale structure is coarse exact if for every uniformly bounded covers U and > 0 there exists a partition of unity (φi)i∈I on X subordinate to a cover V = (Vi)i∈I such that:

P •∀ x, y ∈ X with x, y ∈ U for U ∈ U, i∈I |φi(x) − φi(y)| ≤ .

• The cover V = (Vi)i∈I is a uniformly bounded family.

If X is a metric space, the general deﬁnition of exactness will imply the original deﬁnition.

However, we can now show implications between general coarse properties.

Theorem 3.5. Let X be an ls-space with coarse amenability. Then X is also a coarse exact space.

34 Proof. Let X be a space with coarse amenability. Given a uniformly bounded cover, V and

> 0, deﬁne a uniformly bounded cover U of X such that hor(st(x, V), U) is ﬁnite and

|hor(x, U)| > 1 − |hor(st(x, V), U)| 4

V . Deﬁne ψUs to be the following function:

     0,Us ∩ st(x, V) = ∅  V   ψUs (x) =    1,Us ∩ st(x, V) 6= ∅ 

Then deﬁne the partition of unity to be the following:

ψV (x) φ (x) = Us s P ψV (x) t∈S Ut

Suppose x, y ∈ V for V ∈ V.

X X ψV (x) ψV (y) |φ (x) − φ (y)| = | Us − Us | = s s P ψV (x) P ψV (y) s∈S s∈S t∈S Ut t∈S Ut

V P V V P V V P V V P V X |ψ (x)[ ψ (y)] − ψ (x)[ ψ (x)] + ψ (x)[ ψ (x)] − ψ (y)[ ψ (x)]| = Us t∈S Ut Us t∈S Ut Us t∈S Ut Us t∈S Ut ≤ |(P ψV (x))(P ψV (y))| s∈S t∈S Ut t∈S Ut

35 2 P |ψV (x) − ψV (y)| = s∈S Us Us P ψV (y) t∈S Ut

V V |ψUs (x) − ψUs (y)| = 1 in the following two cases.

• Us ∩ st(x, V) 6= ∅ and Us ∩ st(y, V) = ∅ (x ∈ st(y, V))

• Us ∩ st(x, V) = ∅ and Us ∩ st(y, V) 6= ∅ (y ∈ st(x, V))

Thus either Us ∈ {U | U ∩ st(x, V) 6= ∅, x∈ / U, U ∈ U} or Us ∈ {U | U ∩ st(y, V) 6=

∅, y∈ / U, U ∈ U}. Note that |{U | U ∩ st(x, V) 6= ∅, x∈ / U, U ∈ U}| = |hor(st(x, W), U)| −

|hor(x, U)|.

Without loss of generality let |hor(st(y, V), U)|−|hor(y, U)| ≥ |hor(st(x, V), U)|−|hor(x, U)|.

We then have the following inequality:

And therefore X is an exact space as the support of φs is contained in Us ∈ U, a uniformly

bounded family.

Theorem 3.6. Let X be a large scale space with bounded geometry. If (X, LS) has Property

A, then X is a coarse exact space.

Proof. Let X be a space with a large scale structure and bounded geometry. Let > 0 and

U be a uniformly bounded cover. Thus, for 2 and U, let V be the uniformly bounded cover such that the following holds:

LS N • Ax ⊂ st(x, V) ×

36 LS • (x, 1) ∈ Ax

LS LS LS LS •| Ax ∆Ay | < |Ax ∩ Ay | if y ∈ st(x, U).

Deﬁne a function ψx : X → R to be the following:

ψx(y) = |{x} × N ∩ Ay|

ψx(y) Then deﬁne φx(y) = P . x∈X ψx(y)

We claim that supp(φx) is a uniformly bounded family to prove this, deﬁne Bx = {y ∈ X;(y, n) ∈ Ax} for some n ∈ N Since st({x}x∈X , V)

is a uniformly bounded cover and Ax ⊂ st(x, V) × N, we know the collection {Bx}x∈X is a uniformly bounded cover of X. Next note that supp(φx) = {y ∈ X; x ∈ By} ⊂ st(x, {Bx}x∈X ) since y ∈ By. Therefore each supp(φx) is contained in st(x, {Bx}x∈X ) which implies that {supp(φx)}x∈X is a uniformly bounded family.

Next, we claim that {φx}x∈X is a partition of unity.

X 2 |ψz(x) − ψz(y)| z∈X = X ψz(y) z∈X

37 P First of all, Note that z∈X ψz(x) = |Ax|. Secondly, for a ﬁxed z ∈ X, |ψz(x) − ψz(y)| =

||{z} × N ∩ Ax| − |{z} × N ∩ Ay|| ≤ |{z} × N ∩ Ax∆Ay|

Thus, the above inequality becomes

and therefore, if x, y ∈ U ∈ U, |φi(x) − φi(y)| ≤ .

Therefore, Property A implies Coarse Exactness.

Our next goal is to show that if a large scale space with bounded geometry is a coarse

exact space, then it has coarse ULAµ. However, before we prove this, we will ﬁrst prove some

basic properties of coarse uniform local amenability with respect to all probability measures.

Lemma 3.6.1. Coarse uniform local amenability with respect to probability measures is equivalent to the following deﬁnition:

For each U, > 0 there exists V such that for all probability measures µ on X with ﬁnite support there exists a set E ⊂ X such that:

• E is V-bounded

• the following inequality hold

µ(∂U (E)) < · µ(E)

38 Proof. This follows easily from the proof of coarse equivalence between coarse sparsiﬁcation

and coarse ULAµ.

Lemma 3.6.2. Let X be a space with a large scale structure. If for all uniformly bounded covers U and all > 0 there exists a uniformly bounded cover V such that for all probability

measures µ on X there exists a function l1(X) such that the following holds:

• diam(supp(φ)) is V-bounded.

• the following inequality holds

X X X µ(x) |φ(x) − φ(y)| < µ(x)|φ(x)| x∈supp(µ) y∈supp(µ),y∈st(x,U) s∈supp(µ)

Then X has Coarse Uniform local amenability with respect to probability measures

Proof. Let U be a uniformly bounded cover, > 0 and let µ be a probability measure.

Thus, there exists a uniformly bounded cover, V, independent of µ, and there exists ψ ∈

l1(X) satisfying the conditions above. Deﬁne φ = |ψ| so that φ is a non-negative function.

Take F1 = supp(φ) and deﬁne a sequence of subsets F1 ⊇ F2 ⊇ ... ⊇ Fn such that φ =

Pn i=1 aiχFi for ai a non-negative number, and χ the indicator function. Thus we can rewrite

our inequality to be:

n n X X X X X µ(x) |χFi (x) − χFi (y)| < ai µ(x)|χFi (x)| i=1 x∈supp(µ) y∈supp(µ),y∈st(x,U) i=1 x∈supp(µ)

39 Therefore, for some ﬁxed i, the following inequality must hold.

X X X µ(x) |χFi (x) − χFi (y)| < µ(x)|χFi (x)| = µ(Fi) x∈supp(µ) y∈supp(µ),y∈st(x,U) x∈supp(µ)

Notice that if x ∈ ∂U (Fi), then x ∈ Vj \ Fi with Vj ∩ Fi 6= ∅. Choose y ∈ Vj ∩ Fi and thus

µ(x)|χFi (x) − χFi (y)| = µ(x). If we set E = Fi we get that

X X µ(∂U (E)) ≤ µ(x) |χFi (x) − χFi (y)| < µ(x) x∈supp(µ) y∈supp(µ),y∈st(x,U)

Putting this together with the fact that Fi ⊂ supp(φ), which is V-bounded, we have that X

has coarse ULAµ.

Theorem 3.7. Let X be a space with a large scale structure and bounded geometry. Then

if X is a Coarse Exact space, it has Coarse Uniform Local Amenability for all probability

measures.

Proof. Let X be a space with a large scale structure. Given a uniformly bounded cover U

and > 0. Then there exists a partition of unity on X,(φ)i∈I , subordinated to a cover

V = (Vi)i∈I , such that: P 1. ∀x, y ∈ X with x, y ∈ U for U ∈ U, i∈I |φi(x) − φi(y)| ≤ . 2. The cover V = (Vi)i∈I is a uniformly bounded family.

Let µ be a probability measure with ﬁnite support, and for V let M be the maximum cardinality of any element of st(U, U).

40 We have the following inequalities:

X X X X X X µ(x) |φi(x)−φi(y)| = µ(x) |φi(x)−φi(y)| < i∈I x∈supp(µ) y∈supp(µ),y∈st(x,U) x∈supp(µ) y∈supp(µ),y∈st(x,U) i∈I

X X X X X < µ(x) ≤ M µ(x) = M µ(x)|φi(x)|. x∈supp(µ) y∈supp(µ),y∈st(x,U) x∈supp(µ) i∈I x∈supp(µ)

Therefore, for some ﬁxed i ∈ I the following inequality must hold:

X X X µ(x) |φi(x) − φi(y)| < M µ(x)|φi(x)|. x∈supp(µ) y∈supp(µ),y∈st(x,U) x∈supp(µ)

Corollary 3.7.1. Let X be a large scale space with bounded geometry. If (X, LS) has

Property A, then X has Coarse Uniform Local Amenability with respect to all probability

measures.

41 Chapter 4

Coarse Generalization of Bounded

Geometry

4.1 Bounded Scale Measure

Deﬁnition 30. Given a scale U on X and given a subset A of X, an U-net in A is a maximal subset B of A with respect to the following property: no two elements of B belong to the same element of the scale U.

Deﬁnition 31. X has Bounded Scale Measure if there exists a scale U such that for all scales V there exists a natural number u(V) such that for all V ∈ V any U-net of V has at most u(V) elements.

Theorem 4.1. X has Bounded Scale Measure if and only if there exists a scale U such that for all scales V there exists a natural number n(V) so that for all V ∈ V there is a U-net of

V which has at most n(V) elements.

42 Proof. First suppose there exists a scale U such that for all scales V there exists a natural

number u(V) such that for all V ∈ V any U-net of V has at most u(V) elements. Let V be

a scale of X and let V ∈ V. Take x ∈ V and then choose all points of X x = x1, x2, ..., xk

such that if i 6= j {xi, xj} ∈/ U for all U ∈ U. By construction this is a U-net in V , and

thus k ≤ u(V) as otherwise we have a U-net with more than (V) elements. Thus taking our n(V) = u(V) and the collection {xi}1≤i≤k satisﬁes our condition.

Conversely, suppose there exists a scale W such that for all scales V there exist a natural number n(V) such that for all V ∈ V there is a W-net, B, which has at most n(V) elements.

Deﬁne U = st(W, W), and let V be any scale of X. Let A be any U-net in V and B be the

W-net with |B| ≤ n(V).

We claim that |A \ B| ≤ n(V). By contradiction assume |A \ B| > n(V). By maximality of B, given any xi ∈ A \ B, there exists yi ∈ B such that {xi, yi} ⊂ Wi for Wi ∈ W. If for i 6= j, yi = yj, then {xi, xj} ⊂ st(Wi, W) which is a contradiction. Take a collection {xi} of n(V) elements of A\B, and associate with each xi the unique yi ∈ B and form the collection

{yi} = B. Since |A \ B| > n(V) there exists x ∈ (A \ B) \{xi}1≤i≤n(V), but by construction given any y ∈ B {x, y} ∈/ W for all all W ∈ W. Otherwise, if there exists a y ∈ B with

{x, y} ⊂ W , then y = yi for some i and {x, xi} ⊂ st(Wi, W). This means B is maximal and therefore not a W-net, which is a contradiction. Thus |A \ B| ≤ n(V), and any U-net must have less than 2n(V) + 1 elements which satisﬁes Deﬁnition.

Theorem 4.2. A space X has Bounded Scale Measure if and only if there exists a scale

U such that for all scales V there exists a natural number k(V) such that for all V ∈ V,

k(V) V ⊆ ∪i=1 Ui for a collection Ui ∈ U.

43 Proof. LetX have Bounded Scale Measure, and let V be a scale of X. Let U and u(V) be defined as they are in Definition 1. Define W = st(U, U). Take a U-net of V ∈ V, and call it B. We know |B| ≤ u(V). For each b ∈ B, pick Ub ∈ U such that b ∈ Ub. Define

Wb = st(Ub, U).

u(V) First, we claim that V ⊆ ∪i=1 Wbi .

Let x ∈ V . Then if x ∈ B, by construction there exists a Ux ∈ U such that x ∈ Ux ⊆

u(V) ∪i=1 Wbi . If x∈ / B then by maximality of B there exists b ∈ B such that {x, b} ⊂ U for

u(V) u(V) some U ∈ U. This means x ∈ st(Ub, U) and x ∈ ∪i=1 Wbi . Therefore, V ⊆ ∪i=1 Wbi Letting

W be our designated scale and letting k(V) = u(V) satisﬁes our condition.

Conversely, suppose there exists a scale U such that for all scales V there exists a natural

k(V) number k(V) such that for all V ∈ V, V ⊂ ∪i=1 Ui for a collection Ui ∈ U. Let V be an arbitrary scale of X. LetV ∈ V and let B be a U-net of V .

We claim that |B| ≤ k(V). By contradiction assume |B| > k(V). We know there exists

k(V) Ui ∈ U such that V ⊆ ∪i=1 Ui. Given any x, y ∈ B, then x, y∈ / Ui for all 1 ≤ i ≤ k(V), but

each x ∈ B must be in some Ui as the Ui’s cover V . For each Ui, assign the unique xi ∈ B such that xi ∈ Ui. By assumption |B| > k(V), so there exists x ∈ B \{xi}1≤i≤k(V). Since B is a U-net, then x∈ / Ui for all i with 1 ≤ i ≤ k(V). However this means x∈ / V and thus B is not a U-net which is a contradiction. Therefore for any U-net in V , |B| ≤ k(V). Setting u(V) = k(V) and letting U from Definition 3 be our designated scale satisfies Definition.

Proposition 4.2.1. Let X be a space with bounded scale measure. Then any subset of X

also has bounded scale measure.

44 Proof. Let X be a space with bounded scale measure, and let Y be a subset of X. Let U 0 be the scale of X. Deﬁne a new scale of Y to be U = {U 0 ∩ Y |U 0 ∈ U 0}. Let V be a scale

0 of Y and for V ∈ V, let BV be a U-net of V . Deﬁne V to be the trivial extension of V to

a scale of X and u(V0) be the natural number from the deﬁnition of bounded geometry for

X. If V 0 ⊂ X \ Y , then V must be a point and a U 0-net must also be a point. Otherwise

V 0 is an element of V, and a U-net of V 0 is also a U 0-net of V 0. This is because given any

0 0 0 x, y ∈ BV 0 ⊂ Y , we know that for all U ∈ U , x, y∈ / U ∩ Y . Note that x, y ∈ Y , so therefore

x, y∈ / U 0 for all U 0 ∈ U 0. Therefore, for any scale V of Y , a U-net B is a U 0-net and by assumption, |B| ≤ u(V0). Thus for any scale V of Y , setting u(V) = u(V0), where V0 is the trivial extension of V over X, shows Y has bounded scale measure.

Theorem 4.3. Let X and Y be ls-spaces and let f : X → Y be a large scale equivalence.

Then X has Bounded Scale Measure if and only if Y has Bounded Scale Measure.

Proof. Let X be a space with bounded scale measure. Let U be the scale from the deﬁnition

of bounded geometry. Letf : X → Y be a scale equivalence. Therefore f(U) is a scale in

Y . Now, let V be any scale of Y . For arbitrary V ∈ V let BV be a f(U)-net of V . We

−1 −1 −1 know f (V) is a scale in X. We will show f (BV ) is a U-net of f (V ). By contradiction

−1 assume there exists x, y ∈ f (BV ) such that there is some U ∈ U with x, y ∈ U. However,

this implies f(x), f(y) ∈ f(U), but f(x), f(y) ∈ BV which contradicts that BV is a f(U)-

−1 −1 net. Next assume there is some z ∈ f (V ) \ f (BV ) such that for all x ∈ BV there

does not exist a U ∈ U with x, z ∈ U. However, this implies there is not f(U) ∈ f(U)

with f(x), f(z) ∈ f(U). By maximality of BV , f(z) ∈ B which contradicts that fact that

−1 −1 −1 z∈ / f (BV ). Therefore f (BV ) is a U-net of f (V ). Since X has bounded geometry we

45 −1 −1 −1 know there exists a natural number u(f (V)) such that |f (BV )| ≤ u(f (V)). As well,

−1 −1 since f is a function |BV | ≤ |f (BV )|. Finally, setting u(V) = u(f (V)) gives us that for

any U-net of V ∈ V has at most u(V) elements and thus Y has bounded scale measure.

Conversely, let Y be a space with Bounded Scale Measure and let f : X → Y be a large scale

equivalence function. Let U be a scale in Y such that for all scales V of Y there exists a natural

k(V) number k(V) so that for all V ∈ V, V ⊆ ∪i=1 Ui for a collection Ui ∈ U. Since f is a large scale

equivalence, then f −1(U) is a uniformly bounded cover of X. Take any uniformly bounded

cover V of X. Then f(V) is a cover of Y . From the deﬁnition, there exists a collection of

k(f(V)) −1 k(f(V)) −1 elements Ui ∈ U with f(V) ⊂ ∪i=1 Ui. Thus V ⊂ f (f(V )) ⊂ ∪i=1 f (Ui). Since

V was chosen arbitrarily, f −1(U) with k(V) = k(f(V)) satisﬁes Deﬁnition 3, and X has

bounded scale measure.

Proposition 4.3.1. Let X be a large scale space. If X has bounded geometry, then X has bounded scale measure.

Proof. Deﬁne U to be the trivial cover of X. Let V be any ls-cover of X. Since X has M [V bounded geometry, then for any V ∈ V, |V | ≤ MV . Therefore, V = xi, and letting i=1 k(V) = MV gives us X has bounded scale measure.

4.2 Decomposition of ls-spaces

Decomposition complexity is a generalization of ﬁnite asymptotic dimension. The metric

space deﬁnition of asymptotic dimension, given in [PWY12], was used to show the following

result:

For every R > 0 there exists n + 1 families Ui of subsets of X, i = 0, 1, ..., n and S > 0 such

46 that each Ui is R-disjoint, S-bounded and the families Ui cover X.

This concept gave rise to ﬁnite decomposition complexity and asymptotic property C

amongst other deﬁnitions. Similarly, the large scale generalization gives rise to a pseudo-

metric deﬁnition on a coarse space and the following results:

Proposition 4.3.2. Suppose LS is a large scale structure on a set X. If n ≥ 0, then the

following conditions are equivalent:

1. for every uniformly bounded family B in X there is a uniformly bounded family B0 on

X of which B is a reﬁnement such that the multiplicity on B0 is at most n + 1.

2. for every uniformly bounded family B in X there is a decomposition of X as X0 ∪...∪Xn

such that the family of B-components of each Xi is uniformly bounded.

We now show that large scale spaces that satisfy a nice star property have a similar decomposition of the space.

Theorem 4.4. Let X be a large scale space and U be a large scale cover. If every element

of the third star of U can be covered by n elements of U, then

X = X1 ∪ ... ∪ Xn, where each Xi is a union of U-disjoint subsets that reﬁnes st(U, U)

Proof. This will be proved by induction using the following sets; Deﬁne Kn to be the maximal subset of U such that U, V ∈ Kn if st(U, U) and st(V, U) are U-disjoint. To show such a set exists, we will use Zorn’s lemma with a partial order of inclusion. Suppose we have a totally ordered chain of subsets Ci such that U, V ∈ Ci if st(U, U) and st(V, U) are U-disjoint. We

will show that C = ∪Ci is an upper bound of the chain that satisﬁes our properties. Let

U, V ∈ C.Since the chain of Ci’s are totally ordered, then there exists some Cj such that

47 U, V ∈ Cj. Thus st(U, U) and st(V, U) are U-disjoint. Therefore, a maximal element of X

exists with respect to stars of the elements being U-disjoint. Deﬁne Xn = ∪V ∈Kn st(V, U).

For the base case, suppose every element of the third star of U can be covered by 1 element of U. Deﬁne K1 to be the maximal subset of U such that U, V ∈ Kn if st(U, U) and st(V, U)

are U-disjoint, and deﬁne X1 = ∪V ∈Kn st(V, U).

We will show that X = X1. Let x ∈ X. Since U is a cover of X, then for some U ∈ U, x ∈ U. If U ∈ K1, then x ∈ X1. If U/∈ K1, then for some V ∈ K1, st(U, U) ∩ st(V, U) 6= ∅.

Moreover, st3(U, U) ∩ V 6= ∅. Since st3(U, U) can be covered by 1 element, say U 0 of U, then

0 3 0 U ∩ V 6= ∅, and x ∈ st (U, U) ⊂ U ⊂ X1 and we have shown X = X1.

For the inductive step suppose that every element of the third star of U can be covered by n elements of U. We will show that for Xn as deﬁned above, X \ Xn has the property that each third star of U can be covered by n − 1 elements of U restricted to X \ Xn.

3 n Let U ∈ U restricted to X \ Xn. From the hypothesis we know st (U, U) ⊂ ∪i=1Ui with

Ui ∈ U. Since U/∈ Kn, then there exists V ∈ Kn such that st(U, U) ∩ st(V, U) 6= ∅,which

3 n implies st (U, U) ∩ V 6= ∅. Therefore, (∪i=1Ui) ∩ V 6= ∅ and for some 1 ≤ i ≤ n, Ui ∩ V 6= ∅ or Ui ⊂ st(V, U) ⊂ Xn. Since U ∈ X \ Xn, then U ⊂ U1∪, ..., ∪Ui−1 ∪ Ui+1∪, ... ∪ Un and therefore X \ Xn has the property that each third star of U can be covered by n − 1 elements of U restricted to X \ Xn.

Thus by induction X \ Xn can be split into n − 1 subsets, where each Xi is a union of

U-disjoint subsets that reﬁnes st(U, U). Thus X = X1 ∪...∪Xn−1 ∪Xn with each Xi a union of U-disjoint sets that clearly reﬁne st(U, U).

48 Clearly spaces with bounded scale measure has the property listed above, and therefore a space having bounded scale measure can be decomposed in this way.

This leads us to ask about connections, if any, between bounded scale measure, ﬁnite asymptotic dimension and asymptotic property C.

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52 Vita

Kevin Sinclair was born in Hammond, Indiana to John Sinclair and Vernice Sinclair. He grew up in Los Alamos, New Mexico and Knoxville, Tennessee. He attended the University of Tennessee for undergraduate where decided to be a mathematics major and became enamored with pure mathematics. After graduating and some complications he decided to go back to UT for his Ph.D. in mathematics. There, he went on to focus on Topology under the guidance of Dr. Dydak.