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# Discrete Groups from a Coarse Perspective

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page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTIONANDBASICDEFINITIONS .................. 7 2THEGRIGORCHUKGROUP ........................... 15 3COUNTABLEGROUPS ............................... 17 4THESUBLINEARCOARSESTRUCTURE .................... 22 5THEDIMENSIONOFTHESUBLINEARHIGSONCORONA ......... 29 REFERENCES ....................................... 36 BIOGRAPHICALSKETCH ................................ 38 5

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12 ].Thoughthisquestionisyettoberesolved,wedopresentapartialresult.WeprovethatAN-asdimGZ=AN-asdimG+1fornitelygeneratedgroupsG.Onecanindeedthinkofthisasapartialresult,sinceDydaketal.[ 6 ]showedthat,givenanitelygeneratedgroupG,thereisaleft-invariantpropermetricd0onGsuchthatasdimG=AN-asdim(G;d0). 19 ]Ametricspace(X;d)issaidtohaveasymptoticdimensionn,denotedasdim(X;d)n,ifforeachr>0,thereisanS>0andrdisjoint,SboundedfamiliesU0;U1;:::;UnofsubsetsofXsuchthatU:=[iUiisacoverofX.Notethat,aswillbethecaseforthefollowingdimensions,wesaythatasdimX=nifasdimXnyetasdimXn1doesnothold.IfasdimXnfailstobetrueforeach 8

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23 ].Let(X;dX)and(Y;dY)bemetricspaces.Wesaythatamapf:X!Yis(metrically)properifthepreimageofeachboundedsetisbounded.Itisbornologousif,foreachR>0,thereisanS>0suchthatdY(f(x);f(y))0,C0,andD0suchthat1 11

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23 ].

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23 ]. 15 ].

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17 ],[ 18 ],and[ 21 ].ThisGrigorchukgroup,whichwewilldenoteby,hasmanyinterestingproperties.Itisanitelygenerated2groupwithintermediategrowth,whosewordproblemissolvable,andwhichdoesnotadmitanitedimensionallinearrepresentationthatisfaithful.Also,andarecommensurable,whichmeansthatandhavesubgroupsofniteindexwhichareisomorphic.Adetailedexpositioncanbefoundin[ 21 ].Weprovethathasasymptoticdimensioninnity,asdim=1.IfoneexcludesGromov's\randomgroups"[ 20 ],allpreviouslyknownexamplesofgroupsGwithasdimG=1arebasedonthefactthatGhasafreeabeliansubgroupofarbitrarylargerank.TheGrigorchukgroupisofdierentnature:sinceisa2group,itdoesnothavea(nontrivial)freeabeliansubgroup.WhenGandHaregroupsequippedwithwordmetricsdGanddH,thenthesummetricdG+dHonGHisalsoawordmetric(forthenaturalgeneratingset).IfHGisasubgroupofniteindexof(thenitelygeneratedgroup)G,thentheinclusionmapH!Gisacoarseequivalence.Also,anisomorphismisacoarseequivalence. 25 ]LetGbeanitelygenerated,innitegroupwhichiscommensurablewithitssquareGG.ThenasdimG=1. Proof. 15

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21 ],thereisanisometricembeddingf:Z!G,whereGistakenwithawordmetric.Thus,foreachn1,wehaveanisometricembeddingfff:Zn!Gn,wherewetakethesummetricsonZnandGn.Sinceanisometricembeddingisacoarseembedding,wehavethatasdimGnasdimZn=n.Thus,asdimGnforalln. Proof. 21 ]showthatsatisesthehypothesesofthetheorem. Itisinterestingtonotethatasdim=1,yetdoesnotcontainanisomorphiccopyofZn.However,Zndoescoarselyembedinto.Also,ifonehasanitelygeneratedgroupwhichisknowntobecommensurablewithitssquare,thentheasymptoticdimensioniseither0orinnity,dependingonwhetherthegroupisniteorinnite. 16

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3.1 ,onecanextendtheinvariantasdimtoallcountablegroups(notnecessarilynitelygenerated).Moreexplicitly,foracountablegroupG,wedeneasdimG=asdim(G;d),wheredisaleft-invariant,propermetriconG.Propositions 3.1 and 3.3 showthatasdimGiswell-dened.Forcountablegroups,notethataleft-invariant,propermetricinducesthediscretetopology(BaireCategoryTheorem);consequently,boundedsetsarenite.Thefollowingappearsin[ 23 ,Proposition1.15],[ 14 ,Propostion1.1],and[ 24 ,Proposition1]. 14 ,Proposition1.3],[ 24 ,Theorem1]Aweightfunctiononthecountablegroupinducesapropernormkkgivenbykxk=inffnXi=1w(si)jx=s1s2sn;si2Sg;andsoaweightfunctioninducesaleft-invariant,propermetricd. Proof. 17

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Notethattheinmuminthedenitionofkxkisactuallyaminimum.Toseethis,simplymodifytheargumentinthelastparagraphtoshowthatthesetofelementsoffPni=1w(si)jx=s1s2sn;si2Sglessthankxk+1isaniteset.Thefollowingtheoremgivesanecessaryandsucientconditionforacountablegrouptohaveasymptoticdimensionzero.Thisconditionreliesonlyonthealgebraicstructureofthegroup.Thefollowing(anditscorollaries)canbefoundin[ 24 ]. Proof. 18

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Thefollowingcorollariesareimmediateconsequences. 14 ].

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3.8 impliesinparticularthatasdimQ=1.WerecallthenotionoftheR-stabilizerfrom[ 2 ]. 4 ]. 20

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3.8 incombinationwiththecorrespondingTheoremsforthecasewhenthegroupsarenitelygenerated.Fordetailssee[ 14 ].Sincethenotionofasymptoticdimensionatthetimewhen[ 4 ]waswrittenwasdenedonlyfornitelygeneratedgroups,KistreatedinTheorem7of[ 4 ]asametricspacewiththewordmetricrestrictedfromG. 21

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Proof. 22

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Also,forx0;x12X,itisnotdiculttoshowthatlimx!1supy2Exd(y;x) 15 ,Proposition2.1]LetXandYbepropermetricspaces.Iff:X!Yisaquasi-isometry,thenitisacoarseequivalencewithrespecttothesublinearcoarsestructures. Proof. )[K0.Notethatf(K)isbounded.Supposethat 23

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kzk=00wasarbitrary,(ff)(E)iscontrolledandsofisacoarsemap.Similarly,sincegisaquasi-isometry,itisacoarsemapaswell.Finally,itisclearthatfgandgfareclosetothecorrespondingidentitieswhenXandYareequippedwiththesublinearcoarsestructures.Thus,fisacoarseequivalence. 15 ,Lemma2.3]Let(X;d)beapropermetricspacewithbasepointx0,endowedwiththesublinearcoarsestructureEL.ForanitesystemE1;:::;EnofsubsetsofX,thefollowingareequivalent.1.LX\[\ni=1 Proof. 4mandr0=2m,thenthereisanxmsuchthatkxmk2myetmax1ind(xm;Ei)<1 4mkxmk:

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4mkxmk,andsothereisanaim2Eisuchthatd(xm;aim)<1 4mkxmk.Wehaved(aim;ajm)<1 2mkxmk.Also,kxmkkaimk+d(aim;xm)1 2kxmk.Thus,d(aim;ajm)<1 2mkxmk<1 2kxmkmforall1i;jn.TakeFi;j=f(aim;ajm):m=1;2;:::gforeach1i;jn.Fixingi;j,wetemporarilysetG=Fi;jforconvenience,andshowthatGiscontrolled.Sincekaimk!1andkajmk!1asm!1,itfollowsthatGisproper.Nowlet>0begiven,andtakeMtobeapositiveintegerforwhich1=M<;setK=fajm:1m
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Proof. 4.5 above,istrueifandonlyifLisatleastlinear.

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15 ,Theorem2.11]Let(X;d)beapropermetricspace.Thenthecompactication

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Proof. 15 ]. IfUisacoverofXandAX,thenwewriteUA=fU2U:U\A6=?g.

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13 ]. 15 ,Theorem3.10]Let(X;d)beapropermetricspace.ThendimLXAN-asdimX. Proof. D)ir0fori1.Observethatri+1=C Dri>ri>r0>1.Sinceri>r0>r1fori1,thereisacoverUiofXsuchthatmultUin+1,meshUiri.DeneV1=U1,andnotethatmeshV1
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D)ri1=DriDkxkLU(x);andsothereisaU2UwithWU.WeshowthatLW:X![0;1)isatleastlinear.Seta=3r2,andletxbeanelementofXwithkxka=3r2.Seti=maxfj:3rj+1kxkg,andnotethati1and3ri+1kxk<3ri+2.AsLUi>ri,thereisaU2UisuchthatB(x;ri)U.Sincekxk3ri+1anddiamUmeshUir2,wehave 31

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Sinceanitelygeneratedgroupisquasi-isometrictoitsCayleygraph,acocompactgeodesicmetricspace,onehasthefollowingcorollary. 15 ,3.8]Foranitelygeneratedgroupwithwordmetric,dimL=AN-asdimprovidedAN-asdim<1. 5.1 thatthespacebecocompact. 15 ,Lemma4.2]LetXbeapropermetricspace.Thenthereisanembed-dingLX[0;1],!L(XR+): 32

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kxk+jzjjkwkkxkj kxkkwkjyzj kxk+jkwkkxkj kxkd((x;y);(w;z)) 33

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1.12 thatthereisaclosedsetAXsuchthatz2 4.5 ,thereareC;r0>0suchthatL(u):=maxfd(u;A);d(u;Ot)gCkukwheneveru2Yandkukr0.Inparticular,for(x;y)2A,wehavejytkxkj=d((x;y);(x;tkxk))d((x;y);Ot)maxfd((x;y);Ot);d((x;y);A)gCk(x;y)k=C(kxk+jyj);andsoj(x;y)tj=jy

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4.10 andthecommentsaboveconcerningtheextensionof.Thus,(z)6=t,contradictingz2f1(LXt).Toproveinjectivity,assumef(z)=f(z0)forsomez;z02Y.Lett=(z)(=(z0)).Thenbytheprecedingfact,wehavez;z02f1(LXt)=L(Ot).ButfjLOtisahomeomorphism;consequently,z=z0. 15 ,Theorem4.3]LetXbecocompactconnectedpropermetricspace.ThenAN-asdim(XR)=AN-asdimX+1: 5.3 ,Lemma 5.6 ,theclassicalMoritatheorem,andbyTheorem 5.1 weobtainAN-asdim(XR)dimL(XR)dim(LX[0;1])=dimLX+1AN-asdimX+1:Theoppositeinequalityiswell-established. Againnotingthatanitelygeneratedgroupisquasi-isometrictoitsCayleygraph,wehavethefollowing. 5 ]thatAN-asdimG=dimANGfornitelygeneratedgroups.Consequently,forthesespaces,theformulagivenaboveholdsfordimAN:dimANGZ=dimANG+1.Asmentionedintheintroduction,alongwithresultsofDydaketal.[ 5 ],thisresultinclinesonetobelievethatasdimGZ=asdimG+1fornitelygeneratedgroupsG. 35

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R.I.Grigorchuk,Bernside'sproblemonperiodicgroups,FunctionalAnal.Appl.14(1)(1980)41{43. [18] R.I.Grigorchuk,AnexampleofanitelypresentedamenablegroupnotbelongingtotheclassEG,SbornikMath.189(1)(1998)75{95. [19] M.Gromov,Asymptoticinvariantsofinnitegroups,GeometricGroupTheory,vol.2,CambridgeUniversityPress,1993. [20] M.Gromov,Randomwalkinrandomgroups,Geom.Funct.Anal.13(1)(2003)73{146. [21] PierredelaHarpe,Topicsingeometricgrouptheory,ChicagoLecturesinMathematics,UniversityofChicagoPress,2000. [22] U.Lang,Th.Schlichenmaier,Nagatadimension,quasisymmetricembeddings,andLipschitzextensions,Int.Math.Res.Not.58(2005)3625{3655. [23] J.Roe,Lecturesoncoarsegeometry,UniversityLectureseries,vol.31,AMS,2003. [24] J.Smith,Onasymptoticdimensionofcountableabeliangroups,TopologyAppl.153(12)(2006)2047{2054. [25] J.Smith,TheasymptoticdimensionoftherstGrigorchukgroupisinnity,RevistaMatematicaComplutense,inpress. [26] G.Yu,TheNovikovconjectureforgroupswithniteasymptoticdimension,Ann.ofMath.147(2)(1998)325{355. 37

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Full Text

DISCRETE GROUPS FROM A
COARSE PERSPECTIVE

By
JUSTIN I. SMITH

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

S2007 Justin I. Smith

To the memory of Dawn Rogers Smith

ACKENOWLED GMENTS

I wish to thank my adviser, Alexander Dranishnikov, for his support, numerous

enlightening discussions, and his patience while I was struggling.

I would also like to thank the Department of Mathematics for the supportive

atmosphere they have provided. I am also grateful to the College of Liberal Arts and

Sciences for honoring me with the CLAS Dissertation Fellowship in my final semester.

pagfe

ACK(NOWLEDGMENTS .......... . . 4

ABSTRACT ............ .......... .. 6

CHAPTER

1 INTRODUCTION AND BASIC DEFINITIONS .... .. .. .. 7

2 THE GRIGORCHUK( GROUP ........ .. 15

3 COUNTABLE GROUPS ........ .. .. 17

4 THE SUBLINEAR COARSE STRUCTURE .... .... 22

5 THE DIMENSION OF THE SUBLINEAR HIGSON CORONA .. .. .. .. 29

REFERENCES ......._._.. ......._ .. 36

BIOGRAPHICAL SK(ETCH ....._._. . 38

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DISCRETE GROUPS FROM A
COARSE PERSPECTIVE

By

Justin I. Smith

May 2007

Cl.! I!1-: Alexander N. Dranishnikov
Major: Mathematics

We first use the functorial properties of coarse structures and coarse maps as well as

algebraic properties to deduce that the .I-i-inidllicl~ dimension of the Grigorchuk group

is infinity. Next, by thinking in terms of coarse invariants rather than quasi-isometry

invariants, we are able to extend the notion of a word metric to all countable groups. We

describe nice relations between the .I-i-mptotic dimension of the countable group and the

.I-i-inidlli'lc dimensions of its finitely generated subgroups. Finally, using the so-called

sublinear coarse structure, we are able to show that, for a large class of proper metric

spaces including finitely generated groups, the .I-i-11.1.1..1ic Assouad-N I, ii .. dimension can

be realized as the dimension of the sublinear Higson corona. Using this, we prove that

crossing a space with the real numbers lifts the .I-i-inid l i'lc Assouad-N I, ii .. dimension by

one.

CHAPTER 1
INTRODUCTION AND BASIC DEFINITIONS

Introduction. An early example of neglecting local properties in favor of large-scale

geometry is Legendre's attempted proof of the parallel axiom from Euclidean geometry.

Attempting to show a contradiction in assuming the existence of a triangle whose angles

sum to less than xr, he constructed larger and larger triangles whose angle sums decreased

at a steady rate. Since this process could not go on indefinitely, he concluded that such

triangles could not exist, and consequently that the parallel axiom must hold. The

argument is sketched in more detail in [23] and [16]. Though the argument is ultimately

incorrect, it invites one to further investigate the large-scale properties of spaces.

One of the first invariants of coarse topology was the notion of the ends of a space,

introduced by Fr-eudenthal in 1931. Hopf investigated the ends of finitely generated

groups, showing among other things that such groups have 0, 1, 2, or infinitely many ends.

Stallings later gave an alternate characterization of groups with infinitely many ends.

In this paper we will also investigate coarse and quasi-isometry invariants of finitely

generated (and countable) groups. The quasi-isometry invariants we will look at are the

.I-i-inidlli'lc dimension and the .I-i-mptotic Assouad-N~ I, i I dimension.

The notion of .l-i-in!llantic dimension was introduced by Gromov in order to study

large-scale invariants of discrete groups [19]. A striking application of .l-i-injull'tic

dimension came when G. Yu proved that, for certain finitely generated groups with

finite .I-i-in!llantic dimension, the Higher Novikov Signature Conjecture holds [26]. Other

conjectures were proved for spaces with finite .I-i-mptotic dimension (under various

other conditions), including the coarse Baum-Connes conjecture [26], the K(-theoretic

integral Novikov conjecture [9, 10], and the integral Novikov conjecture [11]. Based on

these results, it is important to have techniques for determining if a space has finite

.I-i-inidlli'lc dimension. One technique for obtaining results about finite dimensionality

is to try to prove theorems analagous to those found in classical dimension theory. For

instance, it has been shown that the product and (finite) union of spaces with finite

.I-i-inidlli'lc dimension also have finite .I-i-mptotic dimension. Included among spaces of

finite .I-i-in!llantic dimension are free groups, free abelian groups of finite rank, hyperbolic

groups, and Coxeter groups.

There are two recurrent themes in this treatise. One is the use of coarse structures

in the investigation of large-scale dimensions on discrete groups. The other is the search

for techniques to find lower bounds on the .I-i-inidlli'lc dimension of the space. The

Grigorchuk group example is an easy instance of this, while the product formula given

towards the end is a more difficult one. A specific problem which motivated some of

the work in this paper, particularly the work with the sublinear coarse structure, is the

following: for a finitely generated group G, is it true that asdimG x Z = asdimG + 1 holds?

It is known that this formula does not hold for all proper metric spaces, i.e. there is a

proper metric space X with asdimX x Z = asdimX [12]. Though this question is yet to be

resolved, we: do present~ a partial result. Wet prove: thlat AN-asdimGlc x Z =AN-asdimGc + 1
forM finitely- generated-- groups--- G One can indeed think of this as a partial result, since

Dydak et al. [6] showed that, given a finitely generated group G, there is a left-invariant

proper metric d' on G such that asdimG = AN-asdim(G, d').

Large-scale dimensions. We start by defining three dimensions, each defined for

metric spaces, which will occupy our attention throughout this treatise. The difference

between these definitions is the restriction placed on the obtained covers. We ;?i- that a

family V of subsets of X is R-disjoint if d(U, V) > R for all U, Ve E with U / V. V is

said to be S-bounded if diamV/ < S for all Ve E and is said to be uniformly bounded if

it is S-bounded for some S.

Definition 1.1. [19] A metric space (X, d) is said to have r onpl~~ .-H.: dimension < n,

denoted asdim(X, d) < n, if for each r > 0, there is an S > 0 and r-disjoint, S-bounded

families Uo,UI,...,Un of subsets of X such that U := UiUi is a cover of X.

Note that, as will be the case for the following dimensions, we ;?i- that asdimX = n if

asdimX < n yet asdimX I n 1 does not hold. If asdimX I n fails to be true for each

n > 0, then we ;?i asdimX = 00. Also, we simply write asdimX rather than asdim(X, d)

if the metric is understood.

Definition 1.2. A metric space (X, d) is said to have r; 1,,1l..H.:: Assonad-Nagata

dimension < n, denoted AN-asdimX < n, if there exists a C > 0 and a D > 0 such that,

for each r > 0, there are r-disjoint, (Cr + D)-bounded families Uo,U1,...,U, of subsets

of X such that U := UiK is a cover of X.

Definition 1.3. A metric space (X, d) is said to have Assonad-Nagata dimension < n,

denoted dimAN X < n, if there exists a C > 0 such that, for each r > 0, there are

r-disjoint, Cr-bounded families Uo,U1,...,U, of subsets of X such that U := UiN is a

cover of X.

This construction was used by Assouad in [1] to resolve a question of T I, ii I Many

properties of asdim have been verified for this .I-i-inph.1ic~ Assouad-N .4 1 dimension and

the Assouad-T I, ii I dimension. For instance, it was shown in [6] that AN-asdimX x Y <

AN-asdimX + AN-asdimY for metric spaces X and Y. They also proved a similar result

for dimAN, which was also proved in [22]. Also, we note that asdimX < AN-asdimX <

dimAN X. For more details, see the papers of Lang and Schlichenmaier [22], Buyalo and

Lebedeva [7], [8], and Dydak, Brodskiy, Levin, Mitra, and Higes [5], [6].

These definitions can be formulated in many v-ars~. In the case of the .I-i-mptotic

Assouad-T I, ii I dimension, the following proposition gives some alternative characterizations.

Proposition 1.4. Let (X, d) be a metric space. The following are equivalent.

1. AN-asdimX < n,

2. there exists a C > 0 and an ro > 0 such that, for each r > ro, there are r-disjoint,
Cr- bounded families Uo, MI, ., of subsets of X such that U := Uisk is a cover of
X;

3. there is a C > 0 and an eo > 0 such that for all e < co (e > 0), there is an
e-Lip~schitz, C/e-cobounded map p : X P to an n-dimensional simp~licial complex;

4.there is a C > 0 and an ro > 0 such that, for all r > ro, there is a cover U of X such
that meshU < Cr, multU < n + 1, and Lu > r,

5. there is a C > 0 and an ro > 0 such that for all r > ro, there is a cover U of X such
that meshU < Cr and B, (x) meets at most n + 1 elements of U for each x EX .

By multU < m we mean that every intersection of m + 1 distinct elements of

U is empty. Also, meshU = supu,, diam(Ui). Here Lu denotes the Lebesgue number

of the cover, Lu = inf~supu,, d(x, X \ U) |x EX }. The relevant definitions (for

the third statement, in particular) can be found in either [2], [3], [4], or [15]. This

proposition can be modified to obtain statements for the .I- ...ph.1ic~ dimension as well

as the Assouad-N I, ii .. dimension.

Word Metrics. We will focus mostly on proper metric spaces in this treatise.

Recall that a proper metric space is a metric space for which closed, bounded sets are

compact. Also, a metric d on a group G is said to be left-invariant if d( fg, f h) = d(g, h)

for all f, g, h E G. More generally, a group acts on a metric space by isometries if

d( fx, fy) = d(x, y) for all f E G, x, y E X.

Let G be a group. We recall that a map |I | | : G [0, 00) is said to be a norm on G

if ||X-l|| = ||X|| for all x e G, ||X|| = 0 if and only if x = e, and |, 11|| < ||X|| + ||y|| for all

x, yE G. Norms on a group G yield left-invariant metrics: given a norm || || on G, define

a metric by d(x, y) = ||X-1y||. We call a norm proper if it is proper as a map (where G

has the topology induced by the associated metric). Note that proper norms correspond to

proper metrics.

Let G be a finitely generated group with finite generating set S. We define ||X|| =

inf~n | x = y172 y,, yi E SU S-l}. || || is a proper norm. If we define ds(x, y) = ||X-1y||,

then ds is a left-invariant, proper metric on G, called the word metric on G associated

with S. It should perhaps be explicitly mentioned here that finitely generated groups with

word metric are discrete, and hence proper here means that bounded sets are finite. On

finitely generated groups, word metrics form an important source of left-invariant, proper

metrics.

Definitions for Metric Spaces. 1\ost of the definitions to follow in this chapter can

he found in [23]. Let (X, dx) and (Y, dy) be metric spaces. We ;?i- that a map f : X Y

is (me l,..~ all~i) proper if the preimage of each hounded set is bounded. It is bornologous if,

f or each R > 0, there is an S > 0 such that d ( f (:), f (Y)) < S whenever dx (:, y) < R. A

map is called corerse if it is both proper and homnologous. If S is a set, then f, y : S X

are said to be close if sup, ad(f(s), g(s)) < oc.

A coarse map f : X Y is said to be a coarse equivalence if there is a coarse map g :

Y X such that gof is close to idx and fo y is close to idy. A coarse map f : X Y is

said to be a coarse embedding if f : (X, dx) (f(X), dvlfx)) is a coarse equivalence. In

particular, an isometric embedding is a coarse embedding. Also, if fi : (Xi, dxz>) ( dy,)

(i = 1, 2) are coarse equivalences, then so is fl x f2 : (X1 x X~, 61) 1 2Y x 62), where

by and 62 are the corresponding sum metrics.

Proposition 1.5. Asymptotic dimension is a coarse invariant. i.e. coarsely equivalent

.spaces have the stone mov'l,1. J.. dimension.

Proof. See ChI Ilpter 9 of [23]. O

We recall that a mapping f : X Y between metric spaces is said to be a quessi-

i~sometry if there are numbers A > 0, C > 0, and D > 0 such that

d(:r, y) C d( f (:), f (y))l < d(:r, y)+ C

and every point of Y is within distance D of ~(X). It is not difficult to see that a

quasi-isometry is a coarse equivalence. Also, for a finitely generated group G with

generating set S, the isometric embedding G L~C(G, S) of the group with word metric

into its C i-ley graph (here equipped with a geodesic metric such that each edge has

length one) is a quasi-isometry.

Since the word metrics associated with any two finite generating sets of a finitely

generated group are coarsely equivalent (even quasi-isometric), we regard .I-i-mptotic

dimension as a group invariant (for finitely generated groups). In section 93 we extend

these notions to all countable groups.

Coarse Structures. Let X he a set. For E, F C X x X, we define the product of E

and F, denoted E o F, by E oF = {((r, x) E X xX | y s.t. (:r, y) E E, (y, x) E F}. For

E C X x X, we define the inverse of E to be E-l = {(y, r) |(:r, y) E E}. For E C X xX

and K C X, we define E[K] = {:rE X |(:r, y) E E and y e K}. Also, for xre X, define

E,. = E[{:r}] and E" = E-l[{:r}]; these are known as E-halls in X (see the example

below for the reason for the terminology).

Definition 1.6. For a set X, a corerse .structure 8 on X is a collection of subsets of X xX

such that the diagonal a belongs to 8 and 8 is closed under the fomation of subsets, finite

unions, products, and inverses. The elements of 8 are called controlled .sets. Also, (X, 8)

(that is, a set X equipped with a coarse structure 8) is called a corerse .space.

Example 1.7. For a metric space (X, d), let 8 consist of all those subsets E of X xX

for which d(E) CRI is bounded (here RW has the usual metric, and we have applied the

distance map d : X xX R I to the set E C X x X); this is a coarse structure,

known as the bounded corerse .structure. In particular, for a finitely generated group

G, since any two word metrics on G are equivalent, we have that the bounded coarse

structure does not depend on the choice of the word metric. Also, if r > 0, and we define

E = {((r, y) |d(:r, y) < r }, then we have E., = E" = B, (:); hence the terminology E-hall.

Indeed, E[K] = NV,(K), the r-neighborhood of the set K.

For a coarse space (X, 8), we ;?i- that B C X is bounded if B x Be S We ;?i- the

coarse space is coarsely connected if {:r, y} is bounded for any pair of points :r, y E X. We

ii- that a map f : (X, 8) (Y, FT) is proper if the preimage of a bounded set is bounded;

it is bornologous if f x f takes controlled sets to controlled sets; a map is called corerse

if it is both proper and homnologous. If S is a set and X is a coarse space (with coarse

structure 8), then f, y: S X are said to be close if {(f(r), g(:r)) :r E X} is a controlled

set. A coarse map f : (X, 8) (Y, FT) is said to be a coarse equivalence if there is a coarse

map y : (Y, FT) (X, 8) such that g of is close to idx and f o y is close to idy. A coarse

map f : (X, 8) (Y, FT) is said to be a coarse embedding if f : (X, 8) ( f(X), FY)

is a coarse equivalence (here FTy = {E E : E C Y x Y}). When dealing with metric

spaces and the bounded coarse structure, our definitions for coarse structures revert to the

definitions for metric spaces given above.

For a coarse structure 8 on X and a coarse structure FT on Y, define a product

structure on X x Y hy E x FT = {E C (X x Y) x (X x Y) : Erx(E) E 8, jTY(E) E F}), where

rx : (X x Y)2 X2 is the projection (similarly for Y). Also, if fi : (X Es) (1 Fe)

(i = 1, 2) are coarse equivalences, then so is fl x f2 : (X1 x X~, 81 x 82 1 2Y 1 Y X F2~ T)

Suppose that X is a topologfical space. E C X xX is said to be proper if, for each

relatively compact set K, both E[K] and E-1[K] are relatively compact. We ;?i- that a

coarse space (X, 8) is consistent with the try.-~I ~ 1.-i on X if it has the property that B C X

is (coarsely) hounded if and only if B is relatively compact (i.e., bounded sets coincide

with relatively compact sets). One can easily show a consistent coarse space (X, 8) is

coarsely connected and each E E is proper.

Definition 1.8. Let X he a topologfical space, and suppose that 8 is a coarse structure

which is consistent with the topology on X. We ;?i- that f : X C is a Hig~son function,

denoted f e Ch(X, 8), if for every E E and every e > 0, there is a compact set

K such that | f (:) f (y)| < e whenever (y, r) E E \ K x K. The Higson functions

form a C*-algebra, and so by the Gelfand-Naimark-Segal (GNS) theorem there is a

compactification hEX of X called the Hig~son *****,l'ra .1.0..rl..>,n such that the algebra of

Higson functions Ciz(X, 8) is isomorphic to C(hEX). The Hig~son coronet is defined by

IZEX = hEX \ X.

The following theorem can he inferred from [23].

Proposition 1.9. Let X and Y be '7... ellti compact. Hau~sdorff .spaces equipped with corerse

structures 8 and F, '' i'..~ 1.:; le; which are consistent with the topologies. If f : X Y is

a coarse, continuous map, then f extends to a continuous map f : hEX hyY such that

f (VEX )C VEY .

For a locally compact, Hausdorff space X equipped with a coarse structure 8 that is

consistent with the topology, then we ;?i that (X, 8) is a proper coarse space if there is a

controlled neighborhood of the diagonal of X x X. The following is Proposition 4.1 of [23].

Proposition 1.10. Let X and Y be proper coarse spaces. A coarse map 4 : X Y

extends to a continuous map v : vX vY. If ~, :X Y are close then v4 = v .

If additionally f is a coarse equivalence, then v( f) is a homeomorphism.

All of the coarse structures in this treatise will be proper coarse structures, and will

be founded upon a proper metric space.

Compactifications. Let X be a compactification of a locally compact space X, and

let V be an open subset of X. Then there is a unique maximal open set V in X such that

V n X = V. In fact, V = X \ X \ V. One can show that V C V.

The following propositions are not difficult. Further details can be found in [15].

Proposition 1.11. Let X be a ***,l~r in. 1.0.rl..>,n of a 'l...arl~lt compact space X, and let

vX = X \ X Then {V n vX : V is open in X } forms a basis for vX .

Proposition 1.12. Let X be a ***,l~r in. 1.0.rl..>,n of a 'l...arl~lt compact space X, and let

vX = X \ X Suppose U is an open subset of vX and x E U. Then there is a set V which

is open in X x E V n vX and V n vX C U.

CHAPTER 2
THE GRIGORCHUK( GROUP

The first Grigorchuk group is described in [17], [18], and [21]. This Grigorchuk group,

which we will denote by P, has many interesting properties. It is a finitely generated

2-group with intermediate growth, whose word problem is solvable, and which does not

admit a finite dimensional linear representation that is faithful. Also, P and E x P are

commensurable, which means that P and E x P have subgroups of finite index which are

isomorphic. A detailed exposition can he found in [21].

We prove that P has .-i-mptotic dimension infinity, asdimP = oc. If one excludes

Gromov's "random groups" [20], all previously known examples of groups G with

asdimG = 00 are based on the fact that G has a free ahelian subgroup of arbitrary

large rank. The Grigforchuk group is of different nature: since P is a 2-group, it does not

have a nontriviall) free ahelian subgroup.

Whnr~ Ge and H are groups equipped with word metrics do and dH, then the sum
metric do + dH on Gx H is also a word metric (for the natural generating set). If H < G

is a subgroup of finite index of (the finitely generated group) G, then the inclusion map

H G is a coarse equivalence. Also, an isomorphism is a coarse equivalence.

Definition 2.1. Two groups P1 and 02 are CO~mmensurable if there exist subgroups

H1 < P1 and H2 2~, each of finite index, such that H1 and H2 are isomorphic.

By the comments above, asdimfI = asdim f2 P1 and P2 are commensurable.

Theorem 2.2. [5/ Let G be a ~nitely generated. infnite II, ey1 which is commensurable

with its .square G x G. Then asdimG = 00.

Proof. We first show that G"z is coarsely equivalent to G for all n > 1. Proceeding

inductively (the n = 1 case is immediate), we assume G"z is coarsely equivalent to G. But

G's+1 is coarsely equivalent to G's x G, which in turn is coarsely equivalent to G x G, and

so by hIypothis~I G'+ is equIivalent~ to G. This3 proves tha adi mG' = asdIIimGUIIC for alll

n >1.

Also, by Exercise IV.A.12 of [21], there is an isometric embedding f : Z G, where
Gc: is.~ taken wit a--M word M- metrc. hus for each a > 1, we have an isometric embedding

fx f -- xf : G, where we take the sum metrics on and G". Since an

isometric embedding is a coarse embedding, we have that asdimG" > 1-.11.11. = n. Thus,

asdimG > n for all n. O

Corollary 2.3. Let P be the G,.lly.-r. 1,;1. II,' T;. Then asdimP = co.

Proof. P is finitely generated by definition. Proposition VIII.14 and Corollary VIII.15

from [21] show that P satisfies the hypotheses of the theorem. O

It is interesting to note that asdimP = co, yet P does not contain an isomorphic <. II

of Z. However, Z. does coarsely embed into P. Also, if one has a finitely generated group

which is known to be commensurable with its square, then the .I-i-mptotic dimension is

either 0 or infinity, depending on whether the group is finite or infinite.

CHAPTER 3
COUNTABLE GROUPS

Below we show that every countable group admits a left-invariant, proper metric.

In view of Proposition 3.1, one can extend the invariant asdim to all countable groups

(not necessarily finitely generated). More explicitly, for a countable group G, we define

asdimG = asdim(G, d), where d is a left-invariant, proper metric on G. Propositions 3.1

and 3.3 show that asdimG is well-defined. For countable groups, note that a left-invariant,

proper metric induces the discrete topology (Baire Category Theorem), consequently,

bounded sets are finite. The following appears in [23, Proposition 1.15], [14, Propostion

1.1], and [24, Proposition 1].

Proposition 3.1. For a countable II,. ;;;-, r:;, two left-invariant, proper metrics are

coarsely equivalent.

Definition 3.2. Let P be a countable discrete group. Let S be a symmetric generating set

(possibly infinite), S = S-l, for P. A weight function w : S [0, 00) on S is any positive,

proper function such that w(s-l) = w(s) for all s e S. The properness can essentially be

viewed as the requirement that lim w = co.

It is not hard to see that for any countable group P, there is a weight function. In

fact, for any symmetric generating set S, there is a weight function with domain S.

Proposition 3.3. [14, Proposition 1.3], [24, Theorem 1] A weight function on the

countable yearty 0;1 induces a proper norm || || given by

i= 1

and so a weight function induces a left-invariant, proper metric d.

Proof. Given a weight function w : S [0, 00), where S is a generating set for the

countabille groupr r, define ||| = inf { w;(as)| = 8182 sn, si E S, }. Note thait if we
i= 1
view 1r as an empty product, ||1rl| = 0. The proof that || || is a norm is left to the reader.

Let R > 0 be given. Let r be a nonzero value that the weight function assumes. So

{S E S|0 < w(S) < r} is nonempty and finite by definition. Thus, there is a t ES such

that w(t) = min~w(S)|S a S,0O < w(S) < r}. It is immediate that 0 < w(t) < w(s) for

all s ES \1Ir. Now, suppose x is such that ||X|| < R and x / 1r. Then ||X|| < R + 1. So

there are sl, s2,.,, n S such that x = 8182 Sn and CE w(aS) < R + 1. Further,

we may assume that aS / 1r for each i. Thus, aS E {S e Slw(S) < R + 1} for all i.
Also, R + 1 > l ~ s wts htn<( )/w(t). Thus, x is an element of

{tlt2 m~t iES \1Ir, w(ti) < R + 1, m < (R + 1)/w(t)}, a finite set. This shows that

{x | ||X|| < R} is finite. O

Note that the infimum in the definition of ||X|| is actually a minimum. To see this,

simply modify the argument in the last paragraph to show that the set of elements of

{E~ W(aS)|IX = 8182 "i,, Si E S} less than ||X|| + 1 is a finite set.
The following theorem gives a necessary and sufficient condition for a countable group

to have .I-i-uspind ilc dimension zero. This condition relies only on the algebraic structure of

the group. The following (and its corollaries) can be found in [24].
Theorem~ 3.4 Let- G-':- beac utbeO'T h nad m if and only if every Anitely

generated ~subgrouup of G is Anite.

Proof. Lt wC :. G 0 0 eawigtfnto nte eeaigstG Let || || and d be

the induced norm and metric, respectively.

First suppose that asdimG = 0. Let T C G be a finite set. Take d > -rn :.I -, ||g ||

As asdimG = 0, there is a d-disjoint, uniformly bounded cover U of G. C!.. .- ~- U e U

with 1 E U. We will show that (T) C U. To do this, we will show by induction that

every product of k (k > 0) elements of T U T-l lies in U. This is true for k = 0, as

1 E U. Now suppose it is true for k 1, k > 1. Consider x = t('ti' t where ti E T

and as E {+1}. Set y = t1982 t "). By the induction assumption, y E U. Since

d(y, x) = ||y-l I || = |iI || = |I, || < d, and because U is a d-disjoint cover, we must have

x E U. Thus, each product of k elements of T U T-l lies in U. Therefore, (T) C U. As U

is uniformly bounded, U is bounded, and so U and (T) are finite.

Conversely, suppose every finitely generated subgroup of G is finite. Let d > 0 be

given. Define T = {s E Glw(s) < d} and H = (T). By definition of weight function, T is

finite. By our assumption, H is finite as well. Let U = {gHlg E G} be the collection of left

costs. So U is a uniformly bounded cover, as multiplication on the left by a fixed element

is an isometry of G. Further, suppose gH / hH. Let x E gH and y e hH. It follows that

y- x ( H. Hence y-lx cannot be written as a product of elements of T U T-l. So if we

take asi EG such that y-lx = 8182 sn and ||y-1, ||I = E w(as), then there is a ) such

that sj ( T. Hence w(sj) > d, and so d(y, x) = ||y-, || 'I> d. Therefore U is a d-dl;id .ilst

uniformly bounded cover. Since d > 0 was arbitrary, asdimG = 0. This completes the

proof. O

The following corollaries are immediate consequences.

Corollary 3.5. Let G be a Anitely generated II,. ;;;.. Then asdimG = 0 iff G is a Anite

Corollary 3.6. Let G be a countable abelian II,. ;;;.. Then asdimG = 0 if and only if G is

a torsion II .- ;,1 .

Example 3.7. The last corollary shows that a 1.. Q/Z, and Z~oo = lim, Zr, all have

.I-i-inidllicl~ dimension 0.

The following theorem reduces the study of .l-i-inidllicl~ dimension of countable groups

to the .I-i-inidlli'lc dimension of finitely generated groups, and can be found in [14].

Theorem 3.8. Let G be a countable II,. ;;;.. Then

asdimG = sup asdimF,

where the supremum varies over Anitely generated ;11, aby, s F of G.

Proof. Fix a weight function w : G [0, 00); let || || and d denote the induced norm and

metric, respectively. If sup asdimF = co, then asdimG = 00, and we are finished. We will

now assume sup asdimF < co. Set m = sup asdimF.

Let d > 0 be given. Set T = {g |w(g) < d} and F = (T). By the definition of

weight function, T is finite and so F is finitely generated. Thus, asdimF I m. So there

exist uniformly bounded, d-dl;id nita families Uo, MI, -- M of subsets of F such that UiN

is a cover of F. Let Z be a system of representatives for the partition by costs G/F. For

0
It is easy to check that Mi is uniformly bounded for i = 0,1i,... m, and that UiN is a

cover of G. We check now that Mi is a d-dl;id .ilst family. For suppose zU / z'U', where

z, z' E Z and U, U' E Ui. Let x E zU and ye z 'U'. First suppose z / z'. Note that

zU c zF and z'U' c z'F, so that x E zF and ye z 'F. Since z / z', zF / z'F. But

xF = zF and yF = z'F. So xF / yF, and hence x-ly Sf F. This means that x-ly cannot

be written as a product of the elements of T, and so d(x, y) = || x-1y|| > d by definition of

|| ||. Now suppose z = z'. So x = zu and y = zu' for some E U and u' E U'. Since we

must have U / U', U and U' are d-disjoint; thus, d(x, y) = d(zu, zu') = d(u, u') > d. It

follows that the family Mi is d-disjoint.

Since d > 0 was arbitrary, asdimG < m. Equality immediately follows. O

Example 3.9. Since every finitely generated subgroup of Q is cyclic, Theorem 3.8 implies

in particular that asdimQ = 1.

We recall the notion of the R-stabilizer from [2].

Definition 3.10. Suppose that a group F acts on a metric space X by isometries. Let

xo E X. For every R > 0 we define the R-stabilizer of xo as follows:

WR(XO) = {g E | d(g(Xo), Xo) < R}.

The following theorems are extensions of earlier results for finitely generated groups,

which can be found in [4].

Theorem 3.11. Suppose that a countable yearty 0;1 acts on a geodesic space X by

isometries. Let xo E X and suppose asdimWR(xo) < n for all R > 0. Then

asdimF < n + asdimX.

Theorem 3.12. (Hurewicz Two~~ Formula) Let : G H be a homomor-phism of

countable (l,. ;;;-- with kernel K. Then

asdimG < asdimH + asdimK.

Both of these Theorems follow from Theorem 3.8 in combination with the corresponding

Theorems for the case when the groups are finitely generated. For details see [14]. Since

the notion of .-i-mptotic dimension at the time when [4] was written was defined only for

finitely generated groups, K is treated in Theorem 7 of [4] as a metric space with the word

metric restricted from G.

CHAPTER 4
THE SUBLINEAR COARSE STRUCTURE

Throughout this section, unless stated otherwise, X will be a proper metric space

with metric d, and xo will be a basepoint of X. We define || || : X [0, 00), sometimes

referred to as a norm, by ||X|| = d(X, xo). Also, we write B, = B,(xo) for the open ball of

Definition 4.1. We define the sublinear coarse structure, denoted SL, on X as follows:

.supy6E, d(,X Supy6E d Z, Y)
St = {E C X xX : E proper, him = 0 = m }.

By ~ ~ ~ ~ ~~11 th ttmn isasp we mean that for each e > 0, there is a
r > 0 such that

for all x with ||X|| > r. It would perhaps be better to think of this as lingl.ll_ In the

event that Ez = 0i, we define supy6EE d(y, X) = 0.

We first check that St is indeed a coarse structure.

Theorem 4.2. St 1. I;,: 4 a proper coarse structure on a proper metric space X.

Proof. It is easy to show that St contains the diagonal and is closed under the formation

of subsets, inverses, and finite unions. Also, the (coarsely) bounded sets are precisely the

relatively compact sets, and as St contains the bounded coarse structure, it contains a

neighborhood of the diagonal. To complete the proof, we show that it is closed under

products.

Let E, Fe S t. Then E oF is proper. Let e > 0 be given. Then there are compact

sets K, L C X (containing the basepoint) such that

supyEF3; d(y, X) supy6EEz (
< mme/3 1}and< /

whenever x ( K and z ( L. Define K' = K U F-1[L]. K' is relatively compact since F is

proper.

Now suppose x ( K', and let y E (E o F),. So (y, x) E Eo F, and thus there is azx

such that (y, z) E E and (z, x) E F. We must have z ( L. For if z E L, then (x, z) E F-l

and so x E F-1[L] C K', a contradiction. Also, x SfK, ye E z, and z e F,. So

d(y, x) < d(y, z) + d(z, x) < 3II 3||X|| I 3d(z, X) + |X||

3 3

supy6(EOF)z y x
Thus, supyE(EOF), d 9, Z) I j .1||, or < e. Since e > 0 was arbitrary, this

completes the proof. O

Also, for xo, xl E X, it is not difficult to show that

.supy6Ez dy x) Supy6Ez y x
him = 0 iff him = 0,
zwoo d(x, xo) zsoo d(x, xi)

where E C X x X. A similar statement holds when E, is replaced by E". This shows that

the coarse structure does not depend on the choice of basepoint.

We often refer to the Higson compactification hE,X (corona ve,X) as the sublinear

Higson ****l'ra. 17.:~ al.>~n (sublinear Higson corona), and it will be denoted using the

simpler notation hLX (VLX).

Proposition 4.3. [15, Proposition 2.1/ Let X and Y be proper metric spaces. If f : X

Y is a quasi-isometry, then it is a coarse equivalence with respect to the sublinear coarse

structures.

Proof. Fix a basepoint xo E X, set yo = f(xo) E Y. Clo~~~--- A > 0 and C > 0 such that

id~, y
and g of are close to the respective identity functions. It is clear that f is proper and

the image under f of a bounded set is bounded. Let E be a controlled set in the sublinear

coarse structure on X. It is not hard to show that f x f(E) is proper.

Let t > 0 be giv~en. There is a bounded Kt' C X such that supV -I ) < henve

x ( K'. Set K = B(xo, 2AC) U B(xo, ) U K'. Note that f (K) is bounded. Suppose that

sup {d(z',zx) : z' E (f x f)(E)z}
z Sf f(K). If (f x f)(E)z = 0, then = < e and we are

finished. Now assume that z' E ( fx f) (E)z; so z' = f(x') and z = f(x) for some x, x' E X

with (x', x) E E. We have that x ( K.

Then

1 2||X|| 2AC ||X|| ||X|| 2AC ||IX||
|| f (x)||> -||x|| C +>
x Ax 2A 22a 2a

and so

d(z', z) d( f(x'), f (x)) Ad(x', x) +C 2 d(x', x) 2AC
< 2A = 2A < e.
|| || || If (x)|| ||x|| ||x|| ||||

sup {d(z', z) : z' E (f x f)(E)z}
Thus, we have < e. Since e > 0 was arbitrary, ( fx f) (E)

is controlled and so f is a coarse map.

Similarly, since g is a quasi-isometry, it is a coarse map as well. Finally, it is clear

that f o g and g of are close to the corresponding identities when X and Y are equipped

with the sublinear coarse structures. Thus, f is a coarse equivalence. O

Corollary 4.4. The sublinear coarse structure St is well-./. It.. ./ on Anitely generated

IIo ;,1- i.e., the sublinear coarse structure for a given pretty 0;1 is independent of the choices

of the Anite generating set and the basepoint.

Lemma 4.5. [15, Lemma 2.3] Let (X, d) be a proper metric space with basep~oint xo,

endowed with the sublinear coarse structure St. For a Anite system El,. ., E, of subsets

of X the following are equivalent.

1. vX n [n" zEi] = 0,;

2. there exist c, ro > 0 such that maxi (||.r|| whenever ||X|| > ro

Proof. We prove 1 implies 2. Assuming that 2 does not hold, then if we let m be a

positive integer, and if we set c =T and ro =2m,2; then there is an xm, such that

| e, || > 2m yet

max d(xm, Ei) < | .||
1

Th~us, for each i:, we have that dl(xm, E ) < |IT I I. ||,and so there is an a~ t Es such that

d~xm, a |e. ||. We,, haved,, a ) ~ < |.Aso,

4m

and hence (1 I) |, I | < |.,, || (all i). Since < 1/2, we have |.,, || >| ., ||. Thus,

1 1 1
d(a'm, Gm) < | |, | and |, || > e |> m
2m m 2

for all 1 < i, j < n. Take Fe,, = { (am, a"m) : m = 1, 2, .. .} for each 1 < i, j < n. Fixing
i, j wetemorailysetG =Fi~j for convenience, and show-"- that G i cotrlle.Snc

|, .I || 00 and |.,,' || 00o as m 00o, it follows that G is proper. Now let e > 0 be

given, and take M~ to be a positive integer for which 1/M~ < e; set K = {a'm : 1 < m < M}).

Suppose that xL B K. If G', = 0, then by: our conlventionl we hlave sup'i ) =0
If y E G,, then there is a positive integer m such that (y, x) = (am, m"), and since

a" = x ( K, we must have m > M~; it follows that "' = < < e and so
sIIupG (I)I < e. Thuls, limsio aI~yC su ) = Similarly, lim,, suptG c ) =: 0 and

hence G = Fi~j is controlled.

Consider the sequence {(a m, a a"m) }" consisting of points of X" C (hLX)".
Regarding this sequence as a net, thereo is asubnet {(k a,..a")}Awhc

converges in (hLX)" (A is a directed set). Set b = (bl, b2, .. ,b") = limAEA @m **>"

So bi e brX anld bi = limxEA tems. By definitions? of the a'm, alnd th~e definition of subnet, we

have that |.,j ||I 00o as A 00o. From this one concludes that bi E VLX for each i. Also,

fixing 1 < i, j < n, we have that (bi, b) = liII\m ('ms L1ms) E Fi3. Since Fi~j is controlled, we

have by Proposition 2.45 of [23] that (bi, ~) E Fi~j \ X xX C AvLx, where A,,x denotes

the diagonal in VLX x VLX. Therefore bi = b So bl = bi = limx a'z E Ei and bl E VLX.

Thus, VLX n [n" zEi] / 0i. So 1 does not hold.

It remains to show that 2 implies 1. Define Fi = Ei \ Bro+cro for 1 < i < n. Let

f : R b deinedby f(x)= E d(x, Fi). Note that f (x) > (||.r|| when ||X|| > ro

since d(x, Fi) > d(x, Ei). Also, f(x) > cro when ||X|| < ro, in particular, f(x) > (||.;-|| for all

x and f(x) > 0 for all x. Define g, : X R I by gi(x) = d(x, Fi)/ f(x).
Let E be a controlled set. Since

1 1 d(y, Fi) d(x, Fi) d(y, Fi)
|ge(y) gi(x)| < d(y, Fi)| | + | <|f (x) f (y)|
f (y) f (x) f (x) f (x)f (y)
d(y, x) ad(y, x) d(x, y) d(x, y)
+ < + < (n + 1)
f (x) f(x) f(x) 0||. -|| '

we have supyEE, Igi ) Si(y)| 0 as x 00o. Since E was an arbitrary controlled set, gi

(viewed as a map to C) is a Higson function for each i. Let Gi : X C be the extension

of gi to the Higson compactification. Since Ci gi = 1, it is immediate that Ci Gi = 1
throughout X. Also, Fi c G 1(0) and it is not hard to see that vX n Fi = vX n Ei. Thus,

vX n (n ,E) = vX n (n zF,) C vX n (ntzG '(0)) = 0

since Ci Gi = 1 on vX. O

A system satisfying one of these two properties is said to diverge. In the case that
n = 2, we can add another condition.

Lemma 4.6. Let A and B be subsets of a metric space X with basepoint xo E X. The

following are equivalent.

1. There exist C, ro > 0 such that max (d(x, A), d(x, B) } > C||.;-||I whenever ||X|| > ro;
2. there exist D, rl > 0 such that d(A \ B,, B \ B,) > Dr whenever r > rl.

Proof. We show 1 implies 2. Given C and ro, take D = C and rl = ro. Let r > rl,

a EA \ B,, and b E B \ B,. So ||a|| > r > ro. Thus,

Cr < C||a|| < max {d(a, A), d(a, B) } = max {0, d(a, B) } = d(a, B) < d(a, b)

by 1. So d(A \ B,, B \ B,) > Dr.
We show 2 implies 1. Let D, rl be positive numbers satisfying 2. Set ro = 2rl and

take C to be a positive number satisfying C < min {1/2, D/4}. Now let x E X be such

that ||X|| > ro = 2rl. To get a contradiction, suppose that max {d(x, A), d(x, B)} < C||. -||.

Then d(x, A) < C||.;-||l and d(x, B) < C||.;-||. Thus, there exist a eA and b e B such that

d(x, a) < C||.;-||l and d(x, b) < C||.;-||. So d(a, b) < 2C||.;-||. We then have

||a|| > ||x|| d(x, a) > ||x|| C||l.-|| = (1 C)||x||I > ||x||/2.

Similarly, ||b|| > ||x||/2. Since ||x||/2 > ri, we have by 2 that

D||X|| D||X||
2 d(A \ B||z||/2, ||z||/2ll ) < d(a, b) < 2C||.;-||l I

a contradiction. Therefore, max {d(x, A), d(x, B)} > C||.;-||l when ||X|| > ro o

Definition 4.7. Let (X, d) be a metric space, and let V be a family of open subsets of X.

We define the Lebesgue function associated with the cover V, denoted L by

L'(x) = sup d(x, X \ V).
vey

Definition 4.8. For a proper metric space (X, d) with basepoint xo, we w?-a function

f : X [0, 00) is (eventually) at least linear if there exist c, ro > 0 such that f (x) > (||. -||

whenever ||X|| > ro

Corollary 4.9. Let (X, d) be a proper metric space endowed with the coarse structure St.

L/et a~ = {01, .. ,0, } be a finite f~r in:l;, of open subsets of X. Then cE = {01, .. ,0, }

covers the corona VLX if and only if the Lebesgue function La is at least linear.

Proof. at = {Oz,..., O,} covers the corona VLX iff VLX \ (Ui0s) = 0i, iff VLX n (X \

UiOi) = 0, iff vX n (ni(X \ Oi)) = 0, iff vX n (niX \ Oi) = 0 by the discussion of

compactifications in ~1, iff the system X \Oi1,...,X \O0, diverges, which, by lemma 4.5

above, is true if and only if La is at least linear. O

Corollary 4.10. Let (X, d) be a proper metric space, and let A be a closed subspace of X

equipped with the restricted metric. Then the embedding A L~X extends to an embedding

hLA LEhX on the -* ~',l~~in.1.0..rl..>,ns and induces an embedding VLA LSVLX on the

coronas.

It follows from this corollary that if A is a closed subset of a proper metric space

X, then hLA is homeomorphic to clhzxA, the closure of A in brX. Further, VLA is

homeomorphic to claz xA \ A.

Algebra of functions. We define a subalgebra U(X) = U(X, xo) of C(X) as follows:

f : X C is in U(X) if and only if f is bounded, continuous, and there exists a c = cf
such that

| f (x)- f (y)|||x|| < cd(x, y).

It is not difficult to check that U(X) is closed under addition, multiplication, and complex

conj ugation.

Remark 4.11. In the definition, the continuity condition is almost unnecessary. It is not

hard to see that the property | f(x) f (y)|||X|| < cd(X, y) implies that f is continuous for

x / xo.

It is easy to show that U(X) separates points and closed sets. It is also clear that

U(X) is, in general, not complete. We set C'(X) = U(X), where the bar represents closure

in C(X) with the uniform metric. So C'(X) is a C*-algebra which separates points and

closed sets. Thus, by the GNS Theorem we can extract a compactification of X, denoted

X, which satisfies C'(X) = C(X).

It is not hard to prove that C'(X) C Ch(X, SL), hence there is a surjective, continuous

map hLX X which extends the identity. This map is actually a homeomorphism.

Theorem 4.12. [15, Theorem 2.11] Let (X, d) be a proper metric space. Then the

, *,1'' I*r.:~ 1. .l..>,n X is homeomorp~hic to the Higson ***l'ra 1. 0. .rl..>,n brX for the sublinear

coarse structure St via a homeomorp~hism extending the .:~:./.;./10 on X .

CHAPTER 5
THE DIMENSION OF THE SUBLINEAR HIGSON CORONA

We recall that a metric space (X, d) is called cocompact if there is a compact subset K

of X such that X = Uelsom(X)y(K), where Isom(X) is the set of all isometries of X. The

proof of the following is longf and requires numerous results concerning extensions, and will

be omitted here.

Theorem 5.1. Let X be a cocompact, connected, proper metric space which has finite

/* cl,~~ .://: Assonad-Nagata dimension. Then dim VLX > AN-asdimX.

Proof. See Theorem 3.7 of [15]. O

If U is a cover of X and A c X, then we write UA = {U E U : Un A / 0}).

Lemma 5.2. Let U and V be covers of X, and suppose that U relin. V. Let K be a

subset of X So for each U E U with U n (X \ K) / 0i, there is a Vu E V with U C Vu.

For V E V, set

V' = [V; n (X \K)] UI U Iii
UE6lX\KV,U=V

W = {UEUIuU CK}U{V'|V E VX\K -
Then

1. W is a cover of X;

2. multW 5 a + 1 if multU 5 a + 1 and multV 5 a + 1;

5. if WE W and W c K, then We E ;

6. if U and V are open covers and K is closed in X, then W is also an open cover;

7. if VE V and V n K = 0i, then V

V' E W.

The proof is straightforward. For the above W we will write

W = U *"K -

The following Theorem is a modification of Lemma 2.9 of [13].

Theorem 5.3. [15, Theorem S.10/ Let (X, d) be a proper metric space. Then dim VLX <

AN-asdimX .

Proof. We write vX = VLX and use B(x, r) to denote a closed ball of radius r. Set

n = AN-asdimX (if AN-asdimX = 00, the inequality is immediate). As {U n vX : U C

X open} is a basis for vX, it suffices to prove that each cover of the form {Ui n vX : 1 <

i < m}, where each Ui C X is open, admits a finite refinement of multiplicity < n + 1.

So let {Ui n vX : 1 < i < m} be a cover of X, and set U = {Ui : 1 < i < m}. Since

AN-asdimX = n, there exist C > 0 and r_l > 0 such that whenever r > r_l, there is

an open cover U(r) of X satisfying multWi(r) < n + 1, meshui(r) < Cr, and Lu(')>r)

Without loss of generality, we take C > 1. Also, there is a D > 0 and an T-2 > 0 such that

Lu(x) > D||X|| whenever x is such that ||X|| > T-2. We may take D < 1.

Now, choose ro > max~ra {T-2, -11. Define ri = ( )iro for i > 1. Observe that

ri+] =" ri > ri > ro > 1. Since ri > ro > r-1 for i > 1, there is a cover Ui of X such that

multUi < n+ 1, meshUi < Cri, and Lu' > ri.

Define V1 = U1, and note that meshV1 < Crl. Now, supposing we have defined

Vl, 7, *i SailSfying meshvy < Crj < rjgy for all 1 < j < i, then 94 refines Mayl, and so

we can define

ji+1 = ji *B(,o,2ri42) Ui+1.

By Lemma 5.2, vji refines Ui 1, and so meshve 1 < Cri 1. Thus, we have constructed vi

for all positive integers i.

Set V = lim infi 94 = U, nt>s v,. We now investigate some properties of V.

Using the definition of M 1~, it is easy to show that if U E Rj and U n B(xo, ri+2) / 0i,
then U E 7441l. We conclude that {U E R : U n B(xo, ri+2) / 0} C V. As Mi is a cover of

X and hence of B,, ,,, we have that V covers B,, ,; as i here is arbitrary, V covers X.

We now show that if Ve E and V n B,, J 0i, then V E 94i_l. First suppose that

V'E Rj anld V n BTL, i 0; then mreshMi < Crg < re Il implies that VC B 2r, and so

V E Psi_l by 5 of the lemma. Now suppose Ve E This means there is an a > i 1 such

that Ve E s. Applying the result we just found and proceeding inductively, one can show

that V E 7,j for all j such that i 1 < j < s.

We show that Mi refines V for all i > 1. We know by 4 of the lemma that Mi refines

VJi+l for all i > 1. Fixing i, let V'E IM. C!.. .- ~- j > i such that Vi n BqJ i 0. As L:

refines vi, there is aU e/ E i such t~hat VC U Also, U~n B~,, V n Bq, 0 Thuls,

V CU E V. So Mi refines V.

Since each Mi has multiplicity I n + 1 for each i, it is clear from the definition that V

has multiplicity I n+ 1.

Set W = VxY\B,. Wle have that (BV)x\Bhg refines W/Z.
We show that W refines U. Let W E W. So there is an x E W such that ||X|| > T2*

Take i = max {j : ||X|| > Ty}, and note that i > 2. Thus, ||X|| < ri 1, or x e B,, z. Hence

W n B,,, J 0. Since We E we have W E Mj_l. So

diamW/l < Cri_i = D( )ri_1 = Dri < D||X|| < Lu(X),

and so there is a U E U with W c U.

We show that LW : X [0, 00) is at least linear. Set a = 3T2, and let x be an

element of X with ||X|| > a = 3r2. Set i = maxj {: 3rjgy < ||X||}, and note that i > 1

and 3ris I < |X|| < 3Ti+2. As Lu' > ri, there is a U E Ui such that B(x, ri) C U. Since

||X|| > 3r il and diamUi < meshUi < Cri < ri 1, we have that U C X \ B(xo, 2ri41). By
definition of Bi, and by 7 of the lemma, we have that U E Mj. In fact, as ||X|| > T2, We have

U: e (B)x\h.3 Since (V,)x:Bh, refines W/, we have that

LW( > d~x X U) > ri.

But~ ~ ~ ~ ~ ~~0 ||| r+2 3FS L()> |X||. Therefore, LW is at least linear.

To summarize, W covers X \ Br,, multW < n + 1, W refines U, and LW is at least

linear. Thus, for W e W, there is a Uw E U for which W C Uw. So for each 1 < i < m,

we define Wi = Uue,=uW

Now set W' = { We}. It follows that W refines W' and W' has multiplicity < n + 1.

Thus, LW/ '> LW and hence LW/ is at least linear. As a consequence, if we define

W' { (Wi n vX }, we have that W' is a cover of vX. Since W' refines U, we have that W'

refines U. Finally, as W' has multiplicity < n + 1, so does W'. O

Since a finitely generated group is quasi-isometric to its Cwiley graph, a cocompact

geodesic metric space, one has the following corollary.

Corollary 5.4. [15, S.8/ For a finitely generated I.-q 0;1 with word metric, dim urf =

AN-asdimF provided AN-asdimF < co.

Example 5.5. Consider the parabolic region X = {(x, y) E RW2 : x > 0,|ly| < z6},

which we will equip with the (restricted) Euclidean metric. Let i : [0, 00) X be the map

i(x) = (x, 0). Taking the usual metric on [0, 00), it is not hard to show that i is a coarse

equivalence for the sublinear coarse structures, and hence there is a homeomorphism

us [0, 00) VLX. In particular, dim vlX = 1. But AN-asdimX = 2. Since X is

a connected proper metric space, this shows that we can not drop the requirement in

Theorem 5.1 that the space be cocompact.

Lemma 5.6. [15, Lemma 4.2] Let X be a proper metric space. Then there is an embed-

ding

VLX x [0, 1] ur V(Xx R I).

Proof. We use || || to denote the norm on both X x R,+ and X. Let Y = {(x, y) E

Xx RW : 0 < y < ||X||, ||X|| > 1}. Consider the maps xr : Y X and 4 : Y [0, 1] given

hy i(.r, y) = and Q(.r, y) =TTT respectively. The reason for ||.r| > 1 in the definition
of Y is the desire to make well-defined in the domain of definition; note that requiring

||.r | > 1 only cuts off a compact set in our cone, and hence leaves the corona unchanged.

We first show that xr is a coarse nmap for the sublinear coarse structures. It is easy to

check that xr is proper. Let E C Y x Y he controlled; we show that xrx xr(E) is controlled.

First notice that xrx xr(E) is proper since E is proper and since (xrx xr(E)) [K] C

x (E [x -K]). Let e > 0. Then there is an R > 0 such that

d((r, y), (w, c))
< e/2

whenever ((.r, y), (w, x)) E E and ||(w, x)||I > R. Now suppose (.r, w) E xrx xr(E) and

|| w|| > R. Then for some y,x ERIW (indeed for any appropriate pair of elements of R4+)

one has ;(.r, y) = .r and xr(w, x) = w. Since ||(w, x)||I > ||w|| > R and |z| < ||w||, we have

d(.r, w) d(.r w) + |y x| d((.r, y), (w, x))
< =2 < e.

Since e was arbitrary, xrx xr(E) is controlled. Since E was an arbitrary controlled set on Y,

we have that xr is a coarse nmap.

We now show that is a sublinear Higson function. For (.r, y), (w, x) E Y, one has

y z y-x 1 1

< +|z| <+

d((.r, y), (w, x)) d((.r, y), (w, x))
< <2

By the alternative characterization of the sublinear Higson compactification given at the

end of 54, we have that is a sublinear Higson function defined on Y.

Define P : VLY iLX to be the nmap induced by the coarse nmap xr, and define

# : VLY [0, 1] to be nmap induced hv the sublinear Higson function ~. We will also have

occasion to consider the extension of to hLY, which we will also denote by ~. That is,

we write : hLY [0, 1] for the extension, and note that #|vex = 0. In particular note

that ~(A) = ~(A) for AC c X, where closure takes place in hLX.

We define f = P x # : VLY VLX x [0, 1]. We show this map is a homeomorphism.

The inverse will be the desired embedding.

Define Ot = { (x, || ||) : x E X, || x|| > 1} for t e [0, 1]. Note that Ot c Y. The map

xrlo, : Ot X \ B1 is bi-Lipschitz since

d(x,yv) < d((x, t||x||), (y, '|| ||)) = d(x,yv) + t|||x|| ||y||| < (1 +t)d(x,yv),

this means that xrlot extends to a homeomorphism vL(xrlo,> : VLOt VLX. Since

|lo, : Ot [0, 1] takes every point of Ot to t, the image of #|sLot : VLOt [0, 1] is also

{t}. This means that the map (xrx ~)|o, : Ot (X \ B1) x t induces a homeomorphism

f |vLot : VLOt i X x t.
It is clear from the discussion above that f is surjective. It remains only to show that

it is injective. Before doing this, we show that f-1(VLX x t) = unOt. It might be helpful to

note that f-1(VLX x t) = 0-l(t). That f-1(VLX x t) > VLOt follows from the definitions

above. To get a contradiction, suppose that there is az E f-1(VLX x t) \ VLOt. It follows

from Proposition 1.12 that there is a closed set A C X such that z E An VLX(= VLA)

and usA n unOt = 0i. Thus, by Lemma 4.5, there are C, ro > 0 such that L(u) :

max~d(u, A), d(u, Ot)} > C||U|| whenever n E Y and ||U|| > ro. In particular, for (x,yv) E A,
we have

|y t||x|||I = d((x, y), (x, t||x||)) > d((x, y), Ot) > max~d((x, y), Ot), d((x,yv), A)}

> C||(x,y)||I = C(||x|| + |y|),

and so

Since this is true for all (x, y) E A, we have that (t C, t + C) n ~(A) = 0, and so

t ( ~(A) = ~(A) > #(vA) by the comments following Corollary 4.10 and the comments

above concerning the extension of ~. Thus, #(z) / t, contradicting z E f-1(VLX x t).

To prove injectivity, assume f (z) = f (z') for some z,zx' E Y. Let t = #(z)(= #(z')).

Then by the preceding fact, we have z, z' E f-1(VLX x t) = VL(Ot). But f|,,sot is a

homeomorphism, consequently, z = z'. O

Theorem 5.7. [15, Theorem 4.3] Let X be cocompact connected proper metric space.

Then

AN-asdim(X x RW) = AN-asdimX + 1.

Proof. By Theorem 5.3, Lemma 5.6, the classical Morita theorem, and by Theorem 5.1 we

obtain

AN-asdim(X x RW) > dim VL(X x RW) > dim(vlX x [0, 1]) = dim VLX + 1 > AN-asdimX + 1.

The opposite inequality is well-established. O

Again noting that a finitely generated group is quasi-isometric to its Cwiley graph, we

have the following.

Corollary 5.8. For a finitely generated II.-q;, G, AN-asdim(G x Z) = AN-asdimG + 1.

It was observed in [5] that AN-asdimG = dimAN G for finitely generated groups.

Consequently, for these spaces, the formula given above holds for dimAN: dimAN G x Z =

dimAN G + 1.

As mentioned in the introduction, along with results of Dydak et al. [5], this result

Iinclinles one to bUelieve: that asdimGlc x Z asdimGlc + 1 for finitely genlerated groups G.

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BIOGRAPHICAL SKETCH

.Justin lan Smith was born on .January 2, 1980 in Avon Park, Florida. He grew up in

Lakeland, Florida, and graduated from George .Jenkins High School in 1998. He earned

his B.S. in physics with a minor in mathematics from Florida Atlantic University in 2002.

.Justin then entered graduate school at the University of Florida in order to continue his

studies in mathematics. At the University of Florida, he earned an 1\.S. in 2004 and

completed his Ph.D. in 2007.